FIXED POINT FREE ACTIONS ON Z-ACYCLIC 2-COMPLEXES1
BOB OLIVER2 AND YOAV SEGEV3
In this paper, we give a complete description of the finite groups which can *
*act on
2-dimensional Z-acyclic complexes without fixed points. One example of such an *
*action
(by the group A5) has been known for a long time, but as far as we know it is t*
*he only
such action constructed earlier. In fact, we construct here actions of this typ*
*e for many
different finite simple groups.
More precisely, our main theorem is the following.
Theorem A. For any finite group G, there is an essential fixed point free 2-d*
*imensional
(finite) Z-acyclic G-complex if and only if G is isomorphic to one of the simpl*
*e groups
P SL2(2k) for k 2, P SL2(q) for q 3 (mod 8) and q 5, or Sz(2k) for odd k 3.
Furthermore, the isotropy subgroups of any such G-complex are all solvable.
Here "G-complex" means a G-CW complex; but the same result holds if one inste*
*ad
uses simplicial complexes with admissible G-action in the sense of [S1] or [AS *
*] (see
Proposition A.4 in the appendix). The word "finite" is in parentheses because *
*the
theorem holds whether or not this condition is included. The condition that the*
* action
be essential was put in to insure that an action of a quotient group G=N does n*
*ot
automatically produce an action of G:
Definition.A G-complex X is essential if there is no normal subgroup 1 6= N C G*
* with
the property that for each H G, the inclusion XHN ! XH induces an isomorphism
on integral homology.
In other words, if there is such a subgroup N C G, then the G-action on X is
"essentially" the same as the G-action on XN , which factors through a G=N-acti*
*on. In
the case of actions on acyclic 2-complexes, the relation between essential acti*
*ons and
arbitrary actions is made precise in the next theorem.
Theorem B. Let G be any finite group, and let X be any 2-dimensional Z-acycli*
*c G-
complex.0Let N be the subgroup generated by all normal subgroups N0 C G such th*
*at
XN 6= ;. Then XN is Z-acyclic; X is essential if and only if N = 1; and if N 6*
*= 1 then
the action of G=N on XN is essential.
The proofs of Theorems A and B rely on the earlier works [O1 ], [O2 ], [S1], *
*and [AS ],
as well as on the classification theorem for finite simple groups. In [S1], Y. *
*Segev proved
that if a finite group G acts on an acyclic 2-complex X, the fixed point set XG*
* is either
Z-acyclic or empty, and is Z-acyclic if G is solvable or G ~=An for n 6. Later*
*, in [AS ],
___________
1Key words: fixed point free, 2-complex, finite simple group
1991 mathematics subject classification: primary 57S17, secondary 20D05, 57M20,*
* 55M20
2Partially supported by the UMR 7539 of the CNRS.
3Partially supported by BSF grant no. 97-00042 and by grant no. 6782-1-95 f*
*rom the Israeli
Ministry of Science and Art.
1
2 Fixed point free actions on acyclic 2-complexes
Aschbacher and Segev extended these results, and proved that XG 6= ; if G is si*
*mple,
except perhaps when G is of Lie type and Lie rank one, or the first Janko group*
* J1 (a
sporadic group).
Techniques for constructing fixed point free actions of finite groups on fini*
*te acyclic or
contractible complexes (without restrictions on dimension) were developed by B.*
* Oliver
in several earlier papers such as [O1 ] and [O2 ]. In particular, in [O2 ], act*
*ions for which
the fixed point set of each subgroup is contractible or empty are studied.
The proof of Theorem A _ both when constructing actions of G and when proving
their nonexistence _ is based on refinements of the techniques developed in the*
*se earlier
papers of both authors. The main new input comes from a more detailed analysis *
*of the
subgroup lattice of G and its orbit space. In particular, necessary and suffici*
*ent condi-
tions for the existence of actions are stated in terms of this lattice in Propo*
*sition 1.9.
Afterwards, the proofs of nonexistence of actions of particular groups require *
*identifying
homology in certain "pieces" of the subgroup lattice of G.
In fact, relatively few solvable subgroups need occur as isotropy groups for *
*the actions
constructed when proving Theorem A, and those which do occur are listed explici*
*tly. It
is possible that these and similarly constructed G-complexes can give new infor*
*mation
about decompositions of BG, and about the cohomology of G.
Theorem A leaves open the question as to whether or not it is possible for a *
*finite group
to act on a 2-dimensional contractible complex without fixed points. Understan*
*ding
actions on acyclic 2-complexes is clearly a first step towards investigating th*
*is question,
but the first author feels that any serious attempt to answer it will require s*
*ome very
different methods than those used here.
This paper is intended for both group theorists and topologists, and we have *
*at-
tempted to write it in a way which will be appealing and readable for both. In*
* par-
ticular, more background material has been included than might normally be the *
*case,
although we have tried to put most of that in the appendix at the end of the pa*
*per.
The paper is organized as follows. In Section 1, conditions are established, *
*in terms
of homological properties of the subgroup lattice of G, which determine the min*
*imal
dimensions of certain "universal" G-complexes. In particular, this section incl*
*udes the
general machinery for constructing such actions. After proving some technical r*
*esults in
Section 2, the constructions of the G-complexes described in Theorem A are carr*
*ied out
in Section 3. In Section 4, we show that any finite group G which acts essentia*
*lly on a
2-dimensional acyclic complex must be almost simple (i.e., there is a nonabelia*
*n simple
group L such that L G Aut(L)). In Section 5, we develop machinery to show the
nonexistence of actions on acyclic 2-complexes; and this is applied in Section *
*6 to prove
Theorem A for simple groups of Lie type and Lie rank one. The sporadic groups *
*are
dealt with in Theorem 7; except for the first Janko group J1 this repeats resul*
*ts already
shown in [AS ]. Theorem B is proven in Section 4, and Theorem A in Section 8. *
* All
of this is preceded by a preliminary "Section 0" where we present some general *
*results
about G-posets and construction of G-complexes; and is followed by an appendix *
*which
includes background material about G-complexes, Z[G]-modules, and simple groups*
* of
Lie type, as well as a sketch of the proofs in [S1] and [AS ] of certain cases *
*of Theorem
A. References of the form A.x, B.x, etc. all refer to the appendix. After the a*
*ppendix,
we attach a list of the notation used throughout the paper.
Bob Oliver and Yoav Segev *
* 3
Bob Oliver would like to thank the Hebrew University for the hospitality duri*
*ng his
visit in 1997. This visit, and the visit of Yoav Segev to Arhus University in 1*
*992, played
an important role in starting our collaboration. We would also like to thank Je*
*an-Pierre
Serre for his letters, which also helped revive our interest in this problem.
0. G-complexes and G-posets
Posets, and in particular families of subgroups considered as posets, will pl*
*ay an
important role as "bookkeeping" devices for controlling dimensions of certain a*
*cyclic
complexes. For any poset S, we let N (S) denote its nerve: the simplicial compl*
*ex with
one vertex for each element of S, and one n-simplex for each chain ff0 < ff1 < *
*. .<.ffn
of elements of S. By a G-poset is meant a poset with G-action which preserves *
*the
ordering. A terminal subposet of a poset S is a subset S0 S such that fi ff *
*2 S0
implies fi 2 S0. For any element ff in a poset S, we set Sff = {fi 2 S | fi ff*
*}. The
next lemma provides a general setting for comparing G-complexes with coverings *
*to the
nerves of the coverings.
Lemma 0.1. Let X be a G-complex, let S be a finite G-poset, and let {Xff}ff2S*
*be a
covering of X by subcomplexes which satisfies the following conditions:
(a) ff fi implies Xff Xfi.
(b) For all x 2 X, the set {ff 2 S | x 2 Xff} has a largest element.
(c) Xg(ff)= g(Xff) for all ff 2 S, g 2 G.
Then there is a G-map fX : X ! N (S) with the property that
fX (Xff) N (Sff) for all ff 2 S. (1)
If, furthermore, Xffis acyclic (contractible) for each ff, then for any map f :*
* X !
N (S) which satisfies (1), and any terminal subposetSS0 S, f restricts to a hom*
*ology
equivalence (homotopy equivalence) fS0: XS0= ff2S0Xff! N (S0).
Proof.For each n 0, let Jn denote the G-set of n-cells of X, and let 'n: Jnx D*
*n ! X
denote the characteristic map for the n-cells. Let : Jn ! S be the map which s*
*ends
j 2 Jn to the largest element in the set {ff 2 S | 'n(j; 0) 2 Xff}; this is wel*
*l defined by
(b) and equivariant by (c). For each ff 2 S, we let [ff] denote the correspondi*
*ng vertex
in N (S).
First define f0: X(0)! N (S) by setting f0('0(j; 0)) = [(j)] for each j 2 J0.*
* This
clearly satisfies condition (1).
Now assume that fn-1: X(n-1)! N (S) has been defined, satisfying (1). For any
j 2 Jn and any v 2 Sn-1, 'n(j; 0) 2 X(j) by construction, and so 'n(j; v) 2 X(j*
*) since
X(j) is a subcomplex of X. So fn-1('n(j; v)) 2 N (S(j) ) by (1), hence it is in*
* some
simplex which contains the vertex [(j)], and the segment from fn-1('n(j; v)) to*
* [(j)]
lies in N (S). So we can define
fn: X(n)-----! N (S)
4 Fixed point free actions on acyclic 2-complexes
by setting fn(x) = fn-1(x) for x 2 X(n-1), and
fn('n(j; tv)) = t.fn-1('n(j; v)) + (1 - t).[(j)] for j 2 Jn, v 2 Sn-1, t *
*2 [0;:1]
This is well defined as a map of sets, since the two definitions agree on 'n(Jn*
*x Sn-1)
X(n-1). So it is continuous by Lemma A.3 (fn|X(n-1)and fn O'n are both continuo*
*us).
Condition (1) still holds for fn, since for all j 2 Jn and v 2 int(Dn), and all*
* ff 2 S,
'n(j; v) 2 Xff () 'n(j; 0) 2 Xff=) ff (j)
=) fn('n(j; v)) 2 N (S(j) ) N (Sff):
And fn is equivariant since is equivariant, since fn-1 is equivariant (by indu*
*ction),
and since the G-action on N (S) is affine.
Finally, define fX : X ! N (S) to be the union of the fn; this is again conti*
*nuous by
Lemma A.3, and condition (1) holds since it holds for each fn.
Now let f be any map which satisfies (1), and assume that Xffis acyclic (cont*
*ractible)
for each ff 2 S. We want to show that f is a homology (homotopy) equivalence.
The group action no longer plays a role here, so we can assume G = 1. We can
assume inductivelySthat for any properly contained terminal poset S0 $ S, f res*
*tricts
to an equivalence ff2S0Xff! N (S0) (since the subspace and subposet still sat*
*isfy
conditions (a) and (b) above). If S has a smallest element oe, then X = Xoeis a*
*cyclic
(contractible) and N (S) is contractible, so any map f : X ! N (S) is a homolo*
*gy
(homotopy) equivalence, and we are done.
Assume now that S contains no smallest element. In this case, we can write S*
* =
S1 [ S2, whereSS1 and S2 are proper terminal subposets of S. Set S0 = S1 \ S2;
and set Xi = ff2SiXfffor each i = 0; 1; 2. Clearly, N (S0) = N (S1) \ N (S2)*
*, and
condition (b) implies that X0 = X1 \ X2. By the inductive assumption, f restric*
*ts to
homology (homotopy) equivalences fi: Xi! N (Si), and so f is a homology (homoto*
*py)_
equivalence by Proposition B.3. |*
*__|
By a family of subgroups of G will here be meant any subset F S(G) which is
closed under conjugation. We do not assume here that subgroups of elements of *
*the
family are also in the family.
For any family F of subgroups of G, a (G; F)-complex will mean a G-CW-complex
all of whose isotropy subgroups lie in F. A (G; F)-complex is universal if the*
* fixed
point set of each subgroup in F is contractible. (The "universality" property *
*of such
spaces is explained in Proposition A.6.) One can, in fact, construct universal *
*(G; F)-
complexes for any family F of subgroups of G, but in most cases any such comple*
*x must
be infinite dimensional. For example, when F = {1} contains only the trivial su*
*bgroup,
a universal (G; F)-complex is just a contractible complex upon which G acts fre*
*ely; and
so its orbit space is a classifying space for G. The results in Section 1 will *
*make it clear
what conditions are needed on F for there to be a finite (or finite dimensional*
*) universal
(G; F)-complex.
The following lemma is the starting point for the constructions of universal *
*(G; F)-
complexes, and of other G-complexes satisfying certain homological conditions. *
*Roughly,
it describes the effect on the homology of X of attaching cells of one orbit ty*
*pe G=H
to X. By "attaching an orbit of cells of type G=Hx Di" to a G-complex X, we mean
Bob Oliver and Yoav Segev *
* 5
replacing X by the complex X ['(G=Hx Dn) for some G-map ': G=Hx Sn-1 ! X(n-1).
We refer to Lemma A.2 for more detail.
Proposition 0.2. Fix a finite G-complex X, and a subgroup H G. Then the follow-
ing hold.
(a) For any n 1, there is a finite G-complex Y X, obtained by attaching to X
orbits of cells of type G=Hx Di for 1 i n, such that Y H is (n - 1)-connected*
* and
Hi(Y H) ~=Hi(XH ) for all i > n. Also, Hn(Y H) is Z-free if Hn(XH ) is Z-free.
(b) Assume n 1, and that XH is (n - 1)-connected. For any homomorphism
': (Z[N(H)=H])k -----! Hn(XH )
of Z[N(H)=H]-modules, there is a finite G-complex Y X, obtained by attaching k
orbits of cells G=Hx Dn+1 to X, such that Hi(Y H) ~=Hi(XH ) for all i 6= n; n +*
* 1, such
that
Hn(Y H) ~=Coker('); (1)
and such that there is a short exact sequence
0 ---! Hn+1(XH ) ----! Hn+1(Y H) ----! Ker (') ---! 0: (2)
(c) Assume, for some n 1, that eH*(XH ) = Hn(XH ) is a stably free Z[N(H)=H]-
module; more precisely that
Hn(XH ) (Z[N(H)=H])k ~=(Z[N(H)=H])m
(where k; m 0). Then there exists a G-complex Y X, obtained by attaching to X
k orbits of cells of type G=Hx Dn and m orbits of cells of type G=Hx Dn+1, such*
* that
Y H is acyclic.
(d) Assume that all connected components of XH are acyclic, and that one of *
*the
components of XH is fixed by the action of N(H)=H and the others are permuted f*
*reely.
Then there exists a G-complex Y X, obtained by attaching to X cells of orbit t*
*ype
G=Hx D1, such that Y H is acyclic.
Proof. (b) Since XH is (n - 1)-connected, the Hurewicz theorem applies to show *
*that
each element h 2 Hn(XH ) is represented by a map ': Sn ! XH , in the sense that
h = '*([Sn]) for some fixed generator [Sn] of Hn(Sn). (See, e.g., [Hu , Theorem*
* II.9.1]
if n > 1, or [Hu , Theorem II.6.1] if n = 1). And we can assume that '(Sn) (XH*
* )(n)
by the cellular approximation theorem [LW , Theorem II.8.5], which says that an*
*y map
Sn ! XH is homotopic to a cellular map, and in particular a map with image in *
*the
n-skeleton.
Now let E = {e1; : :;:ek} denote the canonical basis of (Z[N(H)=H])k, and fix*
* maps
fi: Sn ! (XH )(n)which represent '(ei) 2 Hn(XH ). Define
f : (Ex G=H)x Sn -----! X(n)
by setting f(ei; gH; x) = g.fi(x); and let fH be the restriction of f to the H*
*-fixed
point sets. In particular, for each i and each g 2 N(H), f|eixgHxSn(as a map Sn*
* !
XH ) represents the class g.'(ei) 2 Hn(XH ). In other words, Hn(fH ) = ' under*
* the
identification
Hn((Ex G=H)H xSn) = Hn((Ex N(H)=H)x Sn) ~=(Z[N(H)=H])k:
6 Fixed point free actions on acyclic 2-complexes
Set
Y = X [f (Ex G=H)x Dn+1
(Lemma A.2). Then
n+1
Y H = XH [fH (Ex N(H)=H)x D ;
and (1) and (2) now follow from Lemma B.1.
(a) We prove this inductively. Fix n 0 such that XH is (n - 1)-connected. *
* We
will construct a finite G-complex Y X, obtained by attaching orbits of cells o*
*f type
G=Hx Dn+1 to X, such that Y H is n-connected.
If n = 0 and XH is not connected, then let v-1 and v1 be two vertices in dif*
*ferent
connected components of XH , define f : G=Hx S0 ! X by setting f(gH; t) = gvt, *
*and
set X0 = X [f (G=Hx D1). By construction, (X0)H has fewer connected components
than XH , and by continuing the procedure we obtain a finite G-complex Y such t*
*hat
Y H is connected.
If n = 1 and ss1(XH ) 6= 1, then choose any element 1 6= OE 2 ss1(XH ), repre*
*sent it
by a map f0: S1 ! XH , and extend this to a G-map f : G=Hx S1 ! X by setting
f(gH; v) = g.f0(v). Set X0 = X [f (G=Hx D2). Then ss1((X0)H ) = ss1(XH )=N, whe*
*re
N is a normal subgroup of ss1(XN ) which contains OE (in fact, the normal closu*
*re of ).
Since ss1(XH ) is finitely generated, we can repeat this procedure and obtain a*
* finite
G-complex Y such that Y H is 1-connected.
If n > 1, then the result follows from part (b), where we choose ' to be any *
*surjection
(Hn(XH ) is finitely generated as an abelian group, hence as a Z[N(H)=H]-module*
*).
(c) Upon applying point (b) to the trivial homomorphism
'0: (Z[N(H)=H])k ! Hn-1(XH ) = 0;
we get a finite G-complex Y0 X, obtained by attaching k-orbits of cells G=Hx D*
*n to
X, such that Hi((Y0)H ) ~=Hi(XH ) = 0 for all i 6= n and
Hn((Y0)H ) ~=Hn(XH ) (Z[N(H)=H])k ~=(Z[N(H)=H])m :
If we now apply (b) to any isomorphism ': (Z[N(H)=H])m ! Hn((Y0)H ), we obtain a
finite G-complex Y Y0, constructed by attaching m orbits of cells G=Hx Dn+1, s*
*uch
that Y H is acyclic.
(d) Here, we assume that all connected components of XH are acyclic, and that*
* one is
invariant under the action of N(H)=H and the others are permuted freely. Let X0*
* XH
denote the component which is N(H)=H-invariant, and let X1; X2; : :;:Xk be N(H)*
*=H-
orbit representatives for the other components. (If N(H)=H = 1, then let X0 be *
*any of
the connected components.) Fix vertices xi 2 Xi for i = 0; : :;:k. Set J = {1; *
*: :;:k},
and define ': (G=Hx J)x S0 ! X by setting
'(gH; i; 1) = gxi and '(gH; i; -1) = gx0:
Now set Y = X [' ((G=Hx J)x D1). Then
Y H = XH ['|((N(H)=Hx J)x D1);
and this is acyclic since X0 has been connected (by a unique 1-cell) to each of*
*_the other
connected components of X. |__|
Bob Oliver and Yoav Segev *
* 7
We finish the section with two lemmas which involve elementary properties of *
*nerves
of posets. We first recall the following results of Quillen.
Lemma 0.3 [Q2 , 1.3-1.5].(a) Let T S be posets, and let r : S ! T be any order
preserving map such that r|T = IdT, and such that r(ff) ff for all ff 2 S (or *
*r(ff) ff
for all ff). Then the inclusion of N (T ) in N (S) is a homotopy equivalence.
(b) Let G be a finite group, and let H be any set of subgroups of G. Assume t*
*here
is some H0 2 H such that either H \ H0 2 H for all H 2 H, or 2 H for all
H 2 H. Then N (H) is contractible.
Proof.Point (a) is shown in [Q2 , 1.3]. In fact, N (T ) is a strong deformation*
* retract of
N (S), where r induces the retraction N (S) ! N (T ), and where the homotopy wi*
*th
the identity comes from the assumption that r(x) is always x or always x [Q*
*2 , 1.3].
If H is as in (b), then its nerve is "conically contractible" in the sense of*
* Quillen,_and
hence is contractible [Q2 , 1.4-1.5]. *
* |__|
The following lemma will also be useful, when showing that certain subgroups *
*of G
need not occur as isotropy subgroups in acyclic G-complexes.
Lemma 0.4. Let S be any finite poset, and let S0 S be any subposet with the
property that N (S>ff) ' * for all ff 2 Sr S0. Then N (S0) ' N (S) (the inclus*
*ion
induces a homotopy equivalence).
Proof.It suffices to show this when Sr S0 contains just one element ff. In thi*
*s case,
N (S) is the union of N (S0) with the cone over the subcomplex A N (S0), where
A = N Sff = N (Sff): (1)
*
Note that the nerve of the disjoint union in (1) is identified with the join of*
* the nerves,
since every element in Sff. Then A is contra*
*ctible,_
since N (S>ff) ' * by assumption. *
*|__|
Lemma 0.4 does, in fact, hold without the assumption that S is finite: it fol*
*lows as a
consequence of Quillen's Theorem A [Q1 ] (see also [Q2 , Proposition 1.6]).
A central problem throughout this paper, especially in Sections 5 and 6, is t*
*o find
ways to detect 2-dimensional homology in nerves of certain posets. Given a 2-cy*
*cle in
N (S), the simplest way to show it is nonvanishing in H2(N (S)) is to show that*
* some
2-simplex with nonzero coefficient is maximal in N (S); i.e., not in the bounda*
*ry of any
3-simplex. The following lemma provides a refinement of this observation, and w*
*ill be
used in Section 5.
Lemma 0.5. Let S be a finite poset, and let z be a 2-cycle in the nerve of S.*
* Fix ele-
ments m < M in S, where m is minimal and M is maximal. Set Q = {x2 S | m< x< M},
and let Q0 Q be the set of all x 2 Q such that the simplex (m; x; M) occurs wi*
*th
nonzero coefficient in z. Assume that Q0 6= ;, and that some element of Q0 lie*
*s in a
separate connected component of N (Q) from all of the other elements of Q0. Th*
*en
0 6= [z] 2 H2(N (S)).
Proof.Set X = N (S), for short, and let Y X be the subcomplex of all simplices
which do not contain both vertices m; M. Let C*(X) C*(Y ) be the simplicial ch*
*ain
8 Fixed point free actions on acyclic 2-complexes
complexes; and write
X
z = ax(m; x; M) (mod C2(Y ))
x2Q0
(where 0 6= ax 2 Z for each x).
For any 3-simplex oe in X, either oe is in Y (and so @(oe) 2 C2(Y )), or oe =*
* (m; x; y; M)
for some x; y 2 Q in the same connected component of N (Q) and
@(oe) = (m; x; M) - (m; y; M) (mod C2(Y )):
Thus, if z is a boundary, then the sum of the coefficients ax in the above expr*
*ession
for z, taken over all x 2 Q0 which lie in any given connected component of N (Q*
*), is
zero. And this contradicts the assumption that some element of Q0is in a compon*
*ent_
by itself. |*
*__|
1.Minimal dimensions of universal G-spaces
We will now establish necessary and sufficient conditions for the existence o*
*f universal
complexes satisfying certain dimensional restrictions. These conditions will be*
* expressed
in terms of the homology of the nerves of certain posets.
Throughout this section, G will be a finite group. A nonempty family F S(G)
will be called separating if it has the following three properties: (a) G =2F; *
*(b) any
subgroup of an element of F is in F; and (c) for any H C K G with K=H solvable,
K 2 F if H 2 F. The following property of separating families is immediate.
Lemma 1.1. Each maximal subgroup in a separating family of subgroups of G is_*
*self-
normalizing. |__|
If G is solvable, then it has no separating family of subgroups. If G is not *
*solvable,
then we let SLV denote the family of solvable subgroups: the minimal separatin*
*g family
for G. We also let MAX denote the maximal separating family for G, which can*
* be
described as follows. Let L be the maximal normal perfect subgroup of G; i.e., *
*the last
term in the derived series of G. Then MAX is the family of all subgroups of *
*G which
do not contain L. In particular, if G is perfect, then MAX is the family of *
*all proper
subgroups of G.
A (G; F)-complex will be called H-universal if the fixed point set of each H *
*2 F is
acyclic. The importance of universal, and H-universal, (G; F)-complexes when st*
*udying
2-dimensional actions comes from the following lemma.
Lemma 1.2. Let X be any 2-dimensional acyclic G-complex without fixed points.*
* Let
F be the set of subgroups H G such that XH 6= ;. Then F is a separating family*
* of
subgroups of G, and X is an H-universal (G; F)-complex.
Proof.By [S1, Theorem 3.4], XH is acyclic for each H 2 F; i.e., for each H suc*
*h that
XH 6= ;. (Another proof of this, which does not depend on the odd order theorem*
*, is
given in Theorem 4.1 here.) So by definition, X is an H-universal (G; F)-comple*
*x. Also,
if H C K G are subgroups such that H 2 F and K/H is solvable, then XH is acycl*
*ic,
Bob Oliver and Yoav Segev *
* 9
and so XK = (XH )K=H is acyclic by [S1, Theorem 3.1] (see also Theorem 4.1). T*
*hus,_
F is a separating family. *
*|__|
For any family F of subgroups of G, we consider N (F) as a G-complex via the
conjugation action. Note, however, that N (F) is not itself a (G; F)-complex in*
* general.
For example, when F = {1}, then N (F) is a point, while a (G; F)-complex must h*
*ave
a free G-action.
Recall that for any family F of subgroups of G and any set H of subgroups, FH
denotes the poset of those subgroups in F which contain some element of H. Als*
*o,
for any H G, FH and F>H denote the posets of subgroups in F which contain H,
or strictly contain H, respectively. The following proposition follows immediat*
*ely from
Lemma 0.1.
Proposition 1.3. Fix any family F of subgroups of G. Let N (F) be the nerve of *
*the
poset F, regarded as a G-complex via the action by conjugation. Then for any (G*
*; F)-
complex X, there is a G-map f : X ! N (F) with the property that f(XH ) N (FH *
* )
for all H G. And if X is universal (H-universal), then for any set H of subgro*
*ups of
G, any such map f restricts to a homotopy equivalence (homology equivalence) XH*
* !
N (FH ).
Proof.We apply Lemma 0.1, with S = F (regarded as a poset via inclusion), and
XH = XH for H 2 F. Since X is a (G; F)-complex, every element of X is fixed by
some H 2 F, and so {XH }X2F is a covering of X. Condition (a) of Lemma 0.1 clea*
*rly
holds, and condition (b) holds since the largest element of {H 2 F | x 2 XH } i*
*s the
-1 H *
* __
isotropy group Gx. And condition (c) holds since XgHg = g(X ). *
* |__|
The following lemma, which helps to limit the number of orbit types needed wh*
*en
constructing "minimal" universal (G; F)-complexes, is an easy consequence of Le*
*mma
0.4.
Lemma 1.4. Let F be any family of subgroups of G, and let F0 F be any subfam*
*ily
such that N (F>H ) ' * for all H 2 Fr F0. Then any (H-)universal (G; F0)-comple*
*x is
also an (H-)universal (G; F)-complex; and
N ((F0)H ) ' N (FH ) (1)
for any set H of subgroups of G.
Proof. For any set H of subgroups of G, point (1) follows from Lemma 0.4, appli*
*ed to
the posets S def=FH and S0def=(F0)H .
Let X be an (H-) universal (G; F0)-complex. All isotropy subgroups of X lie *
*in
F0 F, so X is also a (G; F)-complex. For each K 2 F, XK is homotopy (homology)
equivalent to N ((F0)K ) by Proposition 1.3 (applied with H = {K}); this in tu*
*rn is
homotopy (homology) equivalent to N (FK ) by (1); and this last complex is con*
*tractible_
(acyclic). So X is also (H-) universal as a (G; F)-complex. *
* |__|
We are now ready to deal directly with the problem of controlling the dimensi*
*ons of
universal or H-universal (G; F)-complexes. This will be done by attaching cell*
*s, one
orbit type at a time, at each stage arranging for the appropriate fixed point s*
*et to be
contractible or acyclic. The key problem is how to do this with cells in free *
*orbits.
10 Fixed point free actions on acyclic 2-complexes
This will be described in the following three lemmas. The first will be needed*
* when
constructing contractible 1-complexes.
Lemma 1.5. Let X be any finite G-set with the property that |XH | = 1 for each
subgroup 1 6= H G of prime power order. Then X has one fixed point and is
otherwise free.
Proof. We may assume that XG = ;; otherwise the result is clear. We may also as*
*sume
that X has no free orbits (otherwise just remove them). By assumption, each Sy*
*low
subgroup of G acts freely on X away from one fixed point; and so |X| 1 (mod |G*
*|).
Write X = G=H1 q G=H2 q . .q.G=Hk, where 1 6= Hi$ G for all i. In particular,
Xk
[G:Hi] = |X| = r.|G| + 1 (1)
i=1
for some r. Furthermore, for each pair of distinct elements x; y 2 X, the isot*
*ropy
subgroups Gx and Gy have trivial intersection, since otherwise Gx \ Gy contains*
* a
nontrivial p-subgroup (some p) which fixes two points of X. It follows that
X Xk Xk
|G| - 1 (|Gx| - 1) = [G:Hi].(|Hi| - 1) = k.|G| - [G:Hi]:(2)
x2X i=1 i=1
Upon adding (1) and (2), we see that (2) is an equality, and that r = k - 1. Bu*
*t then
after dividing (1) by |G|, we get that
Xk
_1__
> k - 1:
i=1|Hi|
*
* __
Since |Hi| 2 for all i, we must have k = 1, and hence |X| = 1. *
* |__|
A complex X will be called homologically m-dimensional if Hn(X) = 0 for all n*
* > m,
and Hm (X) is Z-free. (Technically, this should be called homologically m-dime*
*nsional,
since it only provides an upper bound on the degrees of homology of X.) We note*
* first
the following properties of subcomplexes of acyclic complexes.
Lemma 1.6. Let X be any m-dimensional acyclic CW complex (m 1). Then any
subcomplex of X is homologically (m- 1)-dimensional. And if A1; : :;:An X are
homologically (m- 2)-dimensional subcomplexes, then their intersection is also *
*homo-
logically (m- 2)-dimensional.
Proof.For any subcomplex A X, eHi(A) ~=Hi+1(X; A) must be zero for i m and
Z-free for i = m - 1. Hence A is homologically (m- 1)-dimensional.
It suffices to prove the last statement when n = 2. For each i m - 2, there*
* is a
Mayer-Vietoris exact sequence
0 ---! Hi+1(A1 [ A2) ----! Hi(A1 \ A2) ----! Hi(A1) Hi(A2):
If i m - 1, then the first and last groups are zero, and so Hi(A1 \ A2) = 0. *
*And
if i = m - 2, then the first and last groups are Z-free, and so Hm-2(A1 \ A2)_i*
*s_also
Z-free. |__|
The next lemma is essentially included in the proof of [O2 , Proposition 6].
Bob Oliver and Yoav Segev *
* 11
Proposition 1.7. Let X be a finite G-complex with the following two properties.
(a) For each 1 6= H G, XH is acyclic or empty, and is acyclic if H has prime*
* power
order.
(b) For some n > 0, eH*(X) = Hn(X), and is Z-free.
Then Hn(X) is stably free as a Z[G]-module.
Proof. For each prime p and each Sylow p-subgroup S G, consider the subcomplex
[
X0= XH = {x 2 X | Sx 6= 1}:
16=HS
By Proposition 1.3, applied with H = {1 6= H S}, X0 is acyclic (N (H) ' * sinc*
*e H
has maximal element S). Hence H*(X; X0) ~=He*(X) also vanishes in degrees diffe*
*rent
from n. Furthermore, since all cells in Xr X0 are permuted freely by S, C*(X; X*
*0) is a
chain complex of finitely generated free Z[S]-modules (Lemma C.1). So by Propos*
*ition
C.2, the unique nonvanishing homology group Hn(X; X0) ~=Hn(X) is Z[S]-stably fr*
*ee.
(Since all but one summand in (1) of Proposition C.2 is stably free, so is the *
*remaining
summand, by definition.) In particular, Hn(X) is a Z[G]-module which is projec*
*tive
after restriction to each Sylow subgroup, and is hence Z[G]-projective by Rim's*
* theorem
[Rim , Proposition 4.9].
Now set Y = Xx X, where X is the unreduced suspension of X (see Lemma
A.5). We identify X with the subcomplex Xx {x0} of Y , where x0 2 X is one of
the suspension vertices. Then H*(X; x0) = Hn+1(X; x0) ~= Hn(X); and so by the
K"unneth formula
8
>:
0 otherwise.
Consider the subcomplexes
[ [
Xs = XH and Ys = Y H:
16=HG 16=HG
We claim that the inclusion map Xs ,! Ys is a homology equivalence. To see this*
*, set
F = {1 6= H G | XH 6= ;}. By Proposition 1.3, there is a map f : Ys ! N (F)
such that f((Ys)H ) N ((F)H ) for all H G; and f|Xs has the same property. S*
*ince
Xs and Ys are both H-universal (G; F)-complexes (Y H = XH xXH is acyclic if XH
is), Proposition 1.3 implies that f restricts to homology equivalences Ys ! N (*
*F) and
Xs ! N (F); and thus that the inclusion Xs Ys is a homology equivalence.
In particular, this shows that H*(Y; X) ~= H*(Y; X [ Ys) (see Lemma B.2). Th*
*us,
C*(Y; X [ Ys) is a chain complex of free Z[G]-modules (by Lemma C.1, since G ac*
*ts
freely on Y r(X [ Ys)) with only two nonzero homology groups. Since Hn(X) Z Hn(*
*X)
is stably free by Proposition C.3, the other homology group Hn(X) must also be *
*stably_
free by Proposition C.2. |*
*__|
For any G-space X and any H G, we write
X>H = {x 2 X | Gx % H}:
12 Fixed point free actions on acyclic 2-complexes
i.e., the union of fixed point sets of subgroups which strictly contain H. Also*
*, for any
family F S(G), F>H denotes the set of elements of F which strictly contain H.
Proposition 1.8. Let G be any finite group, and let F be a separating family fo*
*r G. Let
F0 F be any subfamily with the property that N (F>H ) is contractible (and non*
*empty)
for all H 2 Fr F0. Let d: F0 ! N be any function which is constant on conjugacy
classes of subgroups, such that d(H) = 0 for H maximal in F, such that N ((F0)>*
*H )
is homologically (d(H)- 1)-dimensional for each non-maximal subgroup H 2 F0, and
such that d(H) d(H0) whenever H H0. Then there is a finite H-universal (G; F0*
*)-
complex X with the property that dim(XH ) d(H) for each H 2 F0. Furthermore, X
can be taken to be universal if d(H) 6= 2 for each H 2 F0. Also, X can be chose*
*n such
that every vertex of X is fixed by some maximal subgroup in F.
Proof.Let Fmax be the set of maximal subgroups in F. Set X0 = Fmax, regarded
as a zero-dimensional G-complex. Since the elements of Fmax are all self-norma*
*lizing
(Lemma 1.1), this is a 0-dimensional (G; F)-complex, and (X0)H contains exactly*
* one
point for each H 2 Fmax. Let H1; : :;:Hk = 1 be conjugacy class representative*
*s for
the elements of Fr Fmax, ordered such that d(H1) d(H2) . .d.(Hk), and such th*
*at
i j if Hi contains a subgroup conjugate to Hj. For each i = 0; : :;:k, let Hi *
*be the
set of all maximal subgroups in F, together with all subgroups conjugate to Hj *
*for any
j i. In particular, H0 = Fmax and Hk = F. We construct a sequence of G-complex*
*es
X0 X1 X2 . . .Xk, such that for each i 1,
(a) dim(Xi) d(Hi) and X(0)i= X(0)0,
(b) XirXi-1has only orbit types G=Hi,
(c) Xi= Xi-1if Hi2=F0, and
(d) (Xi)Hi is acyclic, and is contractible if Hi2 F0 and d(Hi) 6= 2.
Note that for each H 2 Fmax, (X0)H = {H} is contractible, and hence (Xi)H will *
*be
contractible for all i > 0. Once the Xi have been constructed, we set X = Xk. *
* This
is a (G; F0)-complex; and for all H 2 F0, dim(XH ) d(H), and XH is acyclic, a*
*nd
contractible if d(H) 6= 2. And by (a), each vertex of X is in X0, and hence fix*
*ed by a
maximal subgroup of F.
It remains to construct the Xi. Assume that Xi-1has been constructed (i 1). *
*Then
Xi-1is an H-universal (G; Hi-1)-complex. By Proposition 1.3 (and by definition *
*of the
Hj),
H >H
H* (Xi-1) i = H* (Xi-1) i ~=H* N ((Hi-1)>Hi) = H* N (F>Hi) :
In particular, by Lemma 1.4, (Xi-1)Hi is homologically (d(Hi)- 1)-dimensional, *
*and is
acyclic if Hi2=F0. Also, dim(Xi-1Hi) d(Hi): this is clear if i = 0 (dim (X0) =*
* 0), and
holds for i 1 by (a) since d(Hj) d(Hi) for j < i by assumption. Thus, if Hi2=*
* F0,
we can set Xi= Xi-1.
Assume now that Hi 2 F0. Write H = Hi and d=d(H) for short. If d = 1, then
(Xi-1)H is 1-dimensional, and its connected components are all acyclic. By Lemm*
*a 1.5,
applied to the N(H)=H-set ss0((Xi-1)H ) (the set of connected components of (Xi*
*-1)H ),
(Xi-1)H has one connected component which is fixed by the action of N(H)=H, and*
* the
other components are permuted freely by N(H)=H. So Proposition 0.2(d) applies *
*to
Bob Oliver and Yoav Segev *
* 13
show that there is a finite G-complex Xi, obtained by attaching orbits of cells*
* G=Hx D1
to Xi-1, such that (Xi)H is acyclic.
If d > 1, then by Proposition 0.2(a), there is a G-complex Y Xi-1, construc*
*ted
by attaching cells G=Hx Dk for 1 k d - 1, such that Y H is (d - 2)-connected
and Hd-1(Y H) is Z-free. In particular, Y H is still homologically (d - 1)-dim*
*ensional,
and dim(Y H) d. For any subgroup 1 6= K=H N(H)=H of prime power order,
(Y H)K=H = Y K = (Xi-1)K is acyclic by (d): K 2 F by definition of a separating*
* family,
and so K 2 Hi-1. Proposition 1.7 now applies to show that Hd-1(Y H) is stably f*
*ree as
a Z[N(H)=H]-module. So by Proposition 0.2(c), we can attach orbits of cells of *
*type
G=Hx Dk for k = d - 1; d to Y , to obtain a finite G-complex Xi Y such that (Xi*
*)H_
is acyclic. |*
*__|
In fact, one can show for any family F of subgroups of G that there is a univ*
*ersal
(G; F)-complex. But such a complex must be infinite dimensional if F is not a s*
*eparating
family.
We can now state necessary and sufficient conditions for the existence of uni*
*versal or
H-universal (G; F)-complexes of a given dimension.
Proposition 1.9. Let G be any finite group, and let F be a separating family for
G. Let F0 F be any subfamily with the property that N (F>H ) is contractible
(and nonempty) for all H 2 Fr F0. Then there is a finite universal (G; F0)-comp*
*lex.
Furthermore, the following four conditions are equivalent for any m 2:
(a) There exists an m-dimensional universal (G; F)-complex (H-universal if m *
*= 2).
(b) There exists a finite m-dimensional universal (G; F0)-complex (H-universa*
*l if m =
2).
(c) N (F>H ) is homologically (m - 1)-dimensional for each subgroup H 2 F0.
(d) N ((F0)H ) is homologically (m - 1)-dimensional for each set H of subgro*
*ups of
G.
Proof. Since the nerve N (F) is finite dimensional, the existence of a finite u*
*niversal
(G; F0)-complex follows from Proposition 1.8.
(a ) d) If X is an m-dimensional H-universal (G; F)-complex, then for any se*
*t of
subgroups H, XH is homologically (m - 1)-dimensional by Lemma 1.6. Since
H*(XH ) ~=H*(N (FH )) ~=H*(N ((F0)H ))
by Proposition 1.3 and Lemma 1.4, N ((F0)H ) is also homologically (m-1)-dimen*
*sional.
(d ) c) Follows immediately from Lemma 1.4.
(c ) b) Follows immediately from Proposition 1.8.
*
*__
(b ) a) Follows immediately from Lemma 1.4. |*
*__|
As an immediate corollary of Proposition 1.9, we get:
Corollary 1.10. Let G be any finite group, and let F be a separating family for*
* G.
Then there is a (finite) 2-dimensional H-universal (G; F)-complex if and only i*
*f N (F>H )
is homologically 1-dimensional for each subgroup H 2 F, if and only if N (FH )*
* is__
homologically 1-dimensional for each set H of subgroups of G. *
* |__|
14 Fixed point free actions on acyclic 2-complexes
2.Numbers of cells
Again, G will always be a finite group throughout this section. We prove here*
* some
results which will be useful for keeping track of Euler characteristics of (uni*
*ons of) fixed
point sets in H-universal G-complexes. The notation used for doing this is defi*
*ned as
follows:
Definition 2.1.For any family F of subgroups of G, define
1
iF (H) = i(G;F)(H) = _________. 1 - O(N (F>H )) :
[N(H):H]
for each H 2 F. Set I(G; F) = i(G;F)(1).
We first note the following elementary relation between Euler characteristics*
* of G-
complexes and of their orbit spaces.
Lemma 2.2. Let X0 X be any pair of finite G-complexes, and assume that all or*
*bits
in Xr X0 are of type G=H for some fixed subgroup H G. Then
0
O(X) - O(X0) = |G=H|. O(X=G) - O(X =G) :
Proof.For each nP 0, let cn denote the number of n-cells in X not in X0. Then
O(X) - O(X0) = n0 (-1)ncn. By assumption, each G-orbit of cells has order exa*
*ctly
|G=H|. So the number of n-cells in X=G not in X0=G is __1_|G=H|.cn for each n, *
*and thus
X cn 1 __
O(X=G) - O(X0=G) = ______ = ______ O(X) - O(X0) : |__|
n0 |G=H| |G=H|
The relation between these indices and Euler characteristics of universal com*
*plexes is
given in the following two lemmas.
Lemma 2.3. Fix a separating family F, a finite H-universal (G; F)-complex X, *
*and a
subgroup H G. ForPeach n, let cn(H) denote the number of orbits of n-cells of *
*type
G=H. Then i(H) = n0 (-1)ncn(H).
Proof.By Proposition 1.3, there is a G-map f : X ! N (F), which restricts to ho*
*mology
equivalences XH ! N (FH ) and X>H ! N (F>H ). Thus, by Definition 2.1, and *
*by
Lemma 2.2 applied to the action of N(H) on the complexes X>H XH ,
1 1 H >H
iF (H) = _________. 1 - O(N (F>H ))= _________. O(X ) - O(X )
[N(H):H] [N(H):H]
= O(XH =N(H)) - O(X>H =N(H)):
Each orbit of cells of type G=Hx Dn in X restricts to one of type (N(H)=H)x Dn *
*in
XH , and hence to exactly one n-cell in the orbit space XH =N(H). These are pre*
*cisely
the cells in XH =N(H) which are not in X>H =N(H), and hence
X __
O(XH =N(H)) - O(X>H =N(H)) = (-1)ncn(H): |__|
n0
Bob Oliver and Yoav Segev *
* 15
Lemma 2.4. Let F be any separating family of subgroups of G, and let X be any
finite H-universal (G; F)-complex. Let H F be any subset with the property th*
*at
K H 2 H and K 2 F implies K 2 H. Then
H X
O(N (H)) = O X = [N(H):H].iF (H): (1)
H2H
If, furthermore, H is a family (i.e., a union of G-conjugacy classes), then
H X
O X =G = iF (H): (2)
H2H=conj
Proof.We prove these formulas by induction on |H|; they clearly (vacuously) hol*
*d when
H = ;. Let H be a minimal subgroup of H, and set H0 = Hr {H}. Then N (H) =
N (H0) [N(F>H) C(N (F>H )); in other words, the union of N (H0) and C(N (F>H ))*
* (the
cone over N (F>H )) with intersection N (F>H ). So by the Mayer-Vietoris sequen*
*ce for
the union,
O(N (H)) = O(N (H0)) + 1 - O(N (F>H )) = O(N (H0)) + [N(H):H].iF (H);
P
and so O(N (H)) = H2H [N(H):H].iF (H) by induction. Since O(N (H)) = O(XH ) by
Proposition 1.3, this proves (1).
Now assume that H is a family. For each n 0 and each H 2 H, let cn(H) be the
number of orbits of n-cells of type G=H. Let cn(H) be the sum, taken over conju*
*gacy
class representatives for all H 2 H, of the cn(H). Then cn(H) is precisely the *
*number
of n-cells in XH =G; and so
H X1 n X X1 n X
O X =G = (-1) cn(H) = (-1) cn(H) = iF (H)
n=0 H2H=conjn=0 H2H=conj
__
by Lemma 2.3. |__|
Corollary 2.5. For any separating family F of subgroups of G,
X
iF (H) = 1:
H2F=conj
Proof.If X is any finite H-universal (G; F)-complex, then in particular X is ac*
*yclic,
and so X=G is acyclic (cf. [Br, Theorem III.7.12]). Thus O(X=G) = 1, and so the*
*_result_
follows from Lemma 2.4 (applied with H = F). |*
*__|
The following relations will be useful later, when manipulating nerves of sub*
*groups
of G.
Lemma 2.6. Fix a separating family F of subgroups of G. Let Fc F be the subf*
*amily
of those subgroups H 2 F such that N (F>H ) is not contractible. Fix a subgroup*
* H 2 Fc
such that H $ N(H) 2 F, and let K1; : :;:Kr be G-conjugacy class representative*
*s for
the subgroups K 2 Fc such that K % H and NK (H) = H. For each j, let aj be the
number of Kj-conjugacy classes of subgroups in Kj which are G-conjugate to H and
self-normalizing in Kj. Then
Xr
i(G;F)(H) = - aj.i(G;F)(Kj): (1)
j=1
16 Fixed point free actions on acyclic 2-complexes
Proof. For any subgroup H 2 Fr Fc, N (F>H ) is contractible, and so iF (H) = 0
by Definition 2.1. So we can assume that the K1; : :;:Kr contain G-conjugacy c*
*lass
representatives for all subgroups K 2 F such that K % H and NK (H) = H (not just
those in Fc), without changing the right-hand side in (1).
Let X be any finite H-universal (G; F)-complex. Set H = FH , and set H0 = {K*
* 2
F | K H; NK (H) % H}. Then N (H) and N (H0) are both contractible by Lemma
0.3(b): the first since H has smallest element H; and the second since N(H) 2 H*
*0, and
N(H) \ K 2 H0 for all K 2 H0.
By Lemma 2.4,
X
[N(K):K].iF (K) = O(N (H)) - O(N (H0)) = 1 - 1 = 0: (2)
K2HrH0
Set R = {K 2 F | K % H; NK (H) = H}; the subgroups K1; : :;:Kr are thus G-
conjugacy class representatives for the elements of R. For each j, set
Sj = {g 2 G | gKjg-1 H; NgKjg-1(H) = H}
= {g 2 G | gKjg-1 H; NKj(g-1Hg) = g-1Hg}:
Then by (2),
X |N(K)|.|H| X |N(K)|
iF (H) = - __________iF (K)= - _________iF (K)
K2R |N(H)|.|K| K2R0|N(H).K| 1
Xr X
1
= - @ ______________-1AiF (Kj);
j=1 g2Sj|N(g Hg).Kj|
and it remains only to show that the sum in parentheses is equal to aj: the num*
*ber
of Kj-conjugacy classes of subgroups g-1Hg for g 2 Sj. And this follows since f*
*or any
g; g0 2 Sj, g-1Hg and g-10Hg0 are Kj-conjugate if and only if there exists a 2 *
*Kj such
that a-1g-10Hg0a = g-1Hg, if and only if g-1g0a 2 N(g-1Hg) for some a 2 Kj, if_*
*and_
only if g-1g0 2 N(g-1Hg).Kj. |_*
*_|
3. Construction of 2-dimensional actions
Again, in this section, G always denotes a finite group. To simplify the stat*
*ements
of results here and later, for any separating family F of subgroups of G, we wr*
*ite
(G; F) 2 U2 whenever there exists a 2-dimensional H-universal (G; F)-complex (a*
*nd
(G; F) =2U2 otherwise).
We are now ready to construct the 2-dimensional acyclic actions of the groups*
* G listed
in Theorem A. But we first must look more closely at the question of which subg*
*roups
of G need not appear as isotropy subgroups in a universal (G; F)-complex.
For any G and any separating family F of subgroups of G, we say that H 2 F is
a critical subgroup in F if N (F>H ) is not contractible. As seen in Propositi*
*on 1.9,
subgroups which are not critical need not occur as isotropy subgroups in (H-) u*
*niversal
(G; F)-complexes. When notation is needed, we will denote by Fc the subfamily *
*of
Bob Oliver and Yoav Segev *
* 17
critical subgroups in F. In the following lemma, we note some conditions which *
*allow
us to show that certain subgroups in F are not critical.
Lemma 3.1. Let F be any family of subgroups of G which has the property that
H H0 H00and H; H002 F imply H0 2 F. Fix a subgroup H 2 F. Then
N (F>H ) ' * if any of the following conditions hold:
(a) H is not an intersection of maximal subgroups in F.
(b) There is a subgroup bH% H, bH2 F, such that K \ bH% H for all H $ K 2 Fc.
Proof. (a) Let F0 F be the subfamily of all intersections of maximal subgroups
in F, and let ff: F ! F0 be the function which sends a subgroup to the intersec*
*tion
of the members of Fmax which contain it. Then ff induces a deformation retract*
*ion
N (F>H ) ! N (F0>H) (Lemma 0.3(a)); and N (F0>H) is contractible since it conta*
*ins the
minimal element ff(H).
(b) Set H = {K 2 F | K \ bH% H}. Then bH2 H, and K \ bH2 H for all K 2 H.
So N (H) is contractible by Lemma 0.3(b).
Now (Fc)>H = (Fc)H by assumption, and so
N (F>H ) ' N ((Fc)>H ) = N ((Fc)H ) ' N (FH ) = N (H) ' *;
*
*__
where the homotopy equivalences follow from Lemma 1.4. |*
*__|
The following lemma provides a simple sufficient condition for the existence *
*of a 2-
dimensional H-universal (G; F)-complex.
Lemma 3.2. Let F be any separating family of subgroups of G. Assume, for eve*
*ry
nonmaximal critical subgroup 1 6= H 2 F, that N(H) 2 F, and that K \ N(H) % H
for all nonmaximal critical subgroups K % H in F. Then (G; F) 2 U2.
More precisely, let M1; : :;:Mn be conjugacy class representatives for the ma*
*ximal
subgroups of F, and let H1; : :;:Hk be conjugacy class representatives for all *
*nonmaximal
critical subgroups of F. Then there is a 2-dimensional H-universal (G; F)-compl*
*ex X
which consists of one orbit of vertices of type G=Mi for each 1 i n, (-iF (H*
*j))-
orbits of 1-cells of type G=Hj for each 1 j k, and free orbits of 1- and 2-ce*
*lls. If,
furthermore, G is simple, then X can be constructed to contain exactly iF (1) f*
*ree orbits
of 2-cells (and no free orbits of 1-cells).
Proof.Fix a nonmaximal critical subgroup H = Hj 2 F. If (Fc)>H Fmax, then
N (F>H ) ' N ((Fc)>H ) is homologically 0-dimensional by Lemma 1.4. Otherwise,*
* let
H be the set of all K 2 F>H such that K \ N(H) % H, and set Hc = H \ Fc. Then
N(H) 2 H, and K \ N(H) 2 H for all K 2 H, so N (H) is contractible (Lemma
0.3(b)). Since H F and Hc Fc are terminal subposets, Lemma 0.4 now applies
to show that N (Hc) ' *. Thus, N ((Fc)>H ) consists of one contractible compon*
*ent
N (Hc), together with some isolated vertices for those maximal subgroups M 2 F>H
such that M \ N(H) = H. In particular, N (F>H ) is homologically 0-dimensional.
Hence, by Proposition 1.8, there is a finite H-universal (G; Fc)-complex X su*
*ch that
dim(XM ) = 0 for each maximal subgroup M 2 F, such that dim(XH ) = 1 for each
nonmaximal subgroup 1 6= H 2 Fc, and such that each vertex of X is fixed by a m*
*aximal
subgroup in F. But by Proposition 1.3 and Lemma 1.4, H*(Xs) ~= H*(N (F>1)), so
18 Fixed point free actions on acyclic 2-complexes
N (F>1) is homologically 1-dimensional since Xs is; and by Proposition 1.8 agai*
*n, X
can be taken to be 2-dimensional.
By the above description of X, we see that all orbits of vertices in X are of*
* type G=M
for maximal M; that all orbits of edges are of type G=Hifor 1 i k or (possibl*
*y) free
(of type G=1); and that all orbits of 2-cells are free. Hence the numbers of or*
*bits of cells
of type G=Mi or G=Hj follows from the formula in Lemma 2.3. (Note that iF (M) =*
* 1
whenever M is maximal.) Also, by Proposition 1.7, H1(Xs) is stably free as a Z*
*[G]-
module, and hence is free by Proposition C.4 if G is simple. So by Proposition *
*0.2(c),
X can be constructed by attaching only free orbits of 2-cells to Xs; and the nu*
*mber_of
orbits of cells is again given by Lemma 2.3. *
* |__|
Lemma 3.2 will be applied to construct 2-dimensional actions of the simple gr*
*oups
L2(q) (= P SL2(q)) for certain q, and of the Suzuki groups. We first list some*
* of the
properties of subgroups of the L2(q) which will be needed here, and also later *
*in Section
6.
Proposition 3.3. Fix q = pk 4, where_p is prime. Then the maximal solvable
subgroups H L2(q) = P SL2(q) and H P GL2(q) are as described in the following
table. (Note that L2(q) = P GL2(q) when q is a power of 2.)
_____________________________________________________________
|| H L2(q) (q odd)|| H P GL2(q) || ||
||____________________||__________________||________________||_
|| H | Nr. classes|| H | Nr. classes|| conditions ||
||_________|__________||_______|__________||________________ ||
|| Fo C(q-1)=2| 1 || Fo Cq-1| 1 || __ ||
||_________|__________||_______|__________||_______________ ||
|| Dq-1 | 1 || D2(q-1)| 1 || __ ||
||_________|__________||_______|__________||_______________ ||
|| Dq+1 | 1 || D2(q+1)| 1 || __ ||
||_________|__________||_______|__________||_______________ ||
|| A4 | 1 || 4 | 1 || q 3 (mod 8) ||
||_________|__________||_______|__________||_______________||
|| 4 | 2 || 4 | 1 || q 1 (mod 8) ||
||_________|__________||_______|__________||_______________||_
__
Here, in all cases (when q is odd), H = NPGL2(q)(H). Furthermore, each nonsol*
*vable
subgroup of L2(q) is conjugate in P GL2(q) to one of the groups L2(q0) for q0 =*
* pk0and
k0|k; or to P GL2(q0) for q0 = pk0 and 2k0|k; or (if q is odd and q 1 (mod 5))*
* is
isomorphic to A5.
Proof.See [Sz2, x3.6]. The subgroups of L2(q) are described in [Sz2, Theorems 3*
*.6.25-
26], and in [H1 , 8.27]. The uniqueness up to conjugacy of the dihedral groups *
*follows
from [Sz2, 3.6.23]; and the uniqueness of the Fqo Cq-1 or Fqo C(q-1)=2follows s*
*ince they
are normalizers of Sylow p-subgroups. The maximal subgroups A4 or 4 are normali*
*zers
of elementary abelian subgroups (C2)2 L2(q), of which there is one or two conj*
*ugacy
classes depending on q (mod 8) (see also [H1 , 8.16]). The fact that any subgr*
*oup
isomorphic to L2(q0) or P GL2(q0) is conjugate (in P GL2(q)) to the standard on*
*e follows
from [Sz2, 3.6.20 and Ex. 3.6.1+3].
Note in particular that B (~=Fqo Cq-1or ~=Fqo C(q-1)=2) is represented by the*
* group of
upper triangular matrices, and that D2(q-1)is the subgroup of monomial matrices*
*. The
other dihedral group D2(q+1)or Dq+1is the subgroup of GL(Fq2) (here Fq2is viewe*
*d as a
2-dimensional vector space over Fq) of all transformations of determinant one g*
*enerated
Bob Oliver and Yoav Segev *
* 19
by multiplying by an element of Fq2or by applying the Frobenius automorphism (x*
* 7!
xq).
Finally, the results about maximal subgroups of P GL2(q) follow from the info*
*rmation_
about subgroups of L2(q2) P GL2(q). |_*
*_|
We first construct actions of the groups L2(2k).
Example 3.4. Set G = L2(q), where q = 2k and k 2. Then there is a a 2-dimensi*
*onal
acyclic fixed point free G-complex X, all of whose isotropy subgroups are solva*
*ble. More
precisely, X can be constructed to have three orbits of vertices with isotropy *
*subgroups
isomorphic to Fqo Cq-1, D2(q-1), and D2(q+1); three orbits of edges with isotro*
*py sub-
groups isomorphic to Cq-1, C2, and C2; and one free orbit of 2-cells.
Proof. Let SLV be the separating family of solvable subgroups of G, and let SL*
*V c
SLV be the subfamily of all critical subgroups in SLV . By Proposition 3.3, th*
*e maximal
solvable subgroups of G are the groups B = Fqo Cq-1, D2(q-1), and D2(q+1), wher*
*e each
occurs with exactly one conjugacy class.
The Borel subgroups of G are those conjugate to B; or equivalently those subg*
*roups
of G which fix a line (a 1-dimensional subspace of (Fq)2). Every subgroup of G*
* of
even order is contained in at most one Borel subgroup, since the subgroup of el*
*ements
fixing any two distinct lines is cyclic of order q- 1. Also, any subgroup cont*
*ained in
both a Borel subgroup and a dihedral subgroup must have order 2. Thus, C2 is t*
*he
only subgroup of even order contained in more than one maximal subgroup in SLV .
Any nontrivial odd order subgroup is contained in a unique maximal dihedral sub*
*group
(its normalizer); and a subgroup Cr for 1 6= r|(q- 1) is contained in exactly t*
*wo Borel
subgroups corresponding to the two lines (eigenspaces) it leaves invariant. Thu*
*s, since
each critical subgroup must be an intersection of maximal subgroups in SLV (Le*
*mma
3.1), the only possible critical subgroups are the maximal subgroups, together *
*with Cq-1,
C2, and 1 (one conjugacy class each).
Computations using Lemma 2.6 (and Corollary 2.5 to determine iSLV(1)) now yie*
*ld
the following table.
_____________________________________
|| H 2 SLV c| K \ N(H) = H | i(H) ||
||____________|______________|______||_
|| B = Fqo Cq-1| _ | 1 ||
||____________|______________|_____||_
|| D2(q-1) | _ | 1 ||
||____________|______________|_____||_
|| D2(q+1) | _ | 1 ||
||____________|______________|_____||__
|| Cq-1 | B | -1 ||
||____________|______________|______||
|| C2 | D2(q 1) | -2 ||
||____________|______________|______||_
|| 1 | _ | 1 ||
||____________|______________|_____||__
Table 1
20 Fixed point free actions on acyclic 2-complexes
From this, it is clear that the hypotheses of Lemma 3.2 are satisfied, and henc*
*e that
(L2(q); SLV ) 2 U2. More precisely, the lemma and table show that there is an *
*H-
universal (G; SLV c)-complex, with three orbits G=B, G=D2(q-1), and G=D2(q+1)of*
* ver-
tices; with three orbits G=C2, G=C2, and G=Cq-1 of 1-cells; and with one free o*
*rbit_of
2-cells. |_*
*_|
Before continuing with the construction of the actions of other groups, we wa*
*nt to
discuss the classical example of an A5-action, and its relationship with the co*
*nstruction
(when G = L2(4) ~= A5) in Example 3.4. We first establish our notation. We wr*
*ite
SO(3) = SO(3; R), and write S3 = SL1(H) ~=SU(2; C) for the group of unit quater*
*nions
(elements of norm one in the quaternion algebra H over R). There is a homomorph*
*ism
S3 ! SO(3), surjective with kernel {1}, which is defined by sending a 2 S3 H to
the matrix of the conjugation map (x 7! axa-1) on the subspace * H. Th*
*us,
we regard S3 as a two fold cover of SO(3).
We now identify A5 ~=L2(5) as the icosahedral subgroup of SO(3), and let A*5~=
SL2(5) (the binary icosahedral group) denote its inverse image in S3. Consider*
* the
action of A5 via left multiplication on the space 3 = SO(3)=A5 ~=S3=A*5of left *
*cosets.
This space is the Poincare sphere, a 3-manifold which has the homology of the 3*
*-sphere,
and whose fundamental group is isomorphic to the perfect group A*5. Then A5 act*
*s with
fixed point set (SO(3)=A5)A5 = N(A5)=A5 = pt. Upon removing an open invariant b*
*all
around the fixed point, we obtain a compact acyclic 3-manifold M (with boundary*
*) upon
which A5 acts without fixed points. This was the starting point for the constru*
*ction by
Floyd and Richardson [FR ] of an action of A5 on a disk without fixed points (s*
*ee also
[Br, xI.8] for more details). Since @M 6= ;, M can now be collapsed to a 2-dime*
*nsional
subcomplex X ' M, upon which A5 still acts without fixed point.
This last step can be made more explicit. Let P denote the regular polytope w*
*ith 120
dodecahedral faces, and let be its symmetry group. Clearly, SO(4) ~=S3x C2S3,
and contains A5 (the group of symmetries leaving one face invariant) with index
120. This implies that ~=A*5xC2A*5; and hence that contains a binary icosahed*
*ral
subgroup A*5which permutes freely the faces of P . So 3 ~=S3=A*5can be identif*
*ied
with the space D=~ , obtained by identifying opposite faces of the solid dodeca*
*hedron
D in an appropriate way. This is in fact Poincare's original construction of th*
*e Poincare
sphere. For more details on the identification, and another way of showing that*
* these
two constructions are equivalent, we refer to [KS , pp. 124-128].
Under this identification of 3 with D=~ , the A5 action on 3 is induced by th*
*e usual
action on the dodecahedron. The fixed point is thus the center of D; and the op*
*eration
of removing the fixed point and collapsing the remaining space to a 2-dimension*
*al sub-
complex corresponds to removing the center of D and then collapsing to its boun*
*dary.
The result is an explicit 2-dimensional complex X = @D=~ with fixed point free *
*action
of A5, which has 6 pentagonal 2-cells, 10 edges, and 5 vertices.
Here's another, quicker way to construct this last complex. Let X0 be the 1-s*
*keleton
of the 4-simplex, with the obvious action of A5 permuting the five vertices. An*
*y 5-cycle
in A5 (in the vertices of X0) tells us how to attach a pentagon to X0; and two *
*such
pentagons will be in the same orbit of A5 if and only if the corresponding 5-cy*
*cles are
conjugate. So by attaching to X0 six pentagons corresponding to one conjugacy c*
*lass of
5-cycles in A5, we obtain a 2-complex X with A5-action. One can check directly *
*that
Bob Oliver and Yoav Segev *
* 21
X is acyclic (and with a bit more work show that ss1(X) ~=A*5); but one also se*
*es easily
that it is identical with the previous construction based on the dodecahedron.
If we now subdivide each pentagon in (either of) these spaces, as a union of *
*ten
2-simplices (by adding extra vertices at the midpoints of edges and centers of *
*faces),
we have constructed an A5-complex of the type constructed in Example 3.4 _ exce*
*pt
that the 2-cells have been attached explicitly. This is also identical to the A*
*5-simplicial
complex constructed in [S1, x3] and in [AS , x9]. We also note here that for k *
* 3, the
L2(2k)-complexes constructed in Example 3.4 have the same 1-skeleton as the com*
*plexes
constructed in [AS , x9] (which were not acyclic); they differ only in the way *
*the 2-cells
are attached.
We now consider G = L2(q), when q 3 (mod 8) is an odd prime power.
Example 3.5. Assume that G = L2(q), where q = pk 5 and q 3 (mod 8). Then
there is a a 2-dimensional acyclic fixed point free G-complex X, all of whose i*
*sotropy
subgroups are solvable. More precisely, X can be constructed to have four orbi*
*ts of
vertices with isotropy subgroups isomorphic to Fqo C(q-1)=2, Dq-1, Dq+1, and A4*
*; four
orbits of edges with isotropy subgroups isomorphic to C(q-1)=2, C22, C3, and C2*
*; and one
free orbit of 2-cells.
Proof. Since L2(5) ~=L2(4) has already been dealt with in Example 3.4, we assum*
*e for
simplicity that q > 5. Let SLV be the separating family of solvable subgroups*
* of G,
and let SLV c SLV be the subfamily of all critical subgroups in SLV . By Propo*
*sition
3.3, the maximal solvable subgroups of G are the groups
Dq-1; Dq+1; A4; and B = Fqo C(q-1)=2;
where each occurs with exactly one conjugacy class.
Any subgroup H 2 SLV of order a multiple of p is contained in a unique subgr*
*oup
conjugate to B (it fixes a unique line in (Fq)2); and is contained in one of th*
*e other
maximal subgroups only if p = 3 and H ~=C3. If 1 6= H 2 SLV has order prime to*
* p,
is not maximal, and is not isomorphic to C2, then either it is cyclic of order *
*dividing
(q - 1)=2 and contained in one dihedral group and two Borel subgroups (correspo*
*nding
to the two lines in (Fq)2 fixed by H), or it is cyclic of order dividing (q + 1*
*)=2 and is
contained in a unique Dq+1 (its normalizer), or H is dihedral and contained in *
*a unique
maximal dihedral subgroup Dq1 (the normalizer of its index 2 subgroup). Since *
*each
critical subgroup must be an intersection of maximal subgroups in SLV (Lemma 3*
*.1),
we have now shown that the only possible critical subgroups are the maximal sub*
*groups,
together with one conjugacy class each of subgroups C(q-1)=2, C3, C22, C2, and *
*1.
In the following table, D+ denotes the maximal dihedral subgroup of order q *
*1 0
(mod 4), and D- the other (conjugacy class of) maximal dihedral subgroup (note *
*that
D+ = N(C2)). Recall that we are assuming that q > 5 (otherwise Dq-1= C22).
22 Fixed point free actions on acyclic 2-complexes
_______________________________________
|| H 2 SLV c | K \ N(H) = H | i(H) ||
||_______________|______________|______||
|| B = Fqo C(q-1)=2| _ | 1 ||
||_______________|______________|_____||
|| Dq-1 | _ | 1 ||
||_______________|______________|_____||
|| Dq+1 | _ | 1 ||
||_______________|______________|_____||
|| A4 | _ | 1 ||
||_______________|______________|_____||_
|| C(q-1)=2 | B | -1 ||
||_______________|______________|_____ ||
|| C2 | D+ | -1 ||
||________2______|______________|_____ ||
|| C3 | A4 | -1 ||
||_______________|______________|_____ ||
|| C2 | D- | -1 ||
||_______________|______________|______||
|| 1 | _ | 1 ||
||_______________|______________|_____||_
Table 2
As before, the computations of iSLV(H) for nonmaximal 1 6= H G all follow from
Lemma 2.6, and the computation of iSLV(1) then follows from Corollary 2.5.
Lemma 3.2 now applies to show that (L2(q); SLV ) 2 U2. More precisely, toget*
*her
with Table 2, it shows that a 2-dimensional H-universal (L2(q); SLV )-complex X*
* can
be constructed with four orbits of vertices of types G=B, G=Dq-1, G=Dq+1, and G*
*=A4;
four orbits of 1-cells of types G=C22, G=C(q-1)=2, G=C3, and G=C2; and one free*
* orbit
of 2-cells. Note that G=C(q-1)=2xD1 always connects the orbits G=B and G=Dq-1, *
*and
G=C2x D1 always connects the orbits G=Dq-1and G=Dq+1. The orbit of cells G=C22x*
*D1
connects G=A4 to G=Dq-1 or G=Dq+1, depending on q modulo 8. And the orbit of ce*
*lls
G=C3x D1 connects G=A4 to one of G=B (if q = 3k), or to G=Dq1 (whichever has or*
*der_
a multiple of 3). |*
*__|
The third family of groups with 2-dimensional actions consists of the Suzuki *
*groups
Sz(q), for all q = 22k+1 8. In order to specify more precisely subgroups of S*
*z(q),
we regard it as a subgroup of GL4(Fq) as described in [HB3 , xXI.3]. The follo*
*wing
properties of Sz(q) and its subgroups will be needed here, as well as in Sectio*
*n 6.
Proposition 3.6. Fix q = 22k+1, and let 2 Aut (Fq) be the automorphism x =
k+1 p__2q 2 *
x2 = x (thus (x ) = x ). For a; b 2 Fq and 2 (Fq) , define elements
0 1
1 0 0 0
B a 1 0 0C
S(a; b) = B@ b a 1 0CA;
a2+ + ab + b a1+ + b a 1
and 0 k 1 0 1
1+2 0 0 0 0 0 0 1
B 0 2k 0 0 C B 0 0 1 0C
M() = B@ -2k C ; o = B C :
0 0 0 A @ 0 1 0 0A
k
0 0 0 -1-2 1 0 0 0
Set S(q; ) = *~~, T = ~= Cq-1, and
B = M(q; ) = S(q; )o T and N = ~=D2(q-1):
Bob Oliver and Yoav Segev *
* 23
Then Sz(q) ~=, and under this identification the following hold:
(a) S(q; ) is a Sylow 2-subgroup of Sz(q).
(b) There are four conjugacy classes of maximal subgroups in Sz(q) which are *
*solvable:
(B), (N), (M+), and (M-), where
M+ ~=Cq+p__2q+1oC4 and M- ~=Cq-p__2q+1oC4:
These are the only maximal solvable subgroups in Sz(q).
(c) Each nonsolvable subgroup of Sz(q) is conjugate to Sz(q0), for some q0 = *
*22m+1
where (2m + 1)|(2k + 1).
(d) Sz(q) is contained in the 4-dimensional symplectic group over Fq:
Sz(q) Sp4(q) def={g 2 GL4(q) | gogt = o};
where gt is the transpose of g and o is as above.
(e) All of the subgroups B; N; T; S(q; ); Sz(q) are invariant under the autom*
*orphisms
of GL4(q) induced by automorphisms of the field Fq.
p __ p __
(f) | Sz(q)| = q2(q- 1)(q2+ 1) = q2.(q- 1).(q+ 2q+1).(q- 2q+1), where the*
* four fac-
tors in the second expression are pairwise relatively prime.
Proof.See [HB3 , xXI.3]. Note in particular the relations
S(a; b).S(c; d) = S(a + c; b + d + a c) and M()-1S(a; b)M() = S(a; 1+ b):
The list of maximal subgroups of Sz(q) (points (b) and (c)) is shown in [Sz1, T*
*heorem
9].
Note that if q0 = 22m+1, where (2m + 1)|(2k + 1), then Sz(q) \ GL4(q0) = Sz(q*
*0) (and
similarly for the other subgroups). The inclusion Sz(q0) Sz(q) follows since 2*
*k 2m
k 2m
(mod 22m+1- 1), and hence x2 = x for all x 2 Fq0. The inclusion Sz(q) \ GL4(q*
*0)_
Sz(q0) then follows from (c). *
*|__|
We are now ready to construct actions of Sz(q) on acyclic 2-complexes.
Example 3.7. Set q = 22k+1, for any k 1. Then there is a 2-dimensional acycli*
*c fixed
point free Sz(q)-complex X, all of whose isotropy subgroups are solvable. More *
*precisely,
X can be constructed to have four orbits of vertices with isotropy subgroups is*
*omorphic
to M(q; ), D2(q-1), Cq+p__2q+1oC4, and Cq-p__2q+1oC4; four orbits of edges with*
* isotropy
subgroups isomorphic to Cq-1, C4, C4, and C2; and one free orbit of 2-cells.
Proof. Set G = Sz(q). By Proposition 3.6, G contains the following maximal solv*
*able
subgroups:
M(q; ); D2(q-1); Cq+p__2q+1oC4; and Cq-p__2q+1oC4;
with one conjugacy class for each isomorphism type. If 1 6= H 2 SLV and (|H|; *
*q2+1) 6=
1, then H is contained in a unique maximal subgroup Cq2p__2q+1oC4: the normaliz*
*er
of its unique maximal odd order subgroup. Likewise, if H is dihedral of order d*
*ividing
2(q-1)f(andi|H| 6= 2), then H is contained in a unique maximal subgroup D2(q-1)*
*; while
if |H|fi(q - 1) then H is contained in the same maximal subgroups as its centra*
*lizer of
order q - 1. Any subgroup of even order which is not dihedral is contained in a*
*t most
one maximal subgroup, conjugate to M(q; ). (The centralizer of any involution i*
*n G is
24 Fixed point free actions on acyclic 2-complexes
a 2-group by [Sz1, Proposition 1], and each involution in the Sylow subgroup S(*
*q; ) is
central. So an involution cannot be in two Sylow subgroups.) Thus, any subgroup*
* which
is an intersection of two or more maximal subgroups is isomorphic to one of the*
* groups
Cq-1, C4, C2, or 1; and these are the only possible critical subgroups by Lemma*
* 3.1(a).
There is just one conjugacy class each of subgroups Cq-1 or C2 (note, for examp*
*le, that
all subgroups of order 2 in S(q; ) are conjugate in M(q; )). By [Sz1, Propositi*
*on 18],
G contains two conjugacy classes of elements of order 4, and it is easy to chec*
*k by direct
calculations that an element of order 4 in G is not conjugate to its inverse. H*
*ence G
contains just one conjugacy class of C4's.
Now let SLV cbe the subfamily of critical subgroups in SLV . Consider the fol*
*lowing
table of values for iSLV(H) for H 2 SLV c:
_____________________________________
|| H 2 SLV c| K \ N(H) = H | i(H) ||
||____________|______________|______||_
|| B = M(q; )| _ | 1 ||
||____________|______________|_____||_
|| D2(q-1) | _ | 1 ||
||____________|______________|_____||_
|| C p__ o C4| _ | 1 ||
||__q+_2q+1___|______________|_____||_
|| C p__ o C4| _ | 1 ||
||__q-_2q+1___|______________|_____||__
|| Cq-1 | M(q; ) | -1 ||
||____________|______________|______||
|| C4 | C p__ o C4| -2 ||
||____________|___q_2q+1_____|______||
|| C2 | D2(q-1) | -1 ||
||____________|______________|______||_
|| 1 | _ | 1 ||
||____________|______________|_____||__
Table 3
When H ~=Cq-1, C4, or C2, then iSLV(H) is computed using Lemma 2.6. (Note that
C2 can never be self-normalizing in any group of order a multiple of 4.) The va*
*lue of
iSLV(1) then follows from Corollary 2.5.
Lemma 3.2 now applies to show that (Sz(q); SLV ) 2 U2. More precisely, there *
*is a
2-dimensional H-universal (Sz(q); SLV )-complex which has four orbits of vertic*
*es and
four orbits of edges (with isotropy subgroups as given in Table 3), and one fre*
*e_orbit of
2-cells. |_*
*_|
4. Reduction to simple groups
Throughout this section, G will be a finite group. Recall that a G-complex X *
*is called
essential if there is no normal subgroup 1 6= N C G, with the property that the*
* inclusion
XN X is a G-Z-equivalence; i.e., such that XNH ! XH is a homology equivalen*
*ce
for all H G. We would like to be able to show directly that all groups which h*
*ave
essential fixed point free actions on acyclic 2-complexes are simple. Instead,*
* in this
section, we prove a slightly weaker result (Proposition 4.4), where we show tha*
*t any
group with such an action is an extension of a simple group by outer automorphi*
*sms.
Bob Oliver and Yoav Segev *
* 25
The proof of this uses the result in [S1] that the fixed point set of any gro*
*up acting
on a 2-dimensional acyclic complex must be acyclic or empty. Since the proof in*
* [S1]
requires the odd order theorem, we give here a different one, which is more ele*
*mentary.
Theorem 4.1 [S1, Theorem 3.4].Let X be any 2-dimensional acyclic G-complex (not
necessarily finite). Then XG is acyclic or empty, and is acyclic if G is solvab*
*le.
Proof.The first half of the following proof is essentially the same as that in *
*[S1], but is
included here for the sake of completeness.
If G is a p-group for some prime p, then XG is Z=p-acyclic by Smith theory (c*
*f. [Br,
Theorem III.7.12]), and homologically 1-dimensional by Lemma 1.6. It follows th*
*at XG
is Z-acyclic in this case.
Now assume that G is a minimal group for which there is a counterexample. The*
*n G
must be simple and nonabelian _ since if N C G were a proper normal subgroup, t*
*hen
XN would be acyclic, and hence XG = (XN )G=N would be acyclic or empty (acyclic
if G isTsolvable) by the minimality of G. Also, XH is acyclic for all H $ G, *
*and
XG = H$G XH is homologically 0-dimensional by Lemma 1.6 again. In other word*
*s,
each connected component of XG is acyclic, and it remains to show that there is*
* at most
one component.
Assume otherwise: let k 2 be the number of connected components of XG . Let F
be the (separating) family of proper subgroups H $ G. Very roughly, we will sho*
*w that
X "looks like" the join of an H-universal (G; F)-complex Y with a set of k poin*
*ts. But
for X to be 2-dimensional, Y would have to be 1-dimensional, i.e., a tree; and *
*this is
impossible.
To make this precise, let F+ denote the poset which consists of F, together w*
*ith k
elements (G; i) for i = 1; : :;:k. Extend the ordering on F by setting (G; i) *
*H for all
H 2 F, and with no inclusion relations between the (G; i). Write XG = F1 q . .q*
*.Fk,
where the Fi are the connected components. We now apply Lemma 0.1, with the
covering of X given by XH = XH for H 2 F, and X(G;i)= Fi. Thus, Xffis acyclic *
*for
each ff 2 F+. So by Lemma 0.1, for each H 2 F, H*(X>H ) ~=H*(N ((F+)>H )), and
thus N ((F+)>H ) is homologically 1-dimensional (Lemma 1.6). But the poset (F+)*
*>H
consists of F>H together with the elements (G; i), and so its nerve is the uni*
*on of k
cones over N (F>H ). This complex contains the suspension of N (F>H ) as a retr*
*act (i.e.,
the case k = 2); and hence N (F>H ) is homologically 0-dimensional. Since this*
* holds
for all H 2 F, Proposition 1.8 now applies to show that there is a finite 1-dim*
*ensional
universal (G; F)-complex Y . But then Y is a tree upon which G acts without f*
*ixed __
points, and this is impossible (cf. [Se, xI.6]). *
* |__|
The following easy consequence of Theorem 4.1 turns out to be very useful. It*
*s proof
involves collapsing out certain subcomplexes of a CW complex to create new fixe*
*d points,
and get a contradiction to Theorem 4.1. In general, if X is a G-complex and A X
is a G-invariant subcomplex, then X=A is defined to be the quotient space X=~ ,*
* where
x~ y if x = y or x; y 2 A. This quotient space has an obvious structure as a G-*
*complex:
where (X=A)(n)= X(n)=~ , and where X=A has one vertex for the identification po*
*int
A=A and otherwise one cell for each cell in X not in A (see [LW , Theorem II.5.*
*11],
taking Y = pt). The homology groups of X, A, and X=A are linked by exact sequen*
*ces
26 Fixed point free actions on acyclic 2-complexes
(coming from the fact that Cn(X=A)=Cn(pt) ~=Cn(X)=Cn(A)). In particular, if A *
*is
acyclic, then H*(X=A) ~=H*(X).
Corollary 4.2. Let X be any 2-dimensional acyclic G-complex. Assume that A; B X
are G-invariant acyclic subcomplexes such that A [ B XG . Then A \ B 6= ;.
Proof.Assume otherwise: that A \ B = ;. Let Y be the G-complex obtained by
identifying the subcomplexes A and B each to a point. Then Y is still acyclic*
*, since
A and B are, and Y G consists of the two identification points. And this contr*
*adicts_
Theorem 4.1, which says that Y G must be acyclic or empty. *
* |__|
As immediate consequences of Corollary 4.2 we get:
Lemma 4.3. Let X be a 2-dimensional acyclic G-complex. Then the following hol*
*d.
(a) [AS , 4.5] If H; N G are such that H NG(K) and XH and XK are nonempty,
then XHK 6= ;.
(b) If H G is such that XH = ;, then XCG(H) 6= ;.
Proof. If XG 6= ;, then (a) and (b) are obvious. So assume XG = ;.
(a) Since H normalizes K, both XH and XK are H-invariant acyclic subcomplex*
*es
of X. So by Corollary 4.2, if XH and XK are nonempty, then XH \ XK = XHK 6= ;.
(b) It suffices to prove this when H is minimal among subgroups without fixed*
* points.
Fix a pair M; M0 H of distinct maximal subgroups (H is nonsolvable). Then XM
0 M M0 H M M0
and XM are nonempty, but X \ X = X = X = ;. Thus X and X
are disjoint CG(H)-invariant acyclic subcomplexes of X, and so CG(H) must have *
*fixed_
points by Corollary 4.2. |*
*__|
As a first consequence of Lemma 4.3, we can now prove:
Theorem B. Let G be any finite group, and let X be any 2-dimensional acyclic *
*G-
complex.0Let N be the subgroup generated by all normal subgroups N0 C G such th*
*at
XN 6= ;. Then XN is acyclic; X is essential if and only if N = 1; and if N 6=*
* 1 then
the action of G=N on XN is essential.
Proof.If XN1 6= ; and XN2 6= ;, where N1; N2 C G, then X6= ; by Lemma
4.3(a). Thus XN is nonempty, and is acyclic by Theorem 4.1. The action of G=N *
*on
XN is always essential, since any nontrivial normal subgroup of G=N has empty *
*fixed
point set.
Now assume that N 6= 1. For all H G, XH and XNH are acyclic or empty by
Theorem 4.1; and XNH is nonempty if XH is by Lemma 4.3(a). So the inclusion XN*
*H ! __
XH is always an equivalence of integral homology, and hence X is not essential.*
* |__|
We are now ready to prove:
Proposition 4.4. If G is a nontrivial finite group for which there exists an es*
*sential
2-dimensional acyclic G-complex X, then G is almost simple. More precisely, the*
*re is
a normal subgroup L C G, such that L is simple, such that XL = ;, and such that
CG(L) = 1 (i.e., G Aut(L)).
Bob Oliver and Yoav Segev *
* 27
Proof.By Theorem B, XN = ; for all proper normal subgroups 1 6= N C G. In
particular, XG = ;.
Fix a minimal normal subgroup 1 6= L C G. Then L is nonsolvable, since XL 6= *
*;.
Hence L is a direct product of isomorphic nonabelian simple groups (cf. [Go , T*
*heorem
2.1.5]).
Assume first that L is not simple. By Lemma 4.3(b), XH 6= ; for some simple
factor H C L; and L = since it is a minimal normal subgroup. Si*
*nce
-1 H L
XgHg = g(X ) 6= ; for all g, X 6= ; by Lemma 4.3(a) (applied to the action *
*of L on
X). And this is a contradiction.
Thus, L is simple. Set H = CG(L). Then H C G, and XH 6= ; by Lemma 4.3(b); *
* __
and so H = 1 (again since the G-action on X is essential). *
* |__|
Using Proposition 4.4, when determining which finite groups have essential fi*
*xed point
free actions on 2-dimensional acyclic complexes, it suffices first to determine*
* which simple
groups have such actions, and then consider automorphism groups only of those s*
*imple
groups which do have them.
5. Some conditions for nonexistence of 2-dimensional actions
Again, throughout this section, G is a finite group. We recall two definition*
*s intro-
duced in Section 3. If F is a separating family for G, then Fc denotes the sub*
*family
of critical subgroups for F: the set of all H 2 F such that N (F>H ) 6' *. An*
*d U2
denotes the class of pairs (G; F) (where F is a separating family for G) for wh*
*ich there
exists a 2-dimensional H-universal (G; F)-complex. We have already constructed *
*some
examples of pairs (G; F) which do lie in U2, and next want to show that they ar*
*e the
only ones. In this section, we develop some general techniques for doing this.
For any G-complex X, and any n > 1, it will be convenient to write X[n]to den*
*ote
the union of fixed pointfsetsiof subgroups of order a multiple of n; or equival*
*ently the set
of all x 2 X for which nfi|Gx|. Also, for any family F of subgroups of G, we wr*
*ite F[n]to
denote the subfamily of those subgroups in F of order a multiple of n. We will *
*see that
if (G; F) 2 U2, then not only is N (F[n]) homologically 1-dimensional for all n*
*, but its
orbit space N (F[n])=G is homologically 0-dimensional (i.e., its connected comp*
*onents
are acyclic).
In Section 5a, conditions are established which allow us to directly detect e*
*lements
in H2(N (F[n])), for appropriate n, via Euler characteristic arguments. The pro*
*perties
of N (F[n])=G are shown in Section 5b, and then another set of criteria are fou*
*nd which
detect elements in H1(N (F[n])=G). Afterwards, conditions on G and F are set u*
*p in
Section 5c which imply that for any 2-dimensional H-universal (G; F)-complex X,*
* the
singular set Xs is itself acyclic (and hence H-universal); and then Section 5d *
*deals with
the problem of showing that this is impossible.
28 Fixed point free actions on acyclic 2-complexes
5a. Detecting_2-cycles_in_nerves_of_posets_of_subgroups_
Our main tool here for directly detecting elements in the second homology of *
*nerves of
posets of subgroups will be certain "coset complexes". We adopt the following n*
*otation:
Definition 5.1.Fix any group G, and any triple K1; K2; K3 of subgroups of G. De*
*fine
K1; K2; K3 = K1; K2; K3 G
to be the G-simplicial complex with vertex set (G=K1) q (G=K2) q (G=K3) (where G
acts by left translation), and with a 1- or 2-simplex for every pair or triple *
*of cosets with
nonempty intersection.
Thus, each edge in K1; K2; K3 has the form [aKi; aKj] for some a 2 G and *
*some
1 i < j 3, and each 2-simplex has the form [aK1; aK2; aK3] for some a 2 G. In
many cases, one can show that H2( K1; K2; K3 ) 6= 0 via an easy counting argu*
*ment:
Lemma 5.2. Fix any group G, and any sequence K1; K2; K3 of subgroups of G. S*
*et
Kij= Ki\Kj, K = K1\K2\K3, and G0= . Assume that
___1___ 1 1
+ _______+ _______ 1; (1)
[K12:K] [K13:K] [K23:K]
or (more generally) that
X 1 X3 1 1
_______< 1 + ______ - ______: (2)
0:K]
i; and so t*
*here are
exactly [G:G0] connected components.
By definition, X has three orbits of vertices of type G=Ki, three orbits of e*
*dges of
type G=Kij, and one orbit of 2-simplices of type G=K. Hence
X X3
O(X) = [G:K] - [G:Kij] + [G:Ki]
i [G:G ] = rk(H0(X));
i.
iK2, and *
*let Q0be
the set of those H 2 Q such that the coefficient in z of {K2; H; K0} is nonzero*
*. By
construction, every element of Q0 is G-conjugate (in fact, N02-conjugate) to K1*
*; and
by condition (c), every element of Q in the same N (Q)-connected component as K*
*1 is
contained in K1. Lemma 0.5 now implies that 0 6= [z] 2 H2(N (H)); and so (G; F)*
*_=2U2_
by Proposition 1.9. |*
*__|
Two n-tuples of subgroups (H1; : :;:Hn) and (H01; : :;:H0n) in G will be call*
*ed G-
conjugate if there is some g 2 G such that H0i= gHig-1 for all i. The normaliz*
*er
NG(H1; : :;:Hn) of such an n-tuple is just the intersection of the normalizers *
*NG(Hi).
The next proposition is a somewhat more complicated application of Lemma 5.2.
Proposition 5.4. Fix a separating family F of G. Let K1; K2; K3 2 F be three
subgroups such that neither K2 nor K3 is conjugate to K1. Set Kij= Ki\ Kj and
K = K1 \ K2 \ K3. Let F0 F denote the subfamily consisting of Fc, together with
all subgroups conjugate to any of the Ki, Kij, or K. Assume the following condi*
*tions
hold:
1 1 1
(a1) _______ + _______+ _______ 1; or more generally
[K12:K] [K13:K] [K23:K]
1 1 1 1 1 1 1
(a2) _______+ _______+ _______< 1 + _______+ _______+ _______- ______, where
[K12:K] [K13:K] [K23:K] [K1:K] [K2:K] [K3:K] [G0:K]
G0= .
(b) K1 is maximal in F.
(c) There is no H 2 F0 such that K $ H $ K12or K12$ H $ K1.
(d) NG(K1; K12; K) = K.
(e) The triples (K1; K12; K) and (K1; K13; K) are not G-conjugate.
Then H2(F(K) ) 6= 0; and so (G; F) =2U2.
Proof.Consider the complex X = K1; K2; K3 of Definition 5.1, and let X* den*
*ote
its barycentric subdivision. To distinguish between simplices of X* and of N (*
*F), we
put parentheses (-) around the former and curly brackets {-} around the latter.*
* The
vertices in X* will be denoted (gKi) (the vertices in X), (gKij) (the midpoint *
*of the
edge (gKi; gKj)), and (gK) (the barycenter of the 2-simplex (gK1; gK2; gK3)).
30 Fixed point free actions on acyclic 2-complexes
We have H2(X) 6= 0 by (a1) or (a2), together with Lemma 5.2. Fix a 2-cycle z*
* in
X such that 0 6= [z] 2 H2(X). We can assume that the coefficient in z of the si*
*mplex
(K1; K2; K3) is nonzero (otherwise compose with the action of some appropriate *
*element
of G). Let z* be the corresponding 2-cycle in the barycentric subdivision X* of*
* X.
Let f : X* ! N ((F0)(K) ) be the G-equivariant simplicial map which sends eac*
*h ver-
tex in X* to its isotropy subgroup. Thus f(gKi) = {gKig-1}, f(gKij) = {gKijg-1}*
*, and
f(gK) = {gKg-1}. By conditions (d) and (e), and since neither K2 nor K3 is conj*
*ugate
to K1, the only simplex in X* which is sent to {K1; K12; K} is (K1; K12; K), an*
*d this sim-
plex has nonzero coefficient in the 2-cycle z*. Hence {K1; K12; K} has nonzero *
*coefficient
in the 2-cycle f(z*). By (b) and (c), {K1; K12; K} is maximal in N ((F0)(K) ) (*
*not in the
boundary of any 3-simplex), and hence [f(z*)] 6= 0 in H2(N ((F0)(K) )) = H2(N (*
*F(K)_))
(Lemma 1.4). And thus (G; F) =2U2 by Proposition 1.9(a ) d). *
*|__|
5b. Detecting_nonzero_elements_in_H1(X[n]=G)_
Recall that for any n and F, F[n] F denotes the subfamily of all subgroups in*
* F of
order a multiple of n. We first show, for (G; F) 2 U2, that the connected compo*
*nents of
the orbit space of N (F[n]) are all acyclic, and then set up some conditions wh*
*ich detect
elements in their first homology groups. The starting point for all of this is *
*the following
result, a consequence of Smith theory.
Proposition 5.5. If X is any finite dimensional acyclic G-complex, then X=G is *
*also
acyclic. If f : X ! Y is any equivariant map between finite dimensional G-compl*
*exes
which induces an isomophism H*(X; Z) ~=H*(Y ; Z), then f=G induces an isomorphi*
*sm
H*(X=G; Z) ~=H*(Y=G; Z).
Proof.The first statement is shown, for example, in [Br, Theorem III.7.12]. The*
* second
statement follows from the first, since f induces an isomorphism in integral ho*
*mology
if and only if its mapping cone Cf is acyclic, and similarly for f=G. (Note tha*
*t_Cf=G ~=
(Cf)=G.) |__|
The following result is similar to one used in [O3 ], but formulated here for*
* acyclic
rather than Fp-acyclic spaces.
Proposition 5.6. Fix a prime p, and let X be a finite dimensional acyclic G-com*
*plex
with the property that XP is acyclic for all p-subgroups P G. Then for any
(nonempty) family P of p-subgroups of G, XP =G is acyclic.
Proof.We assume that any p-group which contains an element of P also lies in P *
*(if
not just add these groups to the family). For the purposes of this proof, we de*
*fine, for
any p-subgroup P G,
[ [
XPs= XQ and X(P)s= G.XPs= X(Q):
Q%P Q%P
Q a p-subgr. Q a p-subgr.
We first claim that for any P 2 P, the inclusion of XP into X(P)induces an isom*
*orphism
of homology groups
~= (P) (P)
H*(XP =N(P ); XPs=N(P )) -----! H*(X =G; Xs =G): (1)
Bob Oliver and Yoav Segev *
* 31
In fact, the inclusion induces an isomorphism
~= (P) (P)
C*() : C*(XP =N(P ); XPs=N(P )) -----! C*(X =G; Xs =G)
between the cellular chain complexes of these pairs. The surjectivity of C*() i*
*s clear,
since any open cell oe X(P)rX(P)slies in the G-orbit of some oe XP rXPs. To s*
*ee its
injectivity, fix open cells oe; a(oe) XP rXPsin the same G-orbit (a 2 G). Then*
* P is a
Sylow p-subgroup of the isotropy subgroups Goeand Ga(oe)= aGoea-1, so P and a-1*
*P a
are both Sylow p-subgroups of Goe, and hence a-1P a = gP g-1 for some g 2 Goe. *
* It
follows that ag 2 NG(P ), and thus that oe and a(oe) = ag(oe) lie in the same N*
*(P )-orbit.
This proves the injectivity of C*(); and finishes the proof that (1) is an isom*
*orphism.
Now set
ff = max {a 0 | pa|[G:P ]; some P 2 P }:
The proposition will be proven by induction on ff. If ff = 0, then for any Syl*
*ow p-
subgroup P of G, X(P)= XP and X(P)s= XPs= ;; and so
H*(XP =G) ~=H*(X(P)=G) ~=H*(XP =N(P ))
by (1). Also, XP =N(P ) is acyclic by Proposition 5.5 (since XP is acyclic by a*
*ssumption);
and thus XP =G is acyclic.
Now assume that ff > 0. Let P0 P be the subfamily of all P such that pff|-|[*
*G:P ].
Then XP0=G is acyclic by the induction hypothesis, and it remains to show that
H*(XP =G; XP0=G) = 0. Let P1; : :;:Pk be conjugacy class representatives for th*
*e sub-
groups in Pr P0, and set Pi= P0 [ (Pi). Then by excision,
Mk Mk
H*(XP =G; XP0=G) ~= H*(XPi=G; XP0=G) ~= H*(X(Pi)=G; X(Pi)s=G):
i=1 i=1
It thus remains to show that H*(X(P)=G; X(P)s=G) = 0 for each P = Pi. By (1), t*
*his
means showing that H*(XP =N(P ); XPs=N(P )) = 0. But XPs=N(P ) is acyclic by t*
*he
induction hypothesis again, and XP =N(P ) is acyclic by Proposition 5.5 (since *
*XP_is
acyclic by assumption). |*
*__|
Proposition 5.6 will be applied in particular to get information about the sp*
*aces
X[n]=G and N (F[n])=G.
Corollary 5.7. Let F be any separating family for G, and let F0 F be a subfami*
*ly
which contains Fc. Let X be a finite dimensional H-universal (G; F)-complex. Th*
*en for
any subfamily H of F,
H*(XH =G) ~=H*(FH =G) ~=H*((F0)H =G):
In particular, H*(Xs=G) ~= H*(N (F>1)=G) ~= H*(N ((F0)>1)=G); and H*(X[n]=G) ~=
H*(N (F[n])=G) ~=H*(N ((F0)[n])=G) for all n > 1. And for any prime power q,
N (F[q])=G is acyclic.
Proof.By Proposition 1.3, for any H F, there is a G-map f : X ! N (F) which
restricts to a homology equivalence fH : XH ! N (FH ). By Lemma 1.4, the incl*
*usion
N ((F0)H ) N (FH ) is a homotopy equivalence. So by Proposition 5.5, these m*
*aps in-
duce homology equivalences in the orbit spaces. The isomorphisms involving H*(X*
*s=G)
32 Fixed point free actions on acyclic 2-complexes
and H*(X[n]=G) now follow from the case where H = F>1 or H = F[n]. In particula*
*r,
*
* __
X[q]=G is acyclic by Proposition 5.6 (and since X exists by Proposition 1.8). *
* |__|
The importance of the families F[n]comes from the following lemma. Note that *
*for
a family F of subgroups of G and a group A of automorphisms of G, the orbit spa*
*ce
N (F)=A need not be a simplicial complex: there could, for example, be two edg*
*es
of N (F) not in the same A-orbit, but whose endpoints are identified pairwise. *
* But
N (F)=A does always have the structure of a CW complex in a natural way (cf. Le*
*mma
A.5).
Lemma 5.8. Let F be a separating family of subgroups of G such that (G; F) 2 *
*U2,
and let F0 F be any subfamily which contains Fc. Then_for all n > 1, N ((F0)[n*
*])=G
is homologically 0-dimensional. More generally,_if G Aut(G) is any subgroup_wh*
*ich
contains Inn(G), and such that F and F0 are G -invariant, then N ((F0)[n])=G i*
*s homo-
logically 0-dimensional for all n > 1.
Proof.Let X be any 2-dimensional H-universal (G; F0)-complex (X exists by Propo-
sition 1.8). Then X=G is Z-acyclic by Proposition 5.5. If n = pk where p is p*
*rime,
then X[n]=G is acyclic by Proposition 5.6. If n is not a prime power,Twrite n =*
* q1. .q.k,
where the qi are prime powers for distinct primes. Then X[n]=G = ki=1X[qi]=G *
*is an
intersection of acyclic subspaces of X=G; and hence is homologically 0-dimensio*
*nal by
Lemma 1.6 again.
Thus, N ((F0)[n])=G is also homologically 0-dimensional by Corollary 5.7, and*
* its
connected components are_all acyclic. The last statement_now follows from Propo*
*sition
*
* __
5.5, since N ((F0)[n])=G is the orbit space of the G= Inn(G)-action on N ((F0)[*
*n])=G. |__|
We end this subsection_with an application of Lemma 5.8: one situation in whi*
*ch we
can show that N (F[n])=G is not homologically 0-dimensional, and thus that (G; *
*F) =2U2.
The argument is based on the following observation: given a 1-cycle OE in a sim*
*plicial
complex K which involves at least one "free" edge (an edge with no higher dimen*
*sional
simplices attached), then 0 6= [OE] 2 H1(K). Here, "simplicial complex" is used*
* in the
more general sense, where there can be two or more n-simplices (n 1) having th*
*e same
set of vertices.
When working with the orbit space N (F)=G, we will let [H] denote the vertex *
*cor-
responding to a conjugacy class (H) F. More generally, for any chain H0 $ H1 $
. .$.Hn of subgroups in F, [H0; H1; : :;:Hn] will denote the corresponding n-si*
*mplex
in N (F)=G.
Proposition 5.9. Let F be a separating family of subgroups of G. Assume that th*
*ere
is a maximal subgroup M 2 F, and a pair of maximal subgroups K; K0 M which are
not conjugate in M, but are conjugate in G. Then_(G; F) =2U2. More generally,*
* the
same conclusion_holds if there is a subgroup G Aut(G) containing Inn(G),_such *
*that
F is G -invariant, and such that K and K0 are in the same orbit of G , but not *
*in the
same orbit of the action of the stabilizer of M.
__
Proof.Set n = |K|. Then F[n]=G contains (at least) two edges which connect the *
*vertices
[K] and [M]._The maximality properties guarantee that the resulting loop is non*
*zero in
*
*__
H1(F[n]=G ). So (G; F) =2U2 by Lemma 5.8. |*
*__|
Bob Oliver and Yoav Segev *
* 33
5c. Acyclicity_of_N_(F>1)_
We now find conditions for showing that N (F>1) is acyclic, under the assumpt*
*ion
that (G; F) 2 U2. This can then be combined with results in Section 5d to obta*
*in
contradictions. We first note the following equivalent conditions on F.
Lemma 5.10. Fix a separating family F of subgroups of G, and assume that (G; *
*F) 2
U2. Then the following are equivalent:
(a) N (F>1)=G is connected and H1(N (F>1)=G) = 0.
(b) N (F>1) is acyclic.
(c) N (F>1)=G is acyclic.
Proof.For any 2-dimensional H-universal (G; F)-complex X, H*(Xs) ~= H*(N (F>1))
by Proposition 1.3, and H*(Xs=G) ~=H*(N (F>1)=G) by Corollary 5.7. So it suffic*
*es to
show the equivalence of the above three conditions after replacing N (F>1) by X*
*s.
Since X=G is acyclic (Proposition 5.5), Xs and Xs=G are homologically 1-dimen*
*sional
by Lemma 1.6. Thus, (a) is equivalent to (c). Also, (b) implies (c) by Proposit*
*ion 5.5
again; and it remains to show that (c) implies (b).
If Xs=G is acyclic, then in particular it has Euler characteristic one. Hence*
* by Lemma
2.2,
1 - O(Xs) = O(X) - O(Xs) = |G|. O(X=G) - O(Xs=G) = |G|(1 - 1) = 0;
and so O(Xs) = 1. Since G acts transitively on the connected components of Xs (*
*Xs=G
being connected), all components of Xs have the same Euler characteristic, and *
*so Xs
must be connected. And since Xs is also homologically 1-dimensional, this shows*
*_that_
Xs is acyclic. |*
*__|
The next proposition provides a tool for showing that condition (a) in Lemma *
*5.10
holds.
Proposition 5.11. Assume G has even order, and let F be a separating family for*
* G.
Assume, for each member M 2 Fmax of even order and each element x 2 M of odd
prime order, that either
(1a) |NM ()| is even; or
(1b) there is an element y 2 M of odd prime order such that |NG()\ NG()*
*| and
|NM ()| are both even.
Let (M1); :::; (Mk) be the conjugacy classes of odd order subgroups in Fmax. F*
*or 1
i k, let F0ibe the set of all subgroups of Miwhich are contained in members of*
* Fmax
of even order or in subgroups conjugate to Mj for j < i; and assume that
(2) the image of N ((F0i)>1) in N (F>1)=G is connected and nonempty for each *
*i.
Then N (F>1)=G is connected and H1(N (F>1)=G) = 0.
Proof.For any x 2 H G, we write NH (x) def=NH (), for short. For each i = 0*
*; :::; k,
let Fi be the family of all subgroups in F contained in even order members of F*
*max,
or in subgroups conjugate to Mj for j i; and set Xi = N ((Fi)>1)=G and X = Xk.
In particular, Fk = F, and F0 is the set of all subgroups in F which are contai*
*ned in
34 Fixed point free actions on acyclic 2-complexes
members of Fmax of even order (k = 0 if all members of Fmax have even order). *
*Set
Y = N (F[2])=G X0. By Corollary 5.7, Y is connected and H1(Y ) = 0. Then X0 is
connected, since each vertex of X0 is joined by an edge to a vertex of Y . And *
*for each
i 1, each vertex of Xi not in Xi-1 is joined to [Mi], which in turn is connect*
*ed to
Xi-1via a vertex in the nonempty set F0i. This shows that the Xiare all connect*
*ed. In
particular, X is connected, and it remains to show that H1(X) = 0.
We first set up some notation for elements of H1(X). The homology class of a *
*loop will
be denoted [H0; H1; : :;:Hn], where (H0) = (Hn), and each Hicontains or is cont*
*ained in
Hi+1. Note that by specifying subgroups rather than just conjugacy classes, we *
*eliminate
all ambiguity as to which edge between two vertices is meant (recall that there*
* can be
more than one edge connecting a pair of vertices of X). Finally, to simplify th*
*e notation,
we will sometimes replace a cyclic group Hi= by xi in this notation.
Step 1 We first show that H1(X0) maps trivially to H1(X). Whenever [H0; H1; : :*
*;:Hn]
is a path in X with endpoints in Y , we write [H0; H1; : :;:Hn]Y 2 H1(X) to den*
*ote the
homology class of the 1-cycle [H0; : :;:Hn] - OE for any path OE from [H0] to [*
*Hn] in Y .
This is well defined since Y is connected and H1(Y ) = 0.
Fix a loop in X0; we can assume that it alternates "peaks" and "valleys" (ver*
*tices
corresponding to larger or smaller subgroups); and furthermore that each peak i*
*s max-
imal in F (hence of even order) and each valley is minimal (i.e., of prime orde*
*r). The
loop thus splits into a sum of elements [M; x; M0]Y , where M and M0 are maxima*
*l of
even order, and where |x| is prime. If |x| = 2, then [M; x; M0]Y 2 Im(H1(Y )) =*
* 0; so
we can assume that x has odd prime order.
In either of cases (1a) or (1b) above, NG(x) has even order. Choose a maximal*
* sub-
group Mx 2 F[2]which contains the extension of by a Sylow 2-subgroup of NG(*
*x)=
(this extension is solvable and hence in F[2]). Then [M; x; M0]Y = [M; x; Mx]*
*Y +
[Mx; x; M0]Y , and we are reduced to showing that [M; x; Mx]Y = 0 in H1(X).
If |NM (x)| is even, let H M be any subgroup which contains with index
2. Then H is conjugate in NG(x) to some H0 Mx (by choice of Mx); and so
[M; x; Mx]Y = [M; H; x; H0; Mx]Y = [M; H]Y + [H0; Mx]Y (the last equality holds*
* be-
cause [H; x] = [H0; x]). But these edges lie in Y = N (F[2])=G, and so [M; x; M*
*x]Y = 0.
Thus, [M; x; Mx]Y = 0 whenever x 2 M satisfies condition (1a).
Now assume that x 2 M satisfies condition (1b), and fix y 2 M as in (1b). Fix
subgroups My; Mxy2 Fmax of even order, such that My contains the extension of <*
*y> by
a Sylow 2-subgroup of NG(y)=, and Mxy contains the extension of by a *
*Sylow
2-subgroup of NG(x)\ NG(y) (this last extension must lie in F since M 2*
* F
and F is separating). Consider the following diagram:
M
x
?
?
Mx --- ---! --- ---! My
?
?
y
Mxy
Bob Oliver and Yoav Segev *
* 35
By construction, condition (1a) is satisfied by each of the pairs x 2 Mxy, y 2 *
*Mxy and
y 2 M, and so
[M; ; y; My]Y = 0 = [Mxy; ; y; My]Y = [Mxy; ; x; Mx]Y*
* :
And hence [M; x; Mx]Y = [M; ; x; Mx]Y = 0.
Step 2 We now prove inductively, for i 1, that H1(Xi) has finite image in H1(X)
if H1(Xi-1) does. Fix a loop in Xi. We can again assume that it alternates "p*
*eaks"
and "valleys"; and that each peak is either equal [Mi] or lies in Xi-1. If any*
* of the
valleys is a vertex [H] =2Xi-1, then it must be connected on both sides to [Mi]*
* (but
possibly by different edges). This forms a loop (two edges each connectingf[H]i*
*to [Mi])
whose homology class lies in the image of H1(F[p])=G for any prime pfi|H|, and *
*this
group vanishes by Corollary 5.7. We are thus reduced to looking at 1-cycles of *
*the form
z = OE - [H; Mi; H0], where H; H02 F0iand OE is a path in Xi-1connecting [H] an*
*d [H0].
And since the image of N ((F0i)>1) in X is connected by (2), the path [H; Mi; H*
*0] is
homotopic to a path in Xi-1(and hence [z] is in the image of H1(Xi-1)), modulo *
*loops_
of the form [K; Mi; K0] for G-conjugate subgroups K; K02 F0i. *
* |__|
The following proposition shows that in certain cases, one can replace F by a*
* different
separating family without changing the homology of N (F>1) or of N (F>1)=G. Not*
*e, in
its statement and proof, that any finite group G contains a (unique) maximal no*
*rmal
perfect subgroup L C G: the last term in the derived series of G. This normal s*
*ubgroup
is also characterized by the properties that L is perfect and G=L is solvable.
Proposition 5.12. Let F0 $ F be two separating families in G, and let H F be a*
*ny
subfamily. Assume that one of the following two conditions holds: either
(a) for each perfect subgroup L 2 Fr F0, there is a solvable subgroup N C CG(*
*L)
with N 2 H; or
(b) the maximal normal perfect subgroup Lmax C G is simple, and CG(L) 2 H for
each perfect subgroup L 6= Lmax in S(G)r F0.
Then the inclusion of N (F0H ) into N (FH ) is a homotopy equivalence, and
H*(N (F0H )=G) ~=H*(N (FH )=G):
Proof.Note that the set of perfect subgroups in Fr F0 is nonempty. Since for a*
*ny
H 2 Fr F0 with maximal normal perfect subgroup L C H, L 2 Fr F0 since H=L is
solvable.
We first check that condition (b) implies condition (a). Fix any perfect sub*
*group
L 2 Fr F0, and let L0 L be the maximal normal perfect subgroup of L.CG(L). Then
CG(L0) CG(L), so L0.CG(L0) L.CG(L), and CG(L0) is solvable since (L0.CG(L0))=*
*L0
is solvable and L0\ CG(L0) = Z(L0) is abelian. Also, CG(L) normalizes L0, and *
*so
CG(L0) C CG(L). If (b) holds, then either L = L0 or L0 is not simple (since L C*
* L0);
and in either case L0 6= Lmax and so CG(L0) 2 H. Condition (a) thus applies, w*
*ith
N = CG(L0).
Now assume that condition (a) holds. Fix a conjugacy class L of maximal perf*
*ect
subgroups in Fr F0. Set F00= Fr (FL ): the family of subgroups in F which do
not contain any subgroup in L. This is a separating family (if H=K is solvable*
* and
H L 2 L then K L); and we can assume inductively that the inclusion of N (F0>*
*1)
36 Fixed point free actions on acyclic 2-complexes
into N (F00>1) is a homotopy equivalence. So upon setting F0 = F00, we are red*
*uced
to the case where Fr F0 contains a single conjugacy class L of perfect subgroup*
*s, and
where F0 is the set of subgroups in F which do not contain any subgroup in L.
For each L 2 L, let KL be the set of all subgroups H N(L) such that HL=L is
solvable, and let K0Lbe the set of all H 2 KL such that L 6 H. Then KL F (HL=L
solvable implies HL 2 F and hence H 2 F), and K0L= KL \ F0. By assumption,
there is a solvable normal subgroup N C CG(L) with N 2 H. Upon replacing N by
the subgroup generated by its conjugates in N(L) (still solvable since it is ge*
*nerated by
solvable normal subgroups of CG(L)), we can assume that N C N(L) (and N 2 FH ).
Then HN 2 KL for all H 2 KL (HNL=L is solvable if HL=L is since HL=L normalizes
NL=L and NL=L is solvable). Also, HN 2 K0Lfor all H 2 K0L: since for H 2 KL,
H=(H \ L) ~=HL=L is solvable, so HN=(H \ L) is solvable (since N is solvable and
centralizes H\L), and thus HN contains L if and only if H does. The nerves N ((*
*KL)H )
and N ((K0L)H ) are thus contractible by Lemma 0.4(b).
For each subgroup H 2 Fr F0, there is a unique L 2 L contained in H: the subg*
*roups
in L are maximal among perfect subgroups in Fr F0, and hence L must be the last
term in the derived sequence for H. Thus, L C H and H=L is solvable; and L is t*
*he
unique element of L for which H 2 KL r K0L. In other words, N (FH ) is the un*
*ion
of N (F0H ) with the contractible complexes N ((KL)H ) for L 2 L, any two of *
*the
complexes N ((KL)H ) and N ((KL0)H ) have intersection contained in N (F0H )*
*, and
N (F0H ) \ N ((KL)H ) = N ((K0L)H ) is also contractible for each L. The inc*
*lusion of
N (F0H ) into N (FH ) is thus a homotopy equivalence; and hence H*(N (F0H )=*
*G)_~=
H*(N (FH )=G) by Proposition 5.5. |*
*__|
5d. Connectivity_of_links_at_vertices_
In Section 5c, conditions were found on a separating family F which imply tha*
*t if
(G; F) 2 U2, then N (F>1) is acyclic, and hence there is a 2-dimensional H-univ*
*ersal
(G; F)-complex with no free orbits. The results of this section amount to showi*
*ng that
if there is such an action, then the links at all of its vertices must be conne*
*cted. This
result, and its proof, are closely related to [S2, Theorem 2.8].
Proposition 5.13. Let F be a nonempty family of subgroups of G, such that G =2F.
Let Fmax be the set of maximal members of F. Assume that
(a) each member of Fmax is self-normalizing;
(b) each member of Fr Fmax is contained in at least two members of Fmax; and
(c) N (F) is connected and H1(N (F)) = 0.
Then for each M 2 Fmax, N (F1(M) = N (F1) = N {H 2 F | 1 6= H $ M} :
Proposition 5.14. Fix a separating family F for G. Let F0 F be any subfamily
which contains Fc, and such that each nonmaximal subgroup in F0 is contained in*
* two
or more maximal subgroups. Assume that F satisfies the following two conditions:
(a) N (F>1)=G is connected and H1(N (F>1)=G) = 0.
(b) There is a maximal subgroup M 2 F such that Lk(F0)>1(M) is not connected.
Then (G; F) =2U2.
Proof.Assume that (G; F) 2 U2. Then by (a) and Lemma 5.10, N (F>1) is acyclic. *
*So
Proposition 5.13, applied to the family (F0)>1, implies that Lk(F0)>1(M) = N ((*
*F0)1)
is connected for all maximal subgroups M 2 F, and this contradicts point (b). (*
*Recall_
that all maximal subgroups in F are self-normalizing by Lemma 1.1.) *
* |__|
38 Fixed point free actions on acyclic 2-complexes
6.Simple groups of Lie rank one
We now focus attention on the simple groups of Lie type and Lie rank one. Th*
*ere
are four families of such groups: the two dimensional projective special linear*
* groups
L2(q), the three dimensional projective special unitary groups U3(q), the Suzuk*
*i groups
Sz(22k+1), and the Ree groups Ree(32k+1) = 2G2(22k+1). We refer to Appendix D *
*for
more detail on these groups, and on the classification of finite groups of Lie *
*type in
general.
We first show that the only 2-dimensional actions which involve the simple gr*
*oups
L2(q) or Sz(q) are the ones constructed in Section 3. This will be done in a se*
*ries of
three lemmas, after which the results will be summarized in Proposition 6.4.
Lemma 6.1. Assume that G = L2(q) or P GL2(q), where q = pk and p is an odd pr*
*ime.
Let F be a separating family for G which contains no nonsolvable subgroups L2(q*
*0) or
P GL2(q0) for q0 a smaller power of p. Assume also that F 6= SLV if G = L2(q) *
*and
q 3 (mod 8). Then (G; F) =2U2.
Proof.We refer to the description of maximal subgroups of G in Proposition 3.3.*
* Note
that if G = L2(q) and q 3 (mod 8), then F must contain a subgroup isomorphic to
A5 _ the only nonsolvable subgroups of G not isomorphic to L2(q0) or P GL2(q0) *
*for q0
a smaller power of p. In particular, q 1 (mod 5) in this case.
Case 1: Assume first that p = 3. If k is odd, then q 3 (mod 8) and q 2 (mod
5); and so G 6~= L2(q) by the above remarks. Thus, either G = L2(3k) for k eve*
*n, or
G = P GL2(3k).
Set K1 = P GL2(3) ~=4 (the subgroup of matrices with entries in F3), let K2 b*
*e the
subgroup of upper triangular matrices (K2 ~=Fqo C(q-1)=2or Fqo Cq-1), and let K*
*3 be
the subgroup of monomial matrices (K3 ~=Dq-1 or D2(q-1)). Set Kij= Ki\ Kj and
K = K1 \ K2 \ K3. Then K12~= D6, K13~= C22, K23~= C(q-1)=2or Cq-1, and K ~=C2.
Since K1 is a maximal subgroup in F (see the list of subgroups in Proposition 3*
*.3),
Proposition 5.4 now applies (using condition (a1), or (a2) if G = L2(9)) to sho*
*w that
(G; F) =2U2.
Case 2: Now assume that p 5. By Proposition 3.3, A4 is a maximal subgroup of G
only if G = L2(q) and q 3 (mod 8), in which case (as noted above) F must conta*
*in
subgroups isomorphic to A5. And since there is only one conjugacy class of A4 *
* G
(Proposition 3.3 again), each such subgroup must be contained in some A5 2 F.
Thus, no maximal subgroup of F is isomorphic to A4. From the lists of maximal
subgroups in Proposition 3.3, we now see that each maximal subgroup in F is iso*
*morphic
to one of the groups Fqo C(q-1)=2or Fqo Cq-1(triangular matrices); Dq-1or D2(q-*
*1); Dq+1
or D2(q+1); or 4 or A5. Also, by hypothesis, if p = 5, then A5 ~=L2(5) is not i*
*n F.
Let M1 G be the (maximal) subgroup of upper triangular matrices, and let T *
*M1
be the subgroup of diagonal matrices. From the above list (and since p > 3) we *
*see that
M1 and its conjugates are the only maximalfsubgroupsiin F of order a multiple o*
*f p.
Furthermore, for any subgroup H 2 F with pfi|H|, H leaves invariant a unique li*
*ne in
(Fq)2, and hence is contained in a unique subgroup conjugate to M1 (and thus a *
*unique
maximal subgroup in F). Also, each nontrivial subgroup H M1 of order prime to p
is contained in a unique subgroup conjugate to T (i.e., CM1(H)).
Bob Oliver and Yoav Segev *
* 39
We first check that N (F>1)=G is connected and H1(N (F>1)=G) = 0, using Propo*
*si-
tion 5.11. From the above list of maximal subgroups in F (and since A4 is not a*
*mong
them), we see that for each maximal subgroup M 2 F of even order, and each x 2 M
of odd prime order, NM () has even order. Thus, condition (1a) in Propositio*
*n 5.11
holds. Also, the only maximal subgroups in G of odd order are those conjugate *
*to
M1 ~=Fqo C(q-1)=2, when G = L2(q) and q 3 (mod 4). Let F01be the set of subgro*
*ups
of M1 which are contained in maximal subgroups in other conjugacy classes; by t*
*he
above remarks each H 2 F01is conjugate to a subgroup of T . The image of N ((F0*
*1)>1)
in N (F>1)=G is thus connected, and so condition (2) in Proposition 5.11 holds.*
* This
finishes the proof that N (F>1)=G is connected and H1(N (F>1)=G) = 0.
Now let F0 F be the subfamily consisting of all maximal subgroups in F, toge*
*ther
with all subgroups in F contained in two or more maximal subgroups. We have seen
that each proper subgroup of M1 contained in F0 is contained in a unique subgro*
*up
conjugate to T . In other words, Lk(F0)>1(M) = N ((F0)1) is not connected: i*
*t has one
connected component for each subgroup of M1 conjugate to T . So Proposition 5.1*
*4_now_
applies to show that (G; F) =2U2. *
*|__|
In each of the next two lemmas, we deal simultaneously with simple groups L =*
* L2(q)
and Sz(q), where q = pk and p is prime (p = 2 if L = Sz(q)). It will be conveni*
*ent to fix
subgroups S; T; B; N L of each of these groups, according to the following tab*
*le:
______________________________________________________
|| L|| L2(q) | Sz(q) ||
||__||___________________________|____________________ ||
|| S|| { 1a | a 2 Fq} ~=Fq | S(q; ) ||
||__||_________01________________|___________________ ||
|| T|| {diag(; -1) | 2 (Fq)*}={I}| {M() | 2 (Fq)*} ||
||__||___________________________|___________________ ||
|| B|| So T | M(q; ) = S(q; )o T ||
||__||___________________________|___________________ ||
|| N|| N(T ) = | N(T ) = ||
||__||___________________-10_____|____________________||
Table 4
When L = Sz(q), we are using the notation in Proposition 3.6 (where Sz(q) is re*
*garded
as a subgroup of GL4(q)). All of these subgroups are invariant under the actio*
*n of
Aut(Fq). In both cases, S is a Sylow p-subgroup, B = N(S) is a Borel subgroup, *
*T is
cyclic (of order q- 1 or (q- 1)=2), and N is dihedral.
Lemma 6.2. Assume that G = L is one of the simple groups L2(q) or Sz(q), where
q = pk and p is prime (p = 2 in the second case). Let F be any separating famil*
*y for G
which contains a nonsolvable subgroup isomorphic to L2(q0) or Sz(q0), where q0 *
*= pk0
(and k0|k). Then (G; F) =2U2.
Proof.Assume that q0 = pk0 is chosen so that F contains a maximal subgroup iso-
morphic to G0 = L2(q0), P GL2(q0), or Sz(q0). Thus, G0 is the subgroup of all *
*ma-
trices in G with entries in Fq0. (More precisely, if G = L2(q) P GL2(q), then
G0 = L2(q) \ P GL2(q0).) By Proposition 3.3 or 3.6, if G0 ~=M 2 F, then there *
*is
an automorphism oe 2 Aut(G) such that oe(M) = G0. Thus, upon replacing F by oe(*
*F),
we can assume that G0 2 F.
We now apply Proposition 5.4, with the subgroups K1 = G0, K2 = B, and K3 = N
(as in Table 4). Then K12 = B0, K13 = N0, K23 = T , and K = K1 \ K2 \ K3 = T0.
40 Fixed point free actions on acyclic 2-complexes
Condition (b) of 5.4 holds by assumption (K1 = G0 is maximal in F). Conditions *
*(d)
and (e) are clear: NG(G0; B0; T0) = T0, and the triples (K1; B0; T0) and (K1; N*
*0; T0) are
not G-conjugate.
We next consider condition (c). Clearly, K12= B0 is a maximal subgroup of K1 *
*= G0.
If G = L2(q), then K = T0 is maximal in K12 = B0. And if G = Sz(q), then T0 is
maximal among critical subgroups of B0. (There is one subgroup T0 $ R $ B0, whe*
*re
R = Z(S(q0; )).T0 ~=Fq0oCq0-1. But using Proposition 3.6(b), it is easy to chec*
*k that
every maximal subgroup of G which contains R also contains B0. So by Lemma 3.1(*
*a),
R is not critical.)
It remains to check that inequality (a1) or (a2) holds. From the above descri*
*ption of
the groups, we see that
(
q0 if L = L2(q) q-1
[K12:K] = [B0:T0] = [K13:K] = 2; [K23:K] = ffl.____q*
*;-1
(q0)2 if L = Sz(q), 0
where ffl = 1_2if G = L2(q),Pp is odd, and 2k0|k (so G0 = P GL2(q0)), and ffl =*
* 1 otherwise.
Inequality (a1) now holds ( i 2 is a power of 2. Then there is*
* no
2-dimensional G-complex without fixed points.
Proof.Assume otherwise: let q be such that U3(q) is the smallest counterexample*
*, and
let F be a separating family of subgroups of G such that (G; F) 2 U2.
Set d = (3; q+ 1). Then
2-q+1
|G| = 1_dq3(q2 - 1)(q3 + 1) = q3.(q - 1).(q + 1)2.(q_____d) (1)
(cf. [Ca , Theorem 14.3.2], who writes U3(q) = 2A2(q2)). Here, the factors in t*
*he second
3+1
formula are pairwise relatively prime. (Note that 3|(q2 - q + 1) = q___q+1if a*
*nd only if
3+1 2
3|(q+ 1), and that q___q+1cannot be divisible by 3 .)
Let be the Frobenius automorphism of order 2 on Fq2; and write x = (x) = xq*
* for
any x.
The following list of maximal subgroups of G can be found in [Ha , p. 158] o*
*r in
[GLS , Theorem 6.5.3(a,b,c,g)]. Note also the thesis of Peter Kleidman [Kl1, x5*
*], where
maximal subgroups are listed for classical groups of low rank, and a general pr*
*ocedure
for determining them is described.
(M1): M1 ~=[q3]o C(q2-1)=d; the stabilizers of isotropic lines (generated by v*
* with (v; v) =
0); the Borel subgroups of G. We choose M1 to be the stabilizer of , or e*
*quivalently
the group of upper triangular matrices with respect to the basis {v1; v2; v3}.
(M2): M2 ~=GU2(q)=Cd ~=C(q+1)=dxL2(q); the stabilizers of anisotropic lines (g*
*enerated
by v with (v; v) 6= 0). We choose M2 to be the subgroup of matrices (aij) (wi*
*th respect
to either of the above bases) for which a22is the only nonzero entry in the s*
*econd row
or column.
44 Fixed point free actions on acyclic 2-complexes
(M3): M3 ~=[(Cq+1)2o 3]=Cd; the stabilizer of (the union of) three pairwise or*
*thogonal
lines. We choose M3 to be the group of monomial matrices with respect to the
orthonormal basis {e1; e2; e3}.
(Mq04), if q = qb0and b is an odd prime: Mq04= N(U3(q0)), isomorphic to U3(q0)*
* (if
(b; d) = 1) or P GU3(q0) (if b = d = 3). There are (b; d) conjugacy classes *
*of such
subgroups (all conjugate in P GU3(q)).
(M5): M5 ~=C(q2-q+1)=doC3. Consider the hermitian form <-; -> on Fq6(viewed as*
* a
3
vector space over Fq2) defined by = Tr(xyq ), where Tr: Fq6! Fq2is the*
* trace
2 *
map. Let oe 2 Aut (Fq6) be the automorphism oe(x) = xq , let H (Fq6) be the
subgroup of order q3 + 1, and set
M = Ho (Fq6)* o Aut(Fq6):
Then M preserves <-; ->, and M5 is the intersection of U3(q) with the image o*
*f M
2
in P GU3(q). In particular, C3 acts on C(q2-q+1)via x ! xq .
We can assume inductively that none of the groups Mq04= N(U3(q0)), for q0 > 2*
*, can
act on an acyclic 2-complex without fixed points. So they must all be contained*
* in F.
Also, by Proposition 6.4, if M2 =2F, then the only subgroups of M2 (and its con*
*jugates)
which are in F are solvable subgroups. So either F = MAX , the family of all *
*proper
subgroups of G, or F = F0, the family of all subgroups whose intersection with *
*any
subgroup in (M2) is solvable _ and this latter only when k is prime or a power *
*of 2.
We first show that N (F>1)=G is connected and H1(N (F>1)=G) = 0, using Propos*
*ition
5.11. Since every perfect subgroup in MAX r F0 is of the form L2(2k0) where *
*1 <
k0|k and has nontrivial centralizer, Proposition 5.12 applies, with H = MAX >*
*1 (and
using condition (b)) to show that H*(N ((F0)>1)=G) ~=H*(N (MAX >1)=G). So we *
*can
assume that F = MAX .
The even order maximal subgroups of G are those conjugate to M1, M2, M3, or M*
*q04.
If M = M2, M3, or Mq04and x 2 M is of odd prime order,fthenione easily sees that
NM () has even order. Also, if x 2 M1 and |x|fi(q + 1)=d, then CM (x) has ev*
*en order:
if M1 is the subgroup of upper triangular matrices with respect to the basis {v*
*1; v2; v3},
thenixjis conjugate to a diagonal matrix diag(; -2; ) and is centralized by the*
* element
100110
0 01 . Thus, condition (1a) of Proposition 5.11 holds in all of these cases.
Now let x 2 M1 be of prime order dividing p - 1. We check that condition (1b)
of Proposition 5.11 holds. Let y 2 CM (x) ~=C(q2-1)=dbe any element of prime o*
*rder
dividing (q+1)=d. We have just seen that NM () has even order, and NG()\N*
*G()
also has even order since NM () ~= M2 ~=C(q+1)=dxL2(q). Thus, condition (1b*
*) of
Proposition 5.11 holds in this case.
It remains to check condition (2) of 5.11. Let F1 be the set of all subgroup*
*s of
M5 ~=C(q2-q+1)=doC3 which are also contained in even order maximal subgroups. By
inspection, F1 contains subgroups of order 3, and all maximal subgroups in F1 a*
*re of the
form Cao C3 ( Mq04) for some a. So the image of N ((F1)>1) in N (F>1)=G is none*
*mpty
and connected. Proposition 5.11 thus applies to show that N (F>1)=G is connecte*
*d and
H1(N (F>1)=G) = 0.
This shows that condition (a) in Proposition 5.14 holds, and it remains to ch*
*eck
condition (b). Set M = M5 ~=C(q2-q+1)=doC3. Let Fc F be the subfamily consisti*
*ng
Bob Oliver and Yoav Segev *
* 45
6-1)(q-1)
of all critical subgroups in F. Fix a prime p|(q2 - q + 1) = (q_______(q3-1)(*
*q2-1)such that
p|-|(q60- 1) when q0 is a smaller power of 2 (such a prime existsfbyiZsigmondy'*
*s theorem
[HB2 , Theorem IX.8.3]). Then for any proper subgroup H M with pfi|H|, M is the
unique maximal subgroup of G which contains H, and so H =2Fc (Lemma 3.1(a)). Let
T C M be the subgroup of order (q2 - q + 1)=dp; then M=T ~=Cpo C3. And C3 is not
2 2 2
normal in M=T : since C3 acts on Cp via (x 7! xq ), and (q - 1; (q - q + 1)=d*
*) = 1.
By Proposition 5.14, we will be done upon showing that the nerve of (Fc)1i*
*s not
connected. For any 1 6= H T , H is not critical by Lemma 3.1(b): N(H) = M 2 F,
and NK (H) % K for all K % H (note that K must be contained in M or in one of t*
*he
subgroups N(U3(q0))). Thus, any critical subgroup properly contained in M must *
*be of
the form Ho C3 for H T ; and such subgroups do exist (any subgroup of M maximal
among those contained in other maximal subgroups in F is critical). The image o*
*f the
poset (Fc)1in S(M=T ) thus consists precisely of the subgroups of order 3. S*
*ince the
continuous image of a connected space must be connected, this shows that N ((Fc*
*)1)__
is not connected, and finishes the proof of the proposition. *
* |__|
We note here that Proposition 6.6 can also be proven using Propositions 5.3 a*
*nd 5.4;
but this involves considering several different cases, and requires complicated*
* arguments
that certain subgroups are not critical.
We are now ready to consider the Ree groups 2G2(q).
Proposition 6.7. When q is any odd power of 3, there is no 2-dimensional 2G2(q)-
complex without fixed points.
Proof.Set G = 2G2(q), where q = 3k and k is odd; and assume that F is a separat*
*ing
family for G such that (G; F) 2 U2. We can assume inductively that q is the sma*
*llest
power of 3 for which this happens. Since 2G2(3) ~=Aut(L2(8)) [Jan], this subgro*
*up has
no fixed point free action on a Z-acyclic 2-complex by Lemma 6.3. Thus, we must*
* have
2G2(3) 2 F.
The order of G is given by the formula
p__ p__
|G| = q3(q - 1)(q3 + 1) = q3.23.(q-1_2).(q+1_4).(q + 3q + 1).(q - 3q + *
*1);
(cf. [Ca , Theorem 14.3.2]), where the factors in the last decomposition are p*
*airwise
relatively prime. The maximal solvable subgroups of G, as listed in [Kl2, Theor*
*em C],
all lie in the following conjugacy classes:
(M1): the Borel subgroups P oCq-1, where |P | = q3 (a Sylow 3-subgroup of G). *
*More
precisely, P = (Fq)3 with multiplication given by
(x1; y1; z1).(x2; y2; z2) = (x1 + x2; y1 + y2 + x1.xoe2; z1 + z2 - x1.y2 + y1.*
*x2 - x1.xoe1.x2):
p__ 2
Here, xoe= x 3q (so xoe= x3). The action of (Fq)* on P is given by
(x; y; z)-1 = (x; oey; 2oez):
(See [HB3 , Theorem XI.13.2].)
(M2): M2 = CG(C2) ~=C2x L2(q) for any C2 G
(M3): M3 = N(C22) = (C22xD(q+1_2))o C3 for any C22 G
46 Fixed point free actions on acyclic 2-complexes
(M+4) and (M-4): M4 ~= Cqp__3q+1oC6, where C6 acts via (x 7! xq). (The action*
* of
C6 is determined by the fact that an element of order 2 or 3 has centralizer *
*of order
prime to q2- q+ 1.)
(Mq05): Mq05~=2G2(q0) whenever q = qp0for some (odd) prime p.
By our inductive assumption, 2G2(q0) 2 F for all q0 = 3k0 where k0|k. So all *
*of the
maximal subgroups must be included in F, except possibly those in (M2).
We first show that N (F>1) is connected and that H1(N (F>1)=G) = 0. By Propos*
*ition
5.12 (arguing as in the proof of Proposition 6.6), it suffices to do this when *
*F = MAX :
the family of all proper subgroups of G. We apply Proposition 5.11. From the ab*
*ove
list, we see that all maximal subgroups of G have even order. If M is maximal *
*and
x 2 M has odd prime order, then NM (x) has even order, except possibly when M is
conjugate to M1 and |x| = 3. And under the above description of P C M1, any x 2*
* P
of order 3 is of the form x = (0; b; c) for b; c 2 Fq; x is normalized by (-1) *
*2 (Fq)* if
b = 0 or c = 0; and if b 6= 0 then x = (0; b; c) is conjugate to (0; b; 0). Con*
*dition (1a)
of Proposition 5.11 thus holds (and condition (2) is empty). It follows that N *
*(F>1) is
connected and H1(N (F>1)=G) = 0.
We have now shown that condition (a) in Proposition 5.14 holds. We claim that
condition (b) holds for one of the maximal subgroups M4 ~= Cqp__3q+1oC6; once t*
*his
has been shown then we can conclude that (G; F) =2U2. By Zsigmondy's theorem [H*
*B2 ,
Theorem IX.8.3], there is a prime p|(q6 - 1) = (36k- 1) such that p|-|(3m - 1) *
*for any
3+1 p__ p__
m < 6k. In particular, p|q___q+1= (q + 3q + 1)(q - 3q + 1) _ and thus divides*
* the
order of M = M+4or M-4_ but does not divide the order of 2G2(q0) for any q0 < q.
We claim that the nerve of the poset of proper subgroups of M which are critica*
*l in F
is not connected. Let T C M be the cyclic subgroup of index 6p, and set
H = Im[(Fc)1---! S(M=T )]:
From the above list of maximal subgroups, we see that for any proper subgroup H*
* $ M
of order a multiple of p, H is contained in no other maximal subgroup in F, and*
* hence
H is not critical (Lemma 3.1(a)). Also, for any 1 6= H T , Lemma 3.1(a) applie*
*s (with
bH= N(H) = M) to show that H =2Fc. Thus, H contains neither the trivial subgroup
nor subgroups of order a multiple of p. Also, H contains the subgroups of orde*
*r 6 in
M=T , since any subgroup of the form Ho C6 M (for H T ) which is maximal among
subgroups of M contained in other maximal subgroups of F must be critical. We h*
*ave
now shown that H consists of the subgroups of order 6 in M=T ~= Cpo C6, as well*
* as
possibly the subgroups of order 2 and 3. Since none of these subgroups is norma*
*l (C6
acts on Cp via (x 7! xq) and p is prime to (q2 - 1) and to (q3 - 1)), this show*
*s that the
nerve of H is not connected. And since the continuous image of a connected spac*
*e must __
be connected, this shows that N ((Fc)1) also fails to be connected. *
* |__|
Proposition 6.7 can also be proven using Proposition 5.4 (when F contains cen*
*tralizers
of involutions), and Proposition 5.12 to reduce the general case to this case.
Bob Oliver and Yoav Segev *
* 47
7. Sporadic simple groups
Aschbacher and Segev proved in [AS ] that no sporadic simple group, with the *
*possi-
ble exception of the first Janko group J1, can act on a 2-dimensional acyclic c*
*omplex
without fixed points. In all cases, this was done by applying the four-subgrou*
*p crite-
rion, presented here in Proposition E.1. Since the arguments in [AS ] use a va*
*riety of
structures and definitions unfamiliar to non-group-theorists, we now describe h*
*ow these
results _ as well as the nonexistence of a J1-action _ can be proven using Prop*
*osi-
tion 5.4 instead. Note however that the arguments presented here, while fairly*
* brief
to present, are not really more elementary than those given in [AS ]. They depe*
*nd on
information about maximal subgroups which has been collected together in [Atl] *
*and
[A2 ], but whose proofs (especially for the ten sporadic groups listed in Table*
* 5) are
scattered widely throughout the literature.
We first repeat some definitions in [A2 , x28]. Fix a finite group G, a subgr*
*oup A
Aut(G), and an A-invariant subgroup B G. A regular (A; B)-basis for G is a set
{Gi| i 2 I} of subgroups containing B which satisfies the following two conditi*
*ons:
T
(1) each subgroup H G containing B is in the A-orbit of GJ def=j2JGj for some
unique J I (in particular, B = GI); and
(2) for each J; K I, if a(GK ) GJ for some a 2 A, then GK GJ and a(GK ) =
a0(GK ) for some a02 NA(GJ).
If G has a regular (A; B)-basis of order at least four (for any A and B), then *
*by [AS ,
6.1] (and using the four-subgroup criterion described in Proposition E.1), (G; *
*F) =2U2
for any separating family F which contains the basis. Using Proposition 5.4, th*
*is can
be shown for bases of order three which satisfy certain additional conditions.
Lemma 7.1. Fix a simple group G and a separating family F of subgroups of G.
Assume, for some A Inn(G) and some A-invariant subgroup 1 6= K G, that there
is a regular (A; K)-basis {Ki| i 2 I}, and indices r; s; t 2 I, such that Kr; K*
*s; Kt 2 F
and
____1____ 1 1
+ _________+ _________ 1: (1)
[Krs:Krst] [Krt:Krst] [Kst:Krst]
Then (G;fF)i=2U2. In particular, (1) holds if K contains a Sylow p-subgroup for*
* any
prime pfi|G|.
Proof.For simplicity, we write I = {1; 2; : :;:k}, and assume that {r; s; t} = *
*{1; 2; 3}.
By [A2 , 28.1], {K1; K2; K3} is a regular (NA(K123); K123)-basis; so we can ass*
*ume k = 3
and K = K123. It is immediate from the definition of a regular (A; K)-basis tha*
*t KJ[{i}
is a maximal subgroup of KJ for any J $ I and any i 2 Ir J.
We claim that the subgroups K1; K2; K3 satisfy the hypotheses of Proposition *
*5.4; it
then follows that (G; F) =2U2. We have just checked conditions (b) and (c) (max*
*imal-
ity of subgroups). Condition (a1) holds by assumption, and condition (e) (the t*
*riples
(K1; K12; K) and (K1; K13; K) are not G-conjugate) is immediate from the defini*
*tion of
a regular (A; K)-basis.
We next show that the Kican be ordered so that NG(K1; K12; K) = K, thus provi*
*ng
condition (d). To see this, note first that NG(K) must be A-conjugate (hence e*
*qual)
48 Fixed point free actions on acyclic 2-complexes
to one of the subgroups Ki, Kij, or K. Also, the Ki are maximal in the simple g*
*roup
G and hence self-normalizing. If NG(K) = K, then we are done. Otherwise, we c*
*an
assume (after switching indices if necessary) that NG(K) = K3 or K23. If NG(K) *
*= K23,
then NG(K1; K12; K) K1 \ K23 = K. So suppose NG(K) = K3. Since K12 is not
normal in G = , K12cannot be normal in both K1 and K2, and we can assume
without loss of generality that K12is not normal in K1. Then NK1(K12) = K12, an*
*d so
N(K1; K12; K) K12\ K3 = K. This finishes the proof that (G; F) =2U2.
It remainsftoishow that (1) always holds if K contains a Sylow p-subgroup for*
* some
prime pfi|G|. By definition of a regular basis, [Kij:K] > 1 for all i; j. If *
*[Kij:K] =
[Kik:K] = 2 for some i, then K C Ki= ([A2 , 28.1(2)]); Ki=K is genera*
*ted by
two elements of order 2 and hence dihedral; and this is a contradiction since i*
*t means
there are other overgroups of K not conjugate to any of the given ones.
Thus, [Kij:K] = 2 for at most one pair of indices i; j. So if (1) does not ho*
*ld, then
the three indices [Kij:K] must be (2; 3; 3), (2; 3; 4), or (2; 3; 5). Since ea*
*ch index is
prime to p (K contains a Sylow p-subgroup), this shows that p 5. If [Kij:K] = *
*m,
then the permutation action of Kijon the set Kij=K restricts to a homomorphism
'ij:K ! m-1 4 whose kernel Rijis normal in Kij. Set H = O{2;3}(K) C K: the
smallest normal subgroup of index a product 2r.3s. Then H is characteristic in*
* any
subgroup of K which contains H, and in particular characteristic in each Rij. S*
*o H is
normal in each Kij, and hence normal in G = . Since G is simple,*
* H = 1,
so 2 and 3 are the only primes dividing |K|. And thisfcontradictsithe assumptio*
*n that_
K contains a Sylow p-subgroup for some p 5 and pfi|G|. *
* |__|
We are now ready to prove:
Proposition 7.2. Let G be any of the sporadic simple groups, or the Tits group *
*2F4(2)0.
Then there is no 2-dimensional acyclic G-complex without fixed points.
Proof.We first prove the proposition for ten of the sporadic groups as well as *
*the
Tits group, by direct application of Proposition 5.4. Since M22 is one of these*
* groups,
Proposition E.3 then applies to prove the proposition for the other four Mathie*
*u groups.
The last twelve sporadic groups are then handled using Lemma 7.1. Throughout t*
*he
proof, whenever two names are given for one of the sporadic groups, the first i*
*s that
used in [Atl], and the second the name used in [A1 ] or [A2 ].
Assume the proposition does not hold, and let G be the smallest such group wh*
*ich has
a fixed-point free action on a 2-dimensional acyclic complex X. Let F be the se*
*parating
family of subgroups H G such that XH 6= ;. Consider first the following table,*
* which
describes how Proposition 5.4 can be applied to these eleven simple groups:
Bob Oliver and Yoav Segev *
* 49
_______________________________________________________________________
|| G | K1 | K2 | K3 | K12| K | K13 | K23 | [Kij:K] ||
||____|______|________|_______|_____|________|_________|_______|_______ ||
|| J1| 23:7:3| 7:6 | C3xD10| 7:3| 3 | 6 | 6 | 7;2;2 ||
|| | | | | | | | | ||
|| | | = N(7)| = C(3)| | | | | ||
||____|______|________|_______|_____|________|_________|_______|______ ||
|| M22| L3(4)| 24:S5| 24:A6| 24:A5| 22+4:3| 24:A5 | 22+4:S3|5;5;2 ||
|| | | | | | | | | ||
|| | (point)|(duad)| (hexad)| | = 24:A4| | = 24:S4| ||
||____|______|________|_______|_____|________|_________|_______|______ ||
|| J2| U3(3)|3.PGL2(9)|21+4:S3|31+2:8| 3:8 | 4.S4 | 3:D16| 9;4;2 ||
|| | | | - | - | | | | ||
|| | | = N(3A)| = N(4)| | | | | ||
||____|______|________|_______|_____|________|_________|_______|______ ||
|| HS| U3(5):2|U3(5):2|C(2x)| 51+2:8:2|5:8:2| 2S5:2 | 2S5:2| 25;6;6 ||
|| | | | | + | | | | ||
|| | | N(Syl5)| | | | | | ||
||____|______|________|_______|_____|________|_________|_______|______ ||
|| J3| L2(16):2|L2(17)| 21+4:S3|17:4| 4 | D8x2 | D16 | 17;4;4 ||
|| | | | - | | | | | ||
|| | | N(17)| = N(4)| | | | | ||
||____|______|________|_______|_____|________|_________|_______|______ ||
|| He| S4xL3(2)|L3(2)x7:3|N(3x)|S4x(7:3)|S4x3| S4xD6 | L3(2)x3| 7;2;7 ||
|| | | | | | | | | ||
|| | | = N(7)| | | | | | ||
||____|______|________|_______|_____|________|_________|_______|______ ||
|| Ru| 2F4(2)|L2(25).22|3.A6.22|L2(25).2|D24:2|(31+2:D8):2|D24:22|o;9;2 ||
|| | | | | | | + | | ||
|| | (point)|(edge)| = N(3)| | | | | ||
||____|______|________|_______|_____|________|_________|_______|______ ||
|| O'N| J1 | L3(7):2|(32xA6).2|19:6| 6 | D6xD10 | S3xS4| 19;10;24 ||
|| | | | | | | | | ||
|| | | N(19)| = N(3)| | | | | ||
||____|______|________|_______|_____|________|_________|_______|______ ||
|| HN| A12| M12:2| C(2x)| M12| 2xS5 | 25:S6 | (22xA5):2|o;96;2 ||
|| | | | | | | | | ||
|| = F5| (point)|(point|pair) | | = C (2A)|= C (2B)| | ||
||____|______|________|_______|_____|____M12_|______A12|_______|______ ||
|| Th| 3D4(2):3|(3x13):12|21+8:A9|13:12|12 | C (2x)| 3x12 | 13;o;3 ||
|| | | | + | | | K1 | | ||
|| = F3| | = N(13)| = C(2)| | | order 9216| | ||
||____|______|________|_______|_____|________|_________|_______|______ ||
|| 2F4(2)0|L2(25)|52:4A4|[29.3]| 52:12| 12 | D24 | 4A4 | 25;2;4 ||
|| | | | | | | | | ||
|| | | = N(Syl5)|C(2B)| | | | | ||
||____|______|________|_______|_____|________|_________|_______|_______||
Table 5
We refer to [Atl] for the existence of subgroups with these properties, and to *
*[GLS ,
Table 5.3] for tables of normalizers of prime order subgroups of the sporadic g*
*roups.
The subgroups in Table 5 are described using mostly the notation of [Atl]. How*
*ever,
we write, for example, N(3) or C(3) to denote the normalizer or centralizer in *
*G of a
subgroup of order 3 when there is a unique G-conjugacy class of such subgroups;*
* and
write N(3A) or N(3B) (or N(3x) when the class is unspecified) only when there i*
*s more
than one class. Also, Sylpalways denotes a Sylow p-subgroup of G.
In all cases, the results of Section 6 and Appendix F and the minimality assu*
*mption
on G imply that Ki 2 F for all i = 1; 2; 3. Note in particular the cases G = HS*
* , He,
and HN: K3 2 F since K13or K23is nonsolvable and in F.
The remarks under the names of the subgroups Ki describe how they are chosen
relative to one another. In all cases except G = M22, K1 and K2 are chosen in o*
*ne of
the following two ways: either
(a) they are the stabilizers of a vertex and an edge (or point pair) of a stand*
*ard action
of G on a graph; or
(b) K1 is a maximal subgroup of G, and K2 is the normalizer of some subgroup X *
* K1
(as indicated in the table), or a maximal subgroup (not conjugate to K1) contai*
*ning
NG(X) and such that K12= NK1(X).
50 Fixed point free actions on acyclic 2-complexes
The subgroup K3 is then chosen as the normalizer or centralizer of a certain su*
*bgroup
Y K12 as indicated. In all cases where K12 contains more than one conjugacy cl*
*ass
of subgroup of the given order, the choice is either specified under K = NK12(Y*
* ), or is
clear from the description of K. In many cases, it is unnecessary to identify K*
*3 more
precisely, since the only thing we need know about it is that it must lie in F.
When G = M22, K3 ~=24:A6 is the subgroup which leaves invariant some hexad in
the Steiner system of order 22, and it has the obvious action on this set of or*
*der 6 (cf.
[Gr , Theorem 6.8]). Then K1 is taken to be the stabilizer of some point x in t*
*he hexad,
and K2 the stabilizer of some pair of points in the hexad including x.
In all cases, each of the subgroups in the sequence K K12 K1 G is maximal
and self-normalizing in the next one. Thus, conditions (b,c,d) in Proposition 5*
*.4 always
hold. Condition (e) ((K1; K12; K) is not G-conjugate to (K1; K13; K)) is clear*
* except
when G = M22; in this case K12and K13are distinct parabolic subgroups in K1 ~=L*
*3(4)
containing the same Borel subgroup K, and hence not conjugate in K1. Inequality*
* (a1)
holds in all cases except when G = J1, as can be checked using the list of indi*
*ces [Kij:K]
in the last column (where "o" means that the index is >10 and hence large enoug*
*h not
to matter).
We give particular attention to the case G = J1: the first Janko group, and *
*the
only sporadic group not handled in [AS ]. Fix some K1 ~=C32o(C7o C3): a maxim*
*al
subgroup of G by [A2 , 16.17] (see also 16.4 and 16.16 in [A2 ]). Let K2 ~=C7o*
* C6 be
the normalizer of a subgroup of order 7 in K1, and let K3 ~=C3x D10be the centr*
*alizer
in G of a subgroup of order 3 in K12. Then K12 ~=C7o C3, K13 ~=C6 ~= K23, and
K = K1 \ K2 \ K3 ~=C3. All of these subgroups are solvable, and hence in F. Als*
*o,
X 1 1 1 1 1 1 1 1
_______= __+ __+ __< 1 + ___+ ___= 1 + _______+ _______;
i 0, 'm (Jm xSm-1 ) X(m-1). Moreover, for each j 2 Jm , there*
* are
finite subsets J0k Jk (0 k m - 1) such that
m-1[
'm (jx Sm-1 ) 'k(J0kxDk):
k=0
(c) A subset U X is open if and only if '-1m(U) is open in Jm xDm for each m*
* 0.
(X has the "weak topology" with respect to its cell structure.)
In the above definition, G is always assumed to act trivially on Dm and Sm-1 *
*. Usually,
a G-CW complex will be called a G-complex for short.
A CW complex is just a G-CW complex in the case where G is the trivial group.
An open cell in a (G-)CW complex X is the image 'm (jx int(Dm )) of one open di*
*sk
under the characteristic_map. Note that if oe = 'm (jx int(Dm ))_is any open ce*
*ll, then
'm (jx Dm ) = oe(the closure of oe) and @oe = 'm (jx Sm-1 ) = oer oe (the bound*
*ary of oe)
are determined by oe itself as a subspace of X. By condition (a) in the definit*
*ion, each
point of X lies in exactly one open cell, and the open m-cells of X are the con*
*nected
components of X(m)r X(m-1).
The following is an alternative way to regard G-complexes, once CW complexes *
*have
been defined. Fix a CW complex X with continuous G-action. Call the action admi*
*ssible
if it permutes the open cells of X, and sends a cell to itself only via the ide*
*ntity. If X
is a G-complex, then by definition the G-action is admissible. Conversely, if t*
*he action
of G on X is admissible, then the characteristic maps of X can be redefined to *
*yield a
G-complex. More precisely, if 'm : Jm xDm ! X is the given characteristic map f*
*or the
m-cells of X, then the action of G on the m-cells of X induces an action on Jm *
*. Also,
for any orbit of G on Jm and any j 2 , one can define '0mon x Dm by setting
'0m(gj; x) = g'm (j; x). Upon doing this for all m 0 and all orbits of Jm , we*
* get the
new characteristic maps which make X into a G-complex.
Note in particular the last part of condition (b). Each cell in a CW complex*
* must
be "closure finite": its boundary must be contained in a finite union of closed*
* cells of
smaller dimensions. To see the importance of this condition, consider the space*
* X = D2,
let J0 be the circle S1 with the discrete topology, let J2 be a set with one el*
*ement, and
set Jm = ; for all m 6= 0; 2. Let '0: J0x D0 ! X be projection to the first fac*
*tor (i.e.,
inclusion of the circle), and let '2: J2x D2 ! X be projection to the second fa*
*ctor.
These sets and maps satisfy all of the conditions for a CW structure on D2 exce*
*pt for
closure finiteness. But this goes against our intuitive expectations (by analo*
*gy with
simplicial complexes) that the 0-skeleton of any CW complex should be discrete,*
* and
that compact CW complexes should be made up of finitely many cells.
The following lemma describes the principal means of constructing G-complexes*
* (see,
e.g., Proposition 0.2).
Lemma A.2. Let X be a G-complex, let J be any discrete set with G-action, and*
* let
': Jx Sn-1 ! X(n-1)be any G-equivariant map. Then the space
Y = X [' (Jx Dn)
Bob Oliver and Yoav Segev *
* 53
is a G-complex.
Proof.Let : Jx Dn ! Y be the obvious map; thus |Jx Sn-1= '. This, together with
the characteristic maps for X, make up the characteristic maps for Y . The othe*
*r details_
are the same as in the nonequivariant case; cf. [LW , Proposition II.2.2]. *
* |__|
If X has been constructed via successively attaching cells, i.e., via success*
*ive repetition
of the construction of Lemma A.2, starting with a discrete set, then the closur*
*e finiteness
condition holds automatically. In fact, this is the basis for an alternative de*
*finition of a
CW complex, described more precisely in [LW , Theorem II.2.4].
A (G-invariant) subcomplex of a G-CW complex is a closed (G-invariant) subspa*
*ce
A X which is a union of closed cells in X; i.e., a union of images of characte*
*ristic
maps. A subcomplex is itself a CW complex in an obvious way. Note in particular*
* that
if X is a G-complex, then for every H G, the fixed point set XH is a subcomple*
*x of X:
if 'm : Jm xDm ! X are the characteristic maps for X, then ('m )H : (Jm )H xDm *
*! XH
are the characteristic maps for XH .
The following proposition is an immediate consequence of condition (c) in Def*
*inition
A.1. Roughly, it says that a function defined on a CW complex is continuous if*
* and
only if its restriction to each closed cell of the complex is continuous.
Lemma A.3. Let X be a CW complex, with characteristic maps 'm : Jm xDm ! X.
Then if Y is any topological space, a function f : X ! Y is continuous if and o*
*nly_if
f O'm is continuous for each m. |*
*__|
Recall (cf. [S1], [AS ]) that a simplicial complex X with G action is called *
*admissible
if the action permutes the simplices linearly, and sends a simplex to itself on*
*ly via the
identity. (If this last condition does not hold, then it does hold for the bar*
*ycentric
subdivision of X.) We claimed in the introduction that Theorem A holds equally *
*well if
one replaces "G-complex" by "admissible G-simplicial complex" in the statement.*
* This
follows from the following proposition, where simplicial complexes are always a*
*ssumed
to have the metric topology (cf. [LW , Definition IV.4.1]).
Proposition A.4. Any finite dimensional admissible G-simplicial complex is G-ho*
*mo-
topy equivalent to a G-complex of the same dimension. Any countable, finite di*
*men-
sional G-complex is G-homotopy equivalent to an admissible G-simplicial complex*
* of
the same dimension.
Proof.For any admissible G-simplicial complex X, one can clearly define skeleta*
* and
characteristic maps for X which satisfy conditions (a) and (b) in Definition A.*
*1; but for
these to also satisfy condition (c) we must replace X with a new space Xcw havi*
*ng the
same underlying set but a finer topology (more open sets). The identity map Xcw*
* !
X is continuous and is a homotopy equivalence by [LW , Proposition IV.4.6] (and*
* the
argument in [LW ] can easily be fixed to cover the equivariant case).
The second statement is shown, in the nonequivariant case, by Whitehead in [W*
*h ,
Theorem 13], and his proof carries over immediately to G-complexes. The idea is*
* the
following: once X(m-1)has been replaced by a G-simplicial complex of the same d*
*imen-
sion, then approximate the characteristic map Jm xSm-1 ! X(m-1)by a simplicial *
*map
(possibly after further subdivision of X(m-1)), and attach the m-cells after gi*
*ving_them
appropriate simplicial structure. *
* |__|
54 Fixed point free actions on acyclic 2-complexes
For any space X, we let X denote its unreduced suspension: X def=(Xx I)=~ ,
where (x; 0)~ (x0; 0) and (x; 1)~ (x0; 1) for all x; x02 X. A G-action on X aut*
*omatically
determines a G-action on X, via the trivial action on the interval I.
Lemma A.5. The orbit space X=G of a G-complex X inherits a structure of a CW
complex, with one n-cell in X=G for each G-orbit of n-cells in X. The unreduced
suspension X of any G-complex X is itself a G-complex in a natural way. And if X
and Y are any two G-complexes, at least one of which is finite, then their prod*
*uct Xx Y
is also a G-complex.
Proof.If X is a G-complex, with skeleta X(m) and characteristic maps 'm : Jm xS*
*m !
X, then X=G is a CW complex with skeleta (X=G)(m)= X(m)=G and characteristic ma*
*ps
'm =G: (Jm =G)x Sm ! X=G. This follows immediately from Definition A.1. Note*
* in
particular that condition (c) holds for X=G by definition of the quotient topol*
*ogy: a
subspace is open in X=G if and only if its inverse image is open in X.
The unreduced suspension of a CW complex is again a CW complex by [LW , Corol*
*lary
II.5.12]. And if X or Y is finite, then Xx Y is a G-complex with the obvious pr*
*oduct
structure by [LW , Theorem II.5.2]. In each of these last two cases, the argum*
*ents in
[LW ] carry over without change to the equivariant case.
We remark here that if X and Y are arbitrary CW complexes,Sthen there is an
obvious way to define skeleta for Xx Y : (Xx Y )(m) = i+j=m(X(i)xY (j)). Al*
*so, if
'm : Jm xDm ! X and m : Km xDm xY are the`characteristic maps for X and Y , th*
*en
one can define characteristic maps !m = i+j=m('ix j) for Xx Y . (This require*
*s fixing
identifications DixDj ~=Di+j.) Conditions (a) and (b) in Definition A.1 always *
*hold;_
what can go wrong is condition (c). *
*|__|
The following lemma is not used in the paper, but does help to motivate the c*
*oncept
of "universal" (G; F)-complexes as defined in Section 0.
Proposition A.6. Fix a family F of subgroups of G, and let Y be any universal (*
*G; F)-
complex. Then for any (G; F)-complex X, any G-invariant subcomplex A X, and any
equivariant map f0: A ! Y , f0 extends to an equivariant map f : X ! Y . Furthe*
*rmore,
f is unique up to homotopy, in the sense that if f0: X ! Y is any other extensi*
*on of f0,
then there is an equivariant homotopy F : Xx I ! Y such that F |Xx0 = f, F |Xx1*
* = f0,
and F |AxI = f0OprojA.
Proof.It suffices to prove the existence of f : X ! Y ; the uniqueness then fol*
*lows by
extending the given map on (Xx {0; 1}) [ (Ax I) to Xx I.
We construct f : X ! Y one skeleton at a time. The construction of f0: X(0)[A*
* ! Y
is easy: let {xi} be orbit representatives for the vertices not in A, set Hi = *
*Gxi (the
isotropy subgroup), choose any yi2 Y Hi, and define f0(gxi) = gyi for all g 2 G*
* and all
i (and f0|A = fA).
Now assume that n 1, and that fn-1: X(n-1)[ A ! Y has been constructed. Let
'n: Jnx Dn ! X be the characteristic map for the n-cells of X (where Jn is a di*
*screte
set with G-action), and let J0n Jn be the subset of those n-cells not in A. Set
u0 = fn-1O'n|J0nxSn-1: J0nxSn-1 -----! Y:
Bob Oliver and Yoav Segev *
* 55
For each j 2 J0n, let Gj = {g 2 G | gj = j} 2 F be its isotropy subgroup. Then
u0(jx Sn-1) Y Gj. Also, Y Gjis contractible (since Y is (G; F)-universal), the*
* identity
map Y Gj! Y Gjis homotopic to a constant map, and hence any map to Y Gjis homo-
topic to a constant map. In particular, u0 can be extended to a (nonequivariant*
*) map
v0j: jx Dn ! Y Gj. This can then be extended to a G-map vj: Gjx Dn ! Y (where G*
*j is
the orbit of j) by setting vj(gj; x) = g.v0j(j; x). Upon repeating this procedu*
*re with one
representative from each G-orbit in J0n, the vj combine to give a G-map u: J0nx*
*Dn ! Y
whose restriction to J0nxSn-1 is u0. If we now set fn(x) = fn-1(x) for x 2 X(n-*
*1)[ A,
and fn('n(j; x)) = u(j; x) for (j; x) 2 J0nxDn, then this is a well defined map*
* of sets_from
X(n)to Y , which is equivariant by construction, and continuous by Lemma A.3. *
* |__|
Note that Proposition A.6 implies in particular that any two universal (G; F)*
*-com-
plexes are G-homotopy equivalent.
Appendix B. Cellular homology of G-complexes
The cellular chain complex (Cn(X); @n)n0 of a CW complex is described in [LW*
* ,
xV.2]. Formally, this is defined using singular homology (in particular, Cn(X)*
* =
Hn(X(n); X(n-1))), as in [LW , Definition V.2.1]. By [LW , Proposition V.1.8],*
* Cn(X)
is the free abelian group with basis the set of (oriented) n-cells in X; and by*
* [LW , xV.3]
each boundary map @n: Cn(X) ! Cn-1(X) can be described via the matrix whose en-
tries are the degrees of maps between (n - 1)-spheres induced by the attaching *
*maps
for the n-cells. By [LW , Theorem V.2.11], the singular homology H*(X) is isomo*
*rphic
to the homology of the complex (Cn(X); @n). Hence, if X is a finite complex, th*
*e Eu-
ler characteristic O(X) is equal to the alternating sum of the numbers of cells*
* in each
dimension.
Note that for a map f : X ! Y between CW complexes to induce a homomorphism
C*(X) ! C*(Y ), it must be a cellular map, in the sense that f(X(n)) Y (n)for *
*all n
0. However, since cellular homology H*(C*(X); @) is isomorphic to singular homo*
*logy,
any continuous map between CW complexes induces a homomorphism between their
cellular homology groups.
More generally, if X is any CW complex and A X is any subcomplex, then the
relative cellular chain complex is defined by setting C*(X; A) def=C*(X)=C*(A).*
* Thus,
Cn(X; A) is the free abelian group with one generator for each n-cell of X not *
*in A. By
[LW , Theorem V.2.11] again, the homology of the complex (C*(X; A); @) is natur*
*ally
isomorphic to H*(X; A).
If X is a G-complex and A X is a G-invariant subcomplex, then the cellular c*
*hain
complexes C*(X) and C*(X; A), and the homology groups H*(X) and H*(X; A), are a*
*ll
Z[G]-modules. In fact, each chain group Ci(X) or Ci(X; A) is a permutation modu*
*le,
in the sense that it has a Z-basis which is permuted by the linear action of G.
Once homology has been defined using the cellular chain complex, then the rel*
*ative
and Mayer-Vietoris exact sequences, and excision, are immediate. (Note, however*
*, that
excision in singular homology is needed to establish the basic properties of ce*
*llular
homology of CW complexes [LW , xV.1-2].) To see this, fix a G-complex X. For *
*any
56 Fixed point free actions on acyclic 2-complexes
G-invariant subcomplexes A0 A X, the short exact sequence of chain complexes
0 ---! C*(A)=C*(A0) ----! C*(X)=C*(A0) ----! C*(X)=C*(A) ---! 0
induces, via the snake lemma, the relative exact sequence
. .-.--!Hi(A; A0) ----! Hi(X; A0) ----! Hi(X; A) --@--!Hi-1(A; A0) ---! . .:.
Similarly, for any pair of G-invariant subcomplexes A; B X with A [ B = X, the*
*re is
a short exact sequence
0 ---! C*(A \ B) ----! C*(A) C*(B) ----! C*(A [ B) ---! 0
which induces the Mayer-Vietoris sequence
. . .---!Hi(A \ B) ---! Hi(A) Hi(B) ---! Hi(X) --@-!Hi-1(A \ B) ---! . .:.
All of these are exact sequences of Z[G]-modules.
Similarly, since Br (A\B) contains exactly the same cells as (A[B)r A, the in*
*clusion
map induces an isomorphism
~=
H*(B; A \ B) -----! H*(A [ B; A) (excision)
since it induces an isomorphism of cellular chain complexes.
The following lemma, used in the proof of Proposition 0.2, is one application*
* of
excision and the relative exact sequence. It describes the effect of attaching *
*cells on the
homology of the complexes involved.
Lemma B.1. Let X be a G-complex, let J be a discrete set with G-action, and l*
*et
f : Jx Sn ! X(n)be any G-equivariant map (n 1). Set Y = X [f (Jx Dn+1). Then
there is an exact sequence of Z[G]-modules
0 ---! Hn+1(X) -incl*---!Hn+1(Y ) ----!
f* incl*
Hn(Jx Sn) ----! Hn(X) ----! Hn(Y ) ----! 0;
and the inclusion X -incl-!Y induces isomorphisms Hi(X) ~=Hi(Y ) for all i 6= n*
*; n + 1.
Proof.Let ff: Jx Dn+1 ! Y be the characteristic map (so ff|Jx Sn= f). This indu*
*ces
an isomorphism C*(Jx Dn+1; Jx Sn) ~=C*(Y; X) of chain complexes, and hence an i*
*so-
morphism in homology in all degrees. The following square
Hn+1(Jx Dn+1; Jx Sn) --@-!~Hn(Jx Sn)
? = ?
ff*?y~= Hn(f)?y
Hn+1(Y; X) --@-! Hn(X)
commutes by the naturality of the relative exact sequences for pairs of CW comp*
*lexes,
and the upper boundary map is an isomorphism since Hi(Jx Dn+1) = 0 for i 1.
The lemma now follows from the relative exact sequence for the pair (Y; X), whe*
*re__
Hn+1(Y; X) is replaced by Hn(Jx Sn) via the above square. *
* |__|
The following more technical application of excision and the relative exact s*
*equences
is needed in the proof of Proposition 1.7.
Bob Oliver and Yoav Segev *
* 57
Lemma B.2. Fix a CW complex Y and subcomplexes B; X Y , and set A = B \ X.
Assume that the inclusion induces an isomorphism H*(A) ! H*(B). Then H*(Y; X) ~=
H*(Y; X [ B).
Proof.It suffices to show that H*(X [ B; X) = 0; the result then follows from t*
*he
relative exact sequence for Y X[ B X. But H*(X [ B; X) ~=H*(B; A) by excision*
*, __
and this last group vanishes since H*(A) ~=H*(B). *
* |__|
The following result says, roughly, that a union of homology or homotopy equi*
*valences
between CW complexes is again a homology or homotopy equivalence.
Proposition B.3. Let f : X ! Y be a map between CW complexes. Fix subcomplexes
A1; A2 $ X and B1; B2 Y such that X = A1 [ A2 and Y = B1 [ B2, and set
A0 = A1 \ A2 and B0 = B1 \ B2. Assume that f restricts to homology (homotopy)
equivalences fi: Ai ! Bi for i = 0; 1; 2. Then f is itself a homology (homotop*
*y)
equivalence.
Proof.If f0, f1, and f2 are all homology equivalences, then f is a homology equ*
*ivalence
by the Mayer-Vietoris sequences for the two unions (and the 5-lemma).
Assume now that f0, f1, and f2 are all homotopy equivalences; we must show th*
*at f
is a homotopy equivalence. By the Van Kampen theorem, f induces an isomorphism *
*of
fundamental groups (on each connected component). The map between the universal
covers is a homology equivalence, hence a homotopy equivalence; and hence f : X*
* ! Y
is itself a homotopy equivalence. For the details of this argument, cf. [Gra *
*, Lemma
16.24 & Theorem 16.22].
Alternatively, and more geometrically, one can show directly that any homotop*
*y in-
verse g0: B0 ! A0 of f0 can be extended (one cell at a time) to homotopy invers*
*es
gi: Bi! Ai(i = 1; 2), while at the same time extending the homotopies of g0Of0 *
*' IdA0
and f0Og0 ' IdB0. The result then follows upon taking g = g1 [ g2: Y ! X (and s*
*imi-
larly for the homotopies). The existence of the gi and the homotopies follows f*
*rom the
proofs of [LW , Theorems IV.3.2-3] (applied to the 2-ads (Ai; A0) and (Bi; B0))*
*; although_
the statements of these theorems are not sufficiently precise to do this. *
* |__|
Appendix C. Projective Z[G]-modules
Recall that for any G-complex X, C*(X) and H*(X) are Z[G]-modules in an obvio*
*us
way. A finitely generated Z[G]-module M will be called stably free if there are*
* finitely
generated free modules F0 and F such that M F0 ~=F . Free Z[G]-modules, and he*
*nce
(as an intermediate step) stably free Z[G]-modules play a key role when constru*
*cting
finite G-complexes in Section 1.
Lemma C.1. If X Y are finite G-complexes such that G acts freely on Y rX, th*
*en
C*(Y; X) is a finite chain complex of free finitely generated Z[G]-modules.
Proof.By assumption, G permutes freely the cells in Y not in X. Thus, G permutes
freely a basis of C*(Y; X); and this is a finite basis since X and Y have only*
*_finitely
many cells. |_*
*_|
58 Fixed point free actions on acyclic 2-complexes
The following lemma says in particular that if C* is a finite chain complex o*
*f finitely
generated free Z[G]-modules all but one of whose homology groups is stably free*
*, then
the remaining homology group is also stably free. This does not hold for module*
*s over
arbitrary noetherian rings, but uses special properties of group rings.
Proposition C.2. Let C* be any finite chain complex of projective Z[G]-modules.*
* As-
sume, for some k, that Hi(C*) is projective as a Z[G]-module for all i 6= k, an*
*d that
Hk(C*) is Z-free. Then Hk(C*) is also a projective Z[G]-module, and
M M M M
Hi(C*) Ci~= Hi(C*) Ci: (1)
i even i odd i odd i even
Proof.We first claim the following: if 0 ! A ! B ! C ! 0 is a short exact seque*
*nce
of finitely generated Z-free Z[G]-modules, and two of the modules A, B, and C a*
*re
projective (stably free), then so is the third. This is clear if C is projectiv*
*e, since in that
case B ~=A C. So assume that A and B are projective (stably free). Since all t*
*hree
groups are Z-free and finitely generated, the dual sequence 0 ! C* ! B* ! A* ! *
*0 is
also exact. Here, for any Z[G]-module M, M* def=HomZ(M; Z) has the obvious stru*
*cture
as a Z[G]-module. Dualization clearly takes finitely generated free Z[G]-module*
*s to free
Z[G]-modules, hence the same for projective modules; and so the dualized sequen*
*ce
splits. Thus B* ~= A* C* as Z[G]-modules; and upon dualizing again we see that
B ~=A C. So C is Z[G]-projective (stably free).
Now fix any m; n 2 Z such that m < k < n, and Ci= 0 for all i < m and all i >*
* n.
For each i, set Zi = Ker[Ci--@! Ci-1] and Bi = Im[Ci+1--@! Ci]. Consider the sh*
*ort
exact sequences
0 --! Zi---! Ci---! Bi-1--! 0 and 0 --! Bi---! Zi---! Hi(C*) --! 0:
By induction starting at i = m, one sees that Zi is projective for each i k, a*
*nd that
Biis projective for each i < k. Similarly, by downward induction starting at i *
*= n + 1,
one sees that Biis projective for each i k, and that Ziis projective for each *
*i > k. In
particular, Bk and Zk are both projective, and so the same holds for Hk(C*).
In particular, theLabove shortLexact sequencesLsplit, since all ofLtheir term*
*s are pro-
jective. Set Cev= (C2i), Cod= (C2i+1), Hev= (H2i(C*)), Hod= (H2i+1(C*));
and similarly for Zev, Zod, Bev, and Bod. Then
Hev Cod~= Hev Bev Zod~= Zev Zod;
Hod Cev~=Hod Bod Zev~=Zod Zev;
*
*__
and this proves (1). |*
*__|
The following property of projective Z[G]-modules is a consequence of a theor*
*em of
Swan.
Proposition C.3. Let P and P 0be any two finitely generated projective Z[G]-mod*
*ules.
Then P Z P 0is stably free as a Z[G]-module.
Proof.Assume first that P is free. Let {ai} be a Z[G]-basis for P , and let {b*
*j} be a
Z-basis for P 0. Then {ai bj} is a Z[G]-basis for P P 0, and this module is fr*
*ee. (Note
that we did not need to know that P 0is projective, only that it is Z-free.)
Bob Oliver and Yoav Segev *
* 59
Now consider the general case. By [Sw , Theorems 7.1 and 8.1], for any n > 0,*
* any
finitely generated projective Z[G]-module contains a free submodule of finite i*
*ndex prime
to n. In particular, we can choose free submodules F P and F 0 P 0, such that *
*[P :F ]
and [P 0:F 0] are finite and relatively prime. Consider the commutative diagram
0 ---! F F 0 -i1--!P F 0 ---! (P=F ) F 0---! 0
? ? ?
j1?y j2?y ff?y~=
0 ---! F P 0 -i2--!P P 0 ---! (P=F ) P 0---! 0;
where all tensor products are taken over Z. The rows are both exact, and ff is*
* an
isomorphism since (P=F ) (P 0=F 0) = 0. So by an easy diagram chase, the seque*
*nce
(i1;j1) 0 0 j2-i2 0
0 ---! F F 0------! (P F ) (F P ) ------! P P ---! 0
is exact. We have just seen that the first two terms in this sequence are free*
*,_and so
P P 0is stably free. |*
*__|
In fact, using stability results of Swan, one can show that the tensor produc*
*t of
any two finitely generated projective Z[G]-modules is free. This is not needed *
*for the
constructions in this paper, but the following much deeper stability result is *
*used. It is
not needed to prove the existence of 2-dimensional acyclic G-complexes, but it *
*is used
in Section 3 to show that all of the complexes we construct can be taken to hav*
*e exactly
one free orbit of 2-cells (and no free orbits of cells in other dimensions).
Proposition C.4. If G is simple, or (more generally) if there is no homomorphism
G ! SU(2) (= SU(2; C)) with nonabelian image, then any stably free Z[G]-module *
*is
free.
Proof.By a theorem of Jacobinski [Jac, Theorem 4.1], if A is any Z-order in a f*
*inite di-
mensional semisimple Q-algebra A which satisfies the Eichler condition, then al*
*l finitely
generated stably free A-modules are free. Here, the algebra A satisfies the Eic*
*hler con-
dition if it has no simple factor B, with center K, for which every embedding K*
* ,! C
has image contained in R and induces an isomorphism R K B ~= H (the quaternion
algebra over R).
If Q[G] does not satisfy the Eichler condition _ if B is a simple summand of *
*Q[G]
and R K A ~=H _ then the composite
proj
Q[G] -----! B -----! H
restricts to a multiplicative homomorphism ff: G ! S3 ~=SU(2; C). Here, S3 deno*
*tes
the group of quaternions of norm 1. And since the image of G in H generates H a*
*s an __
R-vector space, Im(ff) must be nonabelian. See also [Re , x38] for more discuss*
*ion. |__|
Appendix D. Finite simple groups of Lie type
We give here a very short discussion of groups of Lie type. For more detail, *
*we refer
to [St1], [St2], [Ca ], or [GLS ].
60 Fixed point free actions on acyclic 2-complexes
The finite simple groups of Lie type consist of the Chevalley groups and thei*
*r twisted
analogs. The finite Chevalley groups are analogs of the (complex or compact) Li*
*e groups,
but realized over a finite field. They thus include the four families of classi*
*cal groups:
An(q) ~=Ln+1(q) = P SLn+1(q), Bn(q) ~=P 2n+1(q) (the commutator subgroup of the
projective orthogonal group P GO2n+1(q)), Cn(q) ~= P Spn(q), and Dn(q) ~= P +2n*
*(q)
(the commutator subgroup of the projective special orthogonal groups with respe*
*ct to
a quadratic form of "plus type"); as well as the exceptional groups E6(q), E7(q*
*), E8(q),
F4(q) and G2(q). All of these are defined over any finite field; i.e., for any *
*prime power
q.
The finite twisted groups of Lie type were first treated systematically by St*
*einberg
in [St1] and [St2], where (very roughly) they are obtained a s fixed points of *
*certain
automorphisms of the Chevalley groups _ group automorphisms which are associated
with automorphisms of the Dynkin diagram. Let G be one of the symbols An; Bn; C*
*n,
etc. Then mG(q) denotes the fixed subgroup of an automorphism of order m of G(q*
*m )
(or of G(q) when G = B2, G2, or F4). The finite twisted groups thus consist of*
* the
classical groups 2An(q) ~=P SUn+1(q) = Un+1(q) and 2Dn(q) ~=-2n(q) (the commuta*
*tor
subgroup of the projective special orthogonal groups of "minus type"); as well *
*as the
Suzuki groups 2B2(22k+1), the Ree groups 2G2(32k+1) and 2F4(22k+1), and the Ste*
*inberg
groups 2E6(q) and 3D4(q).
To make this more concrete, it is necessary_to work with automorphisms_of the*
* Cheval-
ley groups over the algebraic closure Fp,_where p is prime. Let G(F p) denote_a*
* simple
algebraic group of type G defined over Fp. We will always assume that G(F p) is*
* of ad-
joint type (i.e., with trivial center), or equivalently that it is a group of a*
*utomorphisms
of the corresponding Lie algebra. For q a power of p, the finite Chevalley grou*
*p_G(q)
can (roughly) be thought of as the fixed subgroup of the automorphism 'q of G(F*
* p)
induced_by_the field automorphism (t 7! tq). More generally,_a_Steinberg endomo*
*rphism
of G = G(F p) is_defined to be an algebraic endomorphism of G whose fixed subgr*
*oup
C_G(oe) = {x 2 G | oe(x) = x} is finite. (In fact, the Steinberg endomorphisms *
*are all
__
automorphisms of G as an abstract group, but none of them is invertible as an a*
*lgebraic
endomorphism.) The finite twisted_groups of Lie type are (roughly) the fixed su*
*bgroups
of Steinberg endomorphisms of G, which are field automorphisms (t 7! tq) "twist*
*ed" by
graph automorphisms.
__ __ __
More precisely, if oe is a Steinberg endomorphism of G = G(F p), let_Goedenot*
*e the sub-
group of C_G(oe) generated by its Sylow p-subgroups. Equivalently, Goe= ,
__
where U; V G are subgroups_defined_in the next paragraph. If eGis the universa*
*l cen-
tral extension of G, then Goe~=CGe(oe)=Z, where Z denotes the center. For examp*
*le, if q
is a power of p, then SLn(q) = CSLn(_Fp)('q), while P SLn(q) can be a proper su*
*bgroup
__
of CPSLn(_Fp)('q). For all G and all q = pk, G(q) = G'q.
To describe the Steinberg endomorphisms, we must first establish_notation for*
* certain
elements of the Chevalley groups. Fix a prime p, and let F Fp be any subfield.*
* Set
G = G(F), and let be the system of roots of type G. Let +; - denote the sets
of positive and negative roots, respectively. To each r 2 there corresponds a *
*subgroup
(the root subgroup) Xr = {xr(t) | t 2 F} G, isomorphic to the additive group F*
*. Then
Bob Oliver and Yoav Segev *
* 61
U def= and V def= are_both maximal unipotent subgroups of
G; they are closed and connected if F = Fp, and are Sylow p-subgroups of G if F*
* is
finite._Also,_G = ~~__. The subgroup H def=NG(U)\ NG(V ) is a maximal torus *
*of G
if F = Fp, and is called a Cartan subgroup of G when G is finite. This subgrou*
*p H
is abelian, generated by elements hr(t) for simple roots r and t 2 F*; and its *
*elements
are called "diagonal elements" of G. Also, NG(U) = UH and NG(V ) = V H (the Bor*
*el
subgroups of G).
For example, when G = An(F) ~=Ln+1(F), (of adjoint type), then the roots corr*
*espond
to the pairs (i; j) for i 6= j, and the positive roots correspond to the pairs *
*(i; j) for i < j.
In this case, xij(t) = eij(t), the matrix which has 1's on the diagonal, t in p*
*osition
(i; j), and zeros elsewhere. Thus U and V are the subgroups of (strict) upper a*
*nd lower
triangular matrices, and H is the subgroup of diagonal matrices. Note that whe*
*n we
describe elements and subgroups here in terms of matrices, we mean their images*
* under
the surjection of SLn+1(F) onto Ln+1(F) = P SLn+1(F).
__ __
Let oe be a Steinberg endomorphism of G = G(F p) (still assumed_of adjoint ty*
*pe)._By
the Lang-Steinberg theorem [St2, Theorem 10.1],_for any g 2 G, there exists_h_2*
* G such
that g = oe(h)h-1. Hence, all elements in Inn(G ) Ooe are conjugate in Aut(G_).*
* In other
words, composing_a Steinberg endomorphism oe with an inner automorphism of G, d*
*oes
not change G oe(up to conjugation).
__
Next, Steinberg showed that for any oe, there is some g 2 G such that conj(g*
*) Ooe
leaves U and V invariant, and permutes the root subgroups Xr. It thus suffice*
*s to
consider those oe for which oe(Xr) = Xae(r)for some automorphism ae of the root*
* system
of type G, which preserves the positive roots; i.e., a permutation of which p*
*reserves
angles between the roots, such that ae(+) = +. Hence ae permutes the simple roo*
*ts, and
induces a symmetry of the Dynkin diagram of G. By inspection of the Dynkin diag*
*rams,
one sees that if ae 6= Id, then either G = An, Dn, or E6 and ae is the automorp*
*hism of
order 2 of the root system; or G = D4 and ae is an automorphism of order 3; or *
*G = B2,
F4, or G2 and ae is an automorphism of order 2 which interchanges long and shor*
*t roots.
*
* __
If oe(Xr) = Xae(r)for such ae, then necessarily oe(xr(t)) = xae(r)(fflrtqr) f*
*or some fflr 2 (F p)*
and some qr powers of p. After composing with conjugation by a diagonal element*
*, we
can assume fflr = 1 for all simple roots r (and fflr = 1 for all r). Also, by s*
*tudying the
action of oe on diagonal elements, one can show that the ratio qr.krk=kae(r)k i*
*s constant,
independent of r. In particular, if ae = Id, then oe = 'q (q = qr for all r) i*
*s a field
automorphism.
Assume that ae 6= Id, and that all roots in have the same length. Then oe = *
*'qO ae,
where q = pk >_1_(q = qr for all r); and where ae(xr(t)) = xae(r)(t) for all s*
*imple roots
r and all t 2 Fp (and ae(xr(t)) = xae(r)(t) for arbitrary r). The existence of*
* such an
automorphism aeis shown in [St1, Theorem 29] or [Ca , Proposition 12.2.3]. If *
*m is the
__ def__
order of ae, then oem = 'qm, so G oem= G(qm ), and mG(q) = Goecan be viewed as*
* the
subgroup of CG(qm()o) generated by its Sylow-p subgroups, where o is the restri*
*ction
of oe to G(qm ). In other words, we can regard mG(q) = G(qm )o, where o is the*
* field
automorphism (t 7! tq) "twisted" by the "graph automorphism" of G(qm ).
62 Fixed point free actions on acyclic 2-complexes
As one example, consider the automorphism o(aij) = (((-1)i+jan+2-j;n+2-i)q)-1*
* of
Ln+1(q2). This preserves upper and lower triangular matrices, and sends xij(t)*
* to
xn+2-j;n+2-i(tq). The signs have been chosen so that o(xr(t)) = xae(r)(tq) whe*
*n r is
a simple root (i; i + 1) (but not for all roots). Then 2An(q) def=(Ln+1(q2))o =*
* P SUn+1(q)
is the projectivePspecial unitary group defined with respect to the hermitian f*
*orm
(x; y) = u. (-1)i+1xi(yn+2-i)q on (Fq2)n+1 (where u = 1 if n is even and uq-1*
*= -1 if
n is odd). Note that there can be elements of P SLn+1(q2) fixed by o which are *
*not rep-
resented by unitary matrices, which is why one must define 2An(q) = . If
one works in the universal central extension SLn+1(q2) , the subgroup of elemen*
*ts fixed
by o is SUn+1(q).
If has roots of distinct lengths and ae is nontrivial, then as mentioned abo*
*ve G = B2,
F4, or G2 and ae interchanges long and short roots. Set p0 = 2 if G = B2; F4 an*
*d p0 = 3
if G = G2, so that kae(r)k_krk= (p0)1=2 for each r 2 . Since qr._krk_kae(r)kis *
*independent of r
(and the qr all powers of p), this is possible only if p = p0. Hence, oe = 'qO *
*aefor some
q = pk 1, where
(
xae(r)(tp)if r is a short root
ae(xr(t)) =
xae(r)(t)if r is a long root.
__ def__
Then oe2 = 'q2p, so G oe2= G(q2p) = G(p2k+1), and 2G(p2k+1) = Goecan be regard*
*ed
as the fixed subgroup of an involution on G(p2k+1). This group is sometimes den*
*oted
2G(pk+1_2).
As an example, Ono [On ] carried out this procedure on Sp4(22k+1) = B2(22k+1),
regarded as the group of 4x 4 matrices which preserve the symplectic form (x; y*
*) =
x1y4 + x2y3 + x3y2 + x4y1. He obtained precisely the matrix presentation of Sz(*
*22k+1)
described in Proposition 3.6, as the fixed points (Sp4(22k+1))o, where o is the*
* restriction
of the above oe = 'qO aeto Sp4(22k+1).
__ __
The rank of a Chevalley group G(q) is just the rank of G = G(F p) in the usu*
*al
sense; i.e., the number of simple roots in its root system, or the number of no*
*des in its
Dynkin diagram. The rank of a twisted group mG(q) is equal to the number of orb*
*its
of roots (or of nodes) under the corresponding automorphism of the root system *
*or the
Dynkin diagram of G. There are thus four families of finite simple groups of Li*
*e type
and Lie rank 1: the two dimensional projective special linear groups L2(q) ~=A1*
*(q), the
three dimensional projective special unitary groups U3(q) ~=2A2(q), the Suzuki *
*groups
Sz(q) ~=2B2(22k+1), and the Ree groups Ree(32k+1) ~=2G2(32k+1).
We now return to the internal structure of the groups of Lie type. First let *
*G = G(F )
be a Chevalley group over any field F , and let be a root system of type G. We*
* have
already discussed the root subgroups Xr = {xr(t) | t 2 F } for each root r 2 , *
*and
the subgroups U = and V = . For each root r, there is
a surjection OEr:SL2(F ) i which sends 10t1 to xr(t) and 1t01 to x-*
*r(t).
This allows the definition of elements hr() = OEr 0 -01 and nr = OEr -0110. T*
*he
elements_hr(), for r 2 and 2 F *, generate the subgroup H of diagonal elements
of G , and together with the nr they generate the subgroup N = of
monomial elements. Then N=H ~= W , the Weyl group of G (and of its root system),
and B def=____ = NG(U) is the Borel subgroup of G.
Bob Oliver and Yoav Segev *
* 63
__ __ __
Now set G = G(F p), and let oe be a Steinberg morphism of G . Set Uoe= CU(o*
*e)
and Voe= CV (oe), the subgroups of elements fixed by oe, and let bG= *
*be the
corresponding group of Lie type. Set Hb = CH (oe) \ bG, Nb = CN (oe) \ bG, and*
* Bb =
CB (oe) \ bG. Let ae be the automorphism of the root system associated to oe,*
* as
described earlier. In particular ae permutes the positive roots, and hence the*
* simple
roots. By a root (or simple root) of bGis meant a ae-orbit ^r (or ae-orbit of *
*simple
roots). Note that if ae = Id, then bGis an (untwisted) Chevalley group, and its*
* roots are
the roots in the usual sense. We write b = =ae for the set of roots, -^r= {-r |*
* r 2 ^r};
and (when J b) for the set of ae-orbits of roots which are linear combinat*
*ions of
elementsQr 2 ^r2 J. The root subgroup X^rcorresponding to an orbit ^ris the sub*
*group
( r2^rXr)oeof oe-invariant elements. The Weyl group of bGis the group cW = bN=*
*Hb; or
equivalently the subgroup of W = N=H of elements which commute with oe (cf. [Ca*
* ,
Proposition 13.5.2]) when both are considered as groups of permutations of the *
*roots
(or of the real vector space generated by the roots). The Weyl group is generat*
*ed by
elements w^sof order two, one for each ae-orbit ^sof simple roots, where the w^*
*s-action on
sends s to - s for all s 2 ^s. The root subgroups of bGare discussed in detail*
* in [Ca ,
Proposition 13.6.3] and [GLS , Table 2.4]; in particular, they need not be abel*
*ian. The
Weyl groups of the twisted groups are described in [Ca , x13.3]; each is isomor*
*phic to
that of some Chevalley group except when bG= 2F4(22k+1), in which case cW is di*
*hedral
of order 16.
For notational convenience we now drop the "hat" from our notation for the fi*
*nite
simple groups of Lie type of the previous paragraph. Thus from now through the *
*end
of Appendix D, G = bG, U = bU, etc. Also, we'll abuse notation and write r = ^r*
*for a
ae-orbit in .
Tits has axiomatized the properties of the pairs (B; N) in groups of Lie type*
*. These
permit, for example, uniform proofs of the simplicity of these groups in all ca*
*ses where
they are simple. See, e.g., [Ca , x8.2] or [GLS , x1.11] for more detail about *
*such BN-pairs.
By definition, any group of Lie type is generated by its root subgroups (for *
*a given
choice of root system). In fact, it suffices to take the simple roots.
Lemma D.1. Let G be a finite simple group of Lie type, with root system . The*
*n G
is generated by the root subgroups Xs and X-s for simple roots s 2 +.
Proof.See [Ca , Proposition 13.6.5]. Very briefly, when G is a Chevalley group,*
* this holds
since conjugation by elements of N (or of W = N=H) permutes the root subgroups *
*in
the same way as the Weyl group permutes the roots, and each root is in the W -o*
*rbit of
a simple root. Since N=H is generated by the elements ns 2 for simple*
* roots
s, this shows that contains all of the Xr for r 2 , and henc*
*e_is_all
of G. The same argument works for the twisted groups. *
*|__|
We now turn attention to parabolic subgroups: proper subgroups of G which con*
*tain
a Borel subgroup. For convenience, set B0 = V H (and B = UH as usual). Let be
the root system corresponding to G. For each proper subset J of simple roots of*
* G, let
be as defined above, and set
PJ = __** = ****> and PJ0= = >:
64 Fixed point free actions on acyclic 2-complexes
By [Ca , Theorem 8.3.2], these are precisely the overgroups of B in G (i.e., th*
*e parabolic
subgroups containing B).
Lemma D.2. Let G be a finite simple group of Lie type. Let be the root syst*
*em
associated with G, and let + and - be the sets of positive and negative roots. *
*Fix
a set J of simple roots which does not contain all of them, and let LJ be the s*
*ubgroup
generated by the diagonal subgroup H together with the root subgroups Xr for all
r 2 . Let UJ and VJ be the subgroups generated by all Xr for roots r 2 + or *
*r 2 -,
respectively, which are not in . Then UJ C PJ = UJLJ and VJ C PJ0= VJLJ, UJ
and VJ are nilpotent, and = G.
Proof.When G is a Chevalley group, the nilpotency of U UJ and V VJ follows fr*
*om
[Ca , Theorem 5.3.3], and LJ normalizes UJ and VJ by [Ca , Theorem 8.5.2]. Bot*
*h of
these are consequences of Chevalley's commutator formula, which says that for a*
*ny pair
of roots r; s 2 , [Xr; Xs] is generated by the subgroups Xtfor all roots t = ir*
*+js where
i; j > 0. The twisted group case follows immediately by restriction. And PJ = U*
*JLJ
and PJ0= VJLJ since U and V are generated by their root subgroups: by definitio*
*n when
G is a Chevalley group, and by [Ca , Proposition 13.6.1] when G is a twisted gr*
*oup.
This also shows that = **__ = G. Thus C G, since LJ*
* __
normalizes UJ and VJ; and so G = since G is simple. *
* |__|
The decomposition PJ = UJLJ of Lemma D.2 is called the Levi decomposition of *
*PJ,
and LJ is called the Levi subgroup.
We now return to looking at group actions on 2-dimensional acyclic complexes.
Lemma D.3. Let G be a finite simple group of Lie type, and let P $ G be one o*
*f the
parabolic subgroups PJ or PJ0of Lemma D.2. Then for any action of G on an acycl*
*ic
2-complex X, XP 6= ;.
Proof.We can assume XG = ;. By Lemma D.2, there are subgroups UJ C PJ, VJ C PJ0,
and LJ = PJ \ PJ0, such that UJ and VJ are nilpotent, PJ = UJLJ, PJ0= VJLJ, and
= G. In particular, XUJ and XVJ are acyclic, disjoint, and LJ-invarian*
*t. Then
XLJ 6= ; by Corollary 4.2, applied to the0action of LJ on X with invariant subs*
*paces
A = XUJ and B = XVJ; and so XPJ and XPJ are nonempty by Lemma 4.3(a).
To see this more directly, let Y be the complex obtained by collapsing XUJ an*
*d XVJ
to separate points. Then Y is still acyclic, LJ acts on Y , and Y LJcontains at*
* least the
two collapse points. Thus, Y LJis acyclic by Theorem 4.1, is in particular con*
*nected,
and hence XLJ must intersect0with both subcomplexes XUJ and XVJ. It follows tha*
*t_
XPJ = XLJ \ XUJ 6= ; and XPJ = XLJ \ XVJ 6= ;. |__|
Appendix E. The four-subgroup criterion
In [S1] and [AS ], very strong restrictions were placed on the finite simple *
*groups
which could possibly have actions on 2-dimensional acyclic complexes without fi*
*xed
points. The main tool for doing this was a "four subgroup criterion", which fo*
*r the
sake of completeness we present here as Proposition E.1. To illustrate its use,*
* we then
Bob Oliver and Yoav Segev *
* 65
describe how it was applied to certain multiply transitive groups, and to simpl*
*e groups
of Lie type and Lie rank at least two _ those cases of the proof of Theorem A w*
*hich
were not dealt with in Sections 6 and 7.
Proposition E.1 [S1, Theorem 3.2].Fix a finite group G and a 2-dimensional acyc*
*lic
G-complex X. Let H1; H2; H3; H4 G be subgroups such that X__6= ; for a*
*ny
i; j; k. Then X# 6= ;.
Proof.Assume otherwise: that X# = ;. Set H = {H1; H2; H3; H4}. By Th*
*eo-
rem 4.1, XH is the union of the acyclic subcomplexes XHi, which have the proper*
*ty that
any two or three of them have acyclic intersection, but the four have empty int*
*ersection.
This implies that H2(XH ) ~=H2(S2) ~=Z (see Lemma 0.1, applied using the poset S
of nonempty proper subsets of {1; 2; 3; 4}). But this is impossible, since XH *
*_must_be
homologically 1-dimensional by Lemma 1.6. |*
*__|
The simplest application of Proposition E.1 is to multiply transitive groups.
Corollary E.2. Assume that G acts 4-transitively on a set S with point stabiliz*
*er
H G. Let X be a 2-dimensional acyclic G-complex such that XH 6= ;. Then XG 6= *
*;.
Proof.If |S| = 4, then this follows from Theorem B. So assume |S| 5, and fix f*
*our
elements s1; s2; s3; s4 2 S. For each i = 1; 2; 3; 4, let Hi G be the subgroup *
*of elements
which fix sj for all j 6= i. For each {i; j; k; r} = {1; 2; 3; 4}, is the point
stabilizer of sr, and hence fixes a point in X by assumption. So XG 6= ; by Pro*
*position_
E.1. |__|
This is now applied to the alternating groups, as well as most of the Mathieu*
* groups.
Proposition E.3 [S1, 3.6].If G ~=An for n 6, or if G is one of the Mathieu gro*
*ups
M11, or M12, then every G-action on an acyclic 2-complex has fixed points. The *
*same
holds for M23and M24if it holds for M22.
Proof.Let X be a 2-dimensional acyclic G-complex. If G = An for n 6, then by
Corollary E.2, XG 6= ; if XAn-1 6= ;. By Proposition 6.4 above, A6 ~=L2(9) must*
* have
nonempty fixed point set, and the result now follows by induction on n.
Each of the simple Mathieu groups Mn for n = 11; 12; 23; 24 acts 4-transitive*
*ly on a
set with point stabilizer Mn-1 (cf. [A3 , 18.9-10 & 19.4], [Gr , 5.33 & 6.18],*
* [Mat ], or
[Wt ]). So by Corollary E.2, the proposition holds for Mn if it holds for Mn-1.*
* Since M10
contains a subgroup A6 of index 2, this proves the proposition when n = 11 or 1*
*2; and __
it will follow for the other simple Mathieu groups once it has been shown for M*
*22. |__|
Proposition E.1 can also be applied to simple groups of Lie type of Lie rank *
*at least
two. In this case, the subgroups in question come from the root system of the g*
*roup.
Note that the following proof applies only to groups of Lie type which are them*
*selves
simple. The Tits group 2F4(2)0, which has index two in 2F4(2), is dealt with h*
*ere in
Proposition 7.2, as well as in [AS , 5.2].
Proposition E.4 [AS , x5].If G is a simple group of Lie type and Lie rank at le*
*ast 2,
then every G-action on an acyclic 2-complex has fixed points.
66 Fixed point free actions on acyclic 2-complexes
Proof.We use the notation of Lemma D.2. Fix a root system = + q - for G, and
let J1q J2 be a decomposition of the set of simple roots as a disjoint union of*
* nonempty
subsets. For each i = 1; 2, set
H+i= and H-i= :
The subgroup generated by any three of the Hi is contained in one of the parabo*
*lic
subgroups PJior PJ0i(in the notation of Lemma D.2), and hence has nonempty fixed
point set in X by Lemma D.3. But # = G by Lemma D.1, since it contains *
*all __
subgroups Xs and X-s for simple roots s, and hence XG 6= ; by Proposition E.1. *
* |__|
List of notation:
Groups:_
Cm : a cyclic group of order m
D2m: a dihedral group of order 2m
An: the alternating group on n letters
n: the symmetric group on n letters
P GLn(q) = GLn(q)=(center): the projective general linear group over Fq
Ln(q) = P SLn(q): the projective special linear group over Fq
P GUn(q): the projective general unitary group over Fq2
Un(q) = P SUn(q): the projective special unitary group over Fq2
Topological_spaces:_
I = [0; 1]: the unit interval
fi
Dn = x 2 Rn fikxk 1 : the unit ball in Rn
fi
Sn = x 2 Rn+1fikxk = 1 : the unit sphere in Rn+1
X ~=Y : X and Y are homeomorphic
X ' Y : X and Y are homotopy equivalent
X ' *: X is contractible
H*(X) def=H*(X; Z)
Acyclic_means Z-acyclic: X is acyclic iff H*(X; Z) ~=H*(pt; Z)
Families_and_sets_of_subgroups_of_G:
S(G): the family of all subgroups of G
(H): the conjugacy class of H G
F S(G) is a family_() H 2 F implies (H) F
SLV (G): the family of solvable subgroups of G
MAX (G): the maximal separating family of subgroups of G
(G; F) 2 U2 () 9 a 2-dimensional Z-acyclic (G; F)-complex
Bob Oliver and Yoav Segev *
* 67
For_any_families_F,_F0_of_subgroups_of_G:
Fmax: the set of maximal subgroups of F
FH = {K 2 F | K H} 8H G
F>H = {K 2 F | K % H} 8H G
FH= {K 2 F | H $ K $ M} 8H $ M G
FH = {K 2 F | K H, some H 2 H} 8H S(G)
fi fi
F[n]= H 2 F finfi|H| 8n > 1
F^ F0 = {H \ H0| H 2 F; H02 F0}
H 2 F is critical_in F () N (F>H ) 6' *
Fc = {H 2 F | H critical inF}
If_X_is_a_G-complex:_
Gx = {g 2 G | gx = x} 8x 2 X
X is a (G; F)-complex () Gx 2 F 8x 2 X
XH = {x 2 X | hx = x 8h 2 H}: the fixed point set
S
X>H = {x 2 X | Gx % H} = K%H XK
S
XH = H2HXH 8H S(G)
S fi
X[n]= n||H|XH = {x 2 X fi|Gx 2 nZ} 8n > 1
S -1
X(H)= g2GXgHg
S
Xs = 16=HGXH = {x 2 X | Gx 6= 1}: the "singular set" of X
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Bob Oliver Yoav Segev
UMR de Mathematiques Department of Mathematics
Universite Paris Nord Ben Gurion University
93430 Villetaneuse, France Beer Sheva 84105, Isr*
*ael
bob@math.univ-paris13.fr yoavs@math.bgu.ac.il