RATIONAL SO(3)-EQUIVARIANT COHOMOLOGY THEORIES.
J.P.C.GREENLEES
Abstract. We apply the results of [4, 2] to give an explicit algebraic mo*
*del of the category
of rational SO(3)-spectra. This gives a complete classification of ration*
*al SO(3)-equivariant
cohomology theories.
For J.P.May on his 60th birthday.
Contents
1. Introduction. *
* 1
2. The closed subgroups of SO(3). *
* 2
3. Burnside splitting. *
* 5
4. Stable isotropy groups. *
* 6
5. Free G-spectra. *
* 7
6. Reduction to normalizers. *
* 8
7. The algebraic model of the category of (G; c)-spectra. *
* 11
8. The fibres. *
* 13
9. Mackey functors. *
* 15
10. The ordinary part of the model. *
* 17
11. Reduction to the Weyl group. *
* 19
References *
*20
1.Introduction.
This paper presents an algebraic model for SO(3)-equivariant rational cohomol*
*ogy the-
ories. The main mathematical input is the model for SO(2)-equivariant theories*
* given in
Part I of [4], and the analysis of O(2)-equivariant cohomology theories in [3],*
* together with
special cases of results on rational Mackey functors from [2]. Using these ingr*
*edients, and
the usual formal framework of equivariant homotopy, it is not hard to assemble *
*the model.
Nonetheless, new phenomena occur, and it is worth illustrating them in a case w*
*here we can
be completely explicit.
The new feature of SO(3) is that it is semisimple. Thus the maximal torus is *
*no longer
normal and there is significant extra complication due to conjugacy. An exactly*
* analagous
model can be given with SU(2) as the group of equivariance. For groups of large*
*r rank one
___________
1This paper is in final form, and no substantially similar paper will be subm*
*itted elsewhere.
21991 Mathematics Subject Classification:
Primary 55N91
Secondary 55P42, 55P91.
1
2 J.P.C.GREENLEES
expects a similar reduction to the case of a torus and equivariant sheaves over*
* spaces of
subgroups, however one cannot expect such an explicit analysis.
Before proceeding, we should explain that the cohomology theories we refer to*
* are those
admitting suspension isomorphisms for arbitrary representations, often known as*
* `RO(G)-
graded' or `genuine' cohomology theories. These are represented by G-spectra as*
* in [6], and
more specifically by those indexed on a complete G-universe. The advantages of *
*consider-
ing representing objects rather than the cohomology theories are well understoo*
*d, and we
work with G-spectra without further comment, or more precisely in the homotopy *
*category
of G-spectra, in the sense of Quillen. We say that a cohomology theory is rati*
*onal if its
values are graded rational vector spaces. A spectrum is rational if the cohomol*
*ogy theory
it represents is rational. It suffices to check the values on the homogeneous s*
*paces G=H for
closed subgroups H, since all spaces are built from these. We work exclusively *
*with rational
cohomology theories, and rational G-spectra, without always displaying this in *
*the nota-
tion. Thus G-spectra denotes the homotopy category of rational G-spectra, and *
*[X; Y ]G
denotes the rational vector space of morphisms in this category. The G-spectrum*
* represent-
ing the equivariant cohomology theory YG*(.) is denoted Y and the correspondenc*
*e is given
by YG*(X) = [X; Y ]*Gin the usual way.
Convention 1.1. Henceforth all spaces, groups and spectra are rationalized whet*
*her or not
this is indicated in the notation.
We will outline the nature of the model at the end of Section 2 when we have *
*summarized
the subgroup structure of SO(3) and introduced appropriate terminology.
2. The closed subgroups of SO(3).
We take G = SO(3) to be the group of rotations of R3. We need to describe the*
* space
of closed subgroups H of G. For SO(3), the conjugacy class of a subgroup H is d*
*etermined
by the isomorphism type of H; quite generally the G-space of conjugacy classes *
*of H is
(H)G ~=G=NG(H).
We may choose a maximal torus T = SO(2) consisting of rotations around the z-*
*axis, and
hence its normalizer N = O(2), which includes all half turns with axis in the x*
*y-plane. Any
positive dimensional proper subgroup of G is conjugate to N = O(2) or T = SO(2).
Notation 2.1. For the rest of the paper we let G = SO(3), N = O(2); T = SO(2) a*
*nd
W = O(2)=SO(2).
There are three exceptional conjugacy classes of finite subgroups: the symmet*
*ry group A5
of the dodecahedron, the symmetry group 4 of the cube and the symmetry group A4*
* of the
tetrahedron. We choose the cube to have vertices (1; 1; 1), the dodecahedron to*
* have
these amongst its vertices and the tetrahedron to have its vertices at vertices*
* of the cube.
The group A5 is maximal and equal to its own normalizer. Hence its space of con*
*jugates is
the rational homology 3-sphere G=A5. The cubical group 4 is equal to its own no*
*rmalizer,
and its space of conjugates is the rational homology 3-sphere G=4. The normaliz*
*er of the
tetrahedral group A4 is the cubical group 4, and its space of conjugates is the*
* rational
homology 3-sphere G=4.
All other finite subgroups of G are conjugate to subgroups of N = O(2). The*
* finite
subgroups of O(2) are the finite cyclic groups Cn for n 1, which are normal in*
* O(2) and
RATIONAL SO(3)-EQUIVARIANT COHOMOLOGY THEORIES. 3
the finite dihedral groups. All the dihedral groups of order 2n are conjugate t*
*o the standard
one D2n, which we take to be generated by Cn and reflection in the x-axis. The *
*normalizer
of D2n in O(2) is D4n. Since O(2) is maximal in G, and Cn is not normal in G fo*
*r n 2 the
normalizer of Cn in G is O(2) for n 2, and the space of conjugacy classes is G*
*=N (a copy
of RP 2). Since the normalizer in G of D2n for n 3 must preserve the axis of a*
* rotation of
order n, it must lie in O(2) and hence it remains D4n.
The remaining two dihedral subgroups are exceptional. The normalizer of D4 in*
* G is the
cubical group 4. The group D2 is conjugate in G to C2, and the normalizer of D2*
* in G is
a conjugate of O(2).
We divide these subgroups up into the exceptional conjugacy classes
E = (G) [ (A5) [ (4) [ (A4) [ (D4);
the cyclic conjugacy classes [
C = (Cn) [ (T );
n1
and the typical dihedral conjugacy classes
[
D0= (D2n) [ (N):
n3
Note that the dihedral groups D2 and D4 are excluded from D0, and that the cycl*
*ic group C1
is somewhat exceptional in that its normalizer is G rather than N. These sets o*
*f conjugacy
classes can all be given various natural topologies: the relevant one for us is*
* the f topology
of [2]. This makes each of E and C the topological sum of the conjugacy classes*
* G=NG(H) it
contains. However
D0= G xN D0N
where D0Nis the set of dihedral subgroups of N of order 6. The N-space D0Nis t*
*opologized
so that the conjugacy class N=D4n of D2n in N is the circle S(1=n) of radius 1=*
*n centred at
the origin and the conjugacy class of N is the origin itself. The space of all *
*closed subgroups
with the f topology of [2] is the topological sum
SfG = C q D0q E:
Note that G acts continuously on SfG by conjugation.
We say a subgroup is topologically isolated if its conjugacy class is isolate*
*d in the f-
topology. All subgroups of SO(3) except those conjugate to N are topologically*
* isolated.
We say a subgroup H is torally isolated if WG(H) is finite and there is no subg*
*roup K x H
with H=K a torus. The set of torally isolated subgroups of SO(3) is D0q E. Fi*
*nally, a
subgroup is totally isolated if it is both torally and topologically isolated.
Roughly speaking, our description of G-spectra will be in terms of its geomet*
*ric fixed
points H X, viewed as the stalks of a sheaf on SfG. We say that X is lies over *
*K if these
stalks are only non-trivial when K G H. From this point of view, it is natural *
*to separate
behaviour over the terms C, D0and E. To describe the behaviour, we first recall*
* the analysis
for N given in [3]. The space of all closed subgroups of N is the topological s*
*um
SfN = CN q DN ;
where
CN = {Cn | n 1} [ {T }
4 J.P.C.GREENLEES
is the discrete space of cyclic subgroups of N, and
[
DN = (D2n)N [ {N}
n1
is the space of all dihedral subgroups of N. Thus
DN = D0N[ (D2)N [ (D4)N :
The part of an N-spectrum over DN is described by an N-equivariant sheaf of gra*
*ded rational
vector spaces on DN . The part over CN is given by a T -spectrum with homotopy *
*W -action,
which can in turn be described in algebraic terms as in [4].
Returning to G, the part of the model over D0qE splits as a product of Eilenb*
*erg-MacLane
spectra. Over one of these subgroups, the stalk is simply a graded rational WG(*
*H)-module
(Section 8). Since N is the only accumulation point, it is easy to be explicit*
* about the
algebraic nature of such a sheaf as in [3]. This analysis is carried out in Sec*
*tions 9 and 10.
The other conjugacy classes are the finite cyclic groups and T : this is the *
*interesting part
of the model. Its form is exactly similar to the form of the model for O(2), bu*
*t the forgetful
map from G-equivariance to N-equivariance is not an isomorphism. This is becau*
*se the
finite subgroups are normal in N but not in G, and because D2 and C2 are conjug*
*ate in G.
Instead, passage to geometric fixed points gives an isomorphism
[X; Y ]G -! [H X; H Y ]N=H
when X and Y lie over H. For H = T this identifies spectra over T with graded W*
* -modules.
If H = C is a finite cyclic group then N=C ~=O(2), so the right hand side is kn*
*own. If C
is a non-trivial finite cyclic group the group N=C is the Weyl group WG(C), but*
* if C = 1
reduction to the Weyl group WG(1) = G would give no information, and a quite se*
*parate
argument shows that restriction from G-equivariance to N-equivariance is an iso*
*morphism
for free G-spectra.
The decomposition of SfG gives rise to a corresponding equivalence
G-spectra ' (G; C)-spectra x (G; D0)-spectra x (G; E)-spectra;
and the main theorem gives algebraic models for each of the factors.
Theorem 2.2. The homotopy category of rational G-spectra is described by the eq*
*uivalence
G-spectra ' D(As(G; c)) x D(SheavesG(D0)) x D(SheavesG(E))
of triangulated categories. Here As(G; c) is a certain abelian category of inje*
*ctive dimension
1 to be described in Section 7, for a G-space X, SheavesG(X) denotes the catego*
*ry of G-
equivariant sheaves of vector spaces on X, and for an abelian category A, D(A) *
*denotes its
derived category. Since E is a topological sum of orbits, there is a further eq*
*uivalence
Y
SheavesG(E) = Q[WG(E)]-modules
(E)
where the product is over conjugacy classes of exceptional subgroups.
The paper is layed out as follows. First, Sections 3 and 4 recall basic pheno*
*mena: idem-
potents in the Burnside ring enable us to separate the isotropy groups. We the*
*n give the
analysis of free G-spectra in Section 5, since this is the only completely new *
*ingredient. The
next general tool is reduction to the normalizer in Section 6. This provides al*
*l the ingredients
to finish the analysis of G-spectra concentrated over subgroups of SO(2): we co*
*uld simply
RATIONAL SO(3)-EQUIVARIANT COHOMOLOGY THEORIES. 5
transpose the method of [4], but instead we identify this category as a subcate*
*gory of the
algebraic model of O(2)-spectra identified in [3]. This analysis occupies Sect*
*ions 7 and 8.
We then turn to the complementary set of spectra in Sections 9 and 10, showing *
*that they
split as a product of Eilenberg-MacLane spectra. Finally, in Section 11 we incl*
*ude a section
proving that geometric fixed points give a reduction to the Weyl group: this is*
* not necessary
to the analysis, but nonetheless is illuminating.
3. Burnside splitting.
A characteristic feature of the stable setting is the (rationalized) Burnside*
* ring
A(G) = [S0; S0]G:
Since S0 is a unit object, this acts on all rational G-spectra. For each non-tr*
*ivial idempotent
e 2 A(G) we may consider the full subcategory (G; e)-spectra of rational G-spec*
*tra of the
form eX, and any orthogonal decomposition
Xn
1 = ei
i=1
gives a corresponding decomposition
Yn
G-spectra ' (G; ei)-spectra:
i=1
Let FG denote the subspace of SfG consisting of subgroups of finite index in *
*their nor-
malizers. Consider the equivalence relation ~ on SfG in which any pair of subgr*
*oups H x ^H
with H^=H a torus are ~-related; it turns out that each conjugacy class in this*
* quotient is
~=
represented by a unique conjugacy class of elements of FG so that FG=G -! (SfG*
*= ~)=G.
This is easily checked directly in the present case. We have
FG=G = {(G); (A5); (4); (A4); (D4)} [ {(D2n) | n 3} [ {(N); (T )};
which is compact and totally disconnected with (N) as the only limit point.
For any stable G-map f : S0 -! S0, the function
OE(f) : SfG -! Q
defined by OE(f)(H) = deg(H f) is, continuous and equivariant. Furthermore if H*
* x ^Hwith
H^=H a torus we have OE(f)(H) = OE(f)(H^) so OE passes to the quotient to give *
*tom Dieck's
isomorphism
~= G G
OE : A(G) = [S0; S0]G -! C (SfG= ~; Q) = C (FG; Q);
where CG (X; Q) denotes continuous equivariant functions f : X - ! Q, and Q has*
* the
discrete topology. Thus an idempotent e of A(G) corresponds to the open and cl*
*osed G-
invariant subspaces
Supp(e) = {H | e(H) = 1}
of SfG which are unions of ~-equivalence classes.
It is appropriate to let c be the idempotent supported on C, and we let m = 1*
* - c denote
the complementary idempotent. This gives a correseponding decomposition
G-spectra ' (G; c)-spectra x (G; m)-spectra:
6 J.P.C.GREENLEES
The letter m stands for Mackey, since it turns out that (G; m)-spectra split as*
* products of
Eilenberg-MacLane spectra and are therefore given by graded Mackey functors on *
*E q D0.
Furthermore, each topologically isolated conjugacy class (H)G with NG(H)=H fi*
*nite gives
rise to an idempotent eH 2 A(G). If E+ is any finite collection of conjugacy c*
*lasses from
D0[ E not containing (N), we have an orthogonal decomposition
X
1 = c + m- + eH ;
(H)2E+
where Supp(m-) = D0q E \ E+. This gives a corresponding decomposition
Y
G-spectra ' (G; c)-spectra x (G; m-)-spectra x (G; eH )-spectra:
(H)2E+
In the special case E+ = E we let d0denote the idempotent with support D0and e *
*denote the
idempotent with support E. We give an analysis of (G; eH )-spectra for a total*
*ly isolated
conjugacy class (H) in Section 8, and an analysis of all (G; m)-spectra in Sect*
*ions 9 and
10. The analysis of (G; c)-spectra occupies Section 7 and 8.
4. Stable isotropy groups.
Since our analysis is based on geometric fixed points we introduce some conve*
*nient ter-
minology. For a G-spectrum X we say that H is a stable isotropy group if H X i*
*s non-
equivariantly essential, and we let SI(X) denote the set of stable isotropy gro*
*ups. This set
is closed under conjugation, but not under passage to subgroups or supergroups.
Example 4.1. If F is a family of subgroups we let EF denote the universal F-spa*
*ce, and
EeF denote its unreduced suspension. The defining properties of the universal s*
*pace give
o SI(EF+) = F and
o SI(EeF) = All\ F:
If e 2 A(G) is idempotent then
o SI(eS0) = Supp(e):
We record some basic properties.
Lemma 4.2. (i) The set SI(X) is empty if and only if X ' *.
(ii) If X is built from cells G=K+ with K 2 F then SI(X) consists of subconjuga*
*tes of F.
(iii) If X is rational and e 2 A(G) is an idempotent with X ' eX then SI(X) Su*
*pp(e).
(iv) For any spectra X and Y
SI(X _ Y ) = SI(X) [ SI(Y )
and
SI(X ^ Y ) SI(X) \ SI(Y ):
(v) If X -! Y - ! Z is a cofibre sequence then
SI(Z) SI(X) [ SI(Y ):
RATIONAL SO(3)-EQUIVARIANT COHOMOLOGY THEORIES. 7
It is useful to connect stable isotropy groups to the more familiar notion of*
* an F-spectrum
when F is closed under passage to subgroups. An F-spectrum is one constructed f*
*rom cells
G=K+ with K in F.
Lemma 4.3. If F is closed under passage to subgroups and SI(X) F then X is equ*
*ivalent
to an F-spectrum.
Proof: Use the usual procedure for killing homotopy groups.
For any set H of subgroups of G we let (G; H)-spectra denote the full subcate*
*gory of
G-spectra with stable isotropy in H. Thus for example (G; 1)-spectra are free G*
*-spectra and
(G; G)-spectra X have the property that
X ' X ^ eEP ' GX ^ eEP;
where P is the family of proper subgroups of G. It follows that the geometric f*
*ixed point
functor gives an equivalence (G; G)-spectra ' 1-spectra.
Note that if e is a non-trivial idempotent (i.e. e 6= 0 or 1), (G; Supp(e))-s*
*pectra is another
name for the category of (G; e)-spectra.
5.Free G-spectra.
In this section and the next we consider conditions under which the restricti*
*on map
[X; Y ]G -! [X; Y ]N
is an isomorphism. The type of conditions we have in mind are restrictions on t*
*he isotropy
groups of the spectra X and Y . In this section we deal with the case of free s*
*pectra, which
is rather different from the case when X has isotropy groups with normalizer eq*
*ual to N.
We concentrate on the case in hand, when G = SO(3) and N = O(2), but the method*
* is
quite general.
Theorem 5.1. If X is a free G-spectrum then the forgetful map
U : [X; Y ]G*-! [X; Y ]N*
is an isomorphism.
Proof: Since both sides are cohomology theories in X, it suffices to consider t*
*he case X =
G=1+. In this case we have a diagram
~=
[G=1+;OYO]G___//_[S0;OYO]1
Uooooooo~=indG| |
| 1 |=
wwooooi* | ~= |
[G=1+; Y ]N___//_[N=1+; Y_]N_//_[S0; Y ]1:
Since i*O U O indG1(f) = f, it follows that the forgetful map U is an isomorphi*
*sm if and only
if i* is an isomorphism. Thus it suffices to show that for the N-space Q = (G+)*
*=(N+)
[Q; Y ]N*= 0
for all G-spaces Y .
A general approach might use a transfer argument here, but we give an explici*
*t analysis.
In fact it is simpler to begin with the periodic group eG= SU(2) and pass to th*
*e quotient
by its centre.
8 J.P.C.GREENLEES
We extend the use of a tilde, so that eNis the normalizer of the maximal toru*
*s in eG, and
we consider the quotient space eQ= (Ge+)=(Ne+) as a left and right eN-space.
Lemma 5.2. (i) As a eG-space, eG+= S(V )+ where V is the natural 2-dimensional *
*complex
representation of eG.
(ii) As an eN-space
eG+= eN+[ eN+^ e1 [ eN+^ e2:
Proof: The space S(V ) is free, so the orbit of each point is G=1. Since S(V ) *
*is a connected
3-manifold, the orbit is S(V ). This proves Part (i), and Part (ii) follows eas*
*ily.
To describe the attaching map we observe that passage to homology gives an is*
*omorphism
Ne ~=
[Ne+; eN+] -! Hom ZW (H0(Ne+); H0(Ne+)):
The stable maps on the left are realized unstably after one suspension. A basis*
* for this set
of maps is given by the identity and right multiplicaton Rt where t is any elem*
*ent of eNnot
in the identity component.
The attaching maps in eG+may be taken to be 1 - Rt and 1 + Rt.
Corollary 5.3. There is a cofibre sequence
Ne+ 1+Rt-!Ne+ -! eQ
of left eN-spaces. Factoring out the centre, there is a cofibre sequence
N+ 1+Rt-!N+ -! Q
of left N-spaces.
Remark 5.4. It is tempting to try to analyze eQK= (Ge=K+)=(Ne=K+) by factoring *
*out the
right action on K, and hence to obtain results when X has isotropy K. However i*
*f K is not
central, the cell decomposition of eQK is a little more complicated.
It follows that the forgetful map
U : [X; Y ]G*-! [X; Y ]N*
is an isomorphism for X = G+ provided
1 + R*t: ss*(Y ) -! ss*(Y )
is an isomorphism. However, R*tis multiplication by t on non-equivariant homoto*
*py; since
G is connected R*t= 1.
6.Reduction to normalizers.
It is often more convenient to work with normal subgroups. In this section w*
*e prove a
result allowing us to reduce to this case in favourable circumstances. Combinin*
*g this with
the analysis of free spectra in Section 5 we obtain the key reduction. Recall *
*that c is the
idempotent supported on conjugates of T and its subgroups.
RATIONAL SO(3)-EQUIVARIANT COHOMOLOGY THEORIES. 9
Theorem 6.1. If X is a (G; c)-spectrum then restriction
~= N
[X; Y ]G -! (1 - eD2)[X; Y ]
is an isomorphism.
Remark 6.2. (i) Of course the idempotent (1 - eD2) can be replaced by c since X*
* is a
(G; c)-spectrum. However the more precise statement shows that it is only the f*
*usion of C2
and D2 that causes difficulty.
(ii) The special case X = (G=C2)+ and Y = eE[G 6 C2] shows that the idempotent *
*(1 - eD2)
is necessary; here [G 6 C2] is the family of subgroups containing no conjugate *
*of C2. Indeed
(G=C2)D2 ~=(G=C2)C2 = N=C2, so as an N-space G=C2 is obtained from N=C2 by atta*
*ching
cells with isotropy D2. To see that the cells with isotropy D2 make a non-zero *
*change we
need only show the obstruction group
R = [eD2ssD2*(G=C2)=(N=C2); eE[G 6 C2]]N*
is non-zero. One easy way to do this is to use the theory of [3], which shows t*
*hat since the
domain lies over D2, we need only calculate with homotopy Mackey functors:
R = Hom Mackey(ss_N*((G=C2)=(N=C2)); ss_N*(Ee[G 6 C2]):
This calculation is in turn reduced to calculating W -maps of the W -modules ob*
*tained by
applying
eD2ssD2*(.) = ss*(D2(.)):
Thus
R = Hom W (eD2ssD2*((G=C2)=(N=C2)); eD2ssD2*(Ee[G 6 C2]):
Now evidently
eD2ssD2*(Ee[G 6 C2]) = Q;
so the obstruction group R is non-zero provided the trivial representation occu*
*rs in the
domain. However, since (N=C2)D2 = ;,
eD2ssD2*((G=C2)=(N=C2)) = eD2ssD2*((G=C2)+) = ss*((G=C2)D2+);
which has non-zero W -fixed points in degree 0.
Proof of 6.1: Since both sides are cohomology theories of X, it suffices to con*
*sider the
case X = G=H+ with H a cyclic subgroup of T . By the 5-lemma and the cofibre se*
*quence
EG+ -! S0 -! "EG, it suffices to show the result when Y is 1-contractible and w*
*hen Y is
free.
In the case Y is G-free, it suffices to suppose Y = G+, since X is finite. No*
*w EG+ ^X -!
X induces an isomorphism in domain and range so we may suppose X is also free. *
*This case
was dealt with in 5.1.
The case where Y is 1-contractible will be covered in 6.8 (ii) below.
We are considering the case X = G=H+, especially if H is a non-trivial cyclic*
* group.
Definition 6.3. Suppose H is a closed subgroup of G. We say that K NG(H) is a *
*bad
subgroup for H (or simply that K is H-bad) if there is a g 62 NG(H) so that g-1*
*Kg H.
10 J.P.C.GREENLEES
Remark 6.4. (i) A subgroup K is H-bad if and only if (G=H)K 6= NG(H)=H.
(ii) Only proper subgroups of H can be H-bad and the trivial subgroup 1 is alwa*
*ys H-bad
(unless H is normal in G).
Lemma 6.5. If Y is K-contractible whenever K is H-bad, then the restriction
~= N (H)
[G=H+; Y ]G -! [G=H+; Y ] G
is an isomorphism
Proof: Let M = NG(H) and consider the diagram
~=
[G=H+;OYO]G____//_[S0;OYO]H
Unnnnnn~ | G |
nn = |indM |=
vvnnnni* | ~= |
[G=H+; Y ]M ____//_[M=H+; Y ]M__//_[S0; Y ]H
where
i : M=H+ -! G=H+
is the inclusion of M-spaces. Since i* O U O indGM(f) = f, it suffices to show *
*that i induces
an isomorphism. However, by Remark 6.4 (i), we need only attach cells M=K+ when*
* K is
H-bad, and by hypothesis these make no contribution.
Remark 6.6. There are analogous results if we replace NG(H) by an arbitrary sub*
*group M
normalizing H.
Lemma 6.7. If H is a non-trivial cyclic subgroup, the only H-bad subgroups are *
*1 and D2,
and D2 only occurs if H contains C2.
Proof: Any subgroup K G T is cyclic. If K 6= 1 and Kg H then g takes the axis
of a generator of K to that for H. Thus if K = T or K = Cn for some n we deduce
g 2 N = NG(H). The only other cyclic subgroups K are conjugate to D2.
This gives a useful conclusion.
Corollary 6.8. (i) If X is a (G; c)-spectrum and Y is C2-contractible then rest*
*riction
~= N
[X; Y ]G -! [X; Y ]
is an isomorphism.
(ii) If X is a (G; c)-spectrum and Y is 1-contractible then restriction
~= N
[X; Y ]G -! (1 - eD2)[X; Y ]
is an isomorphism.
Proof: Part (i) is a special case of Part (ii), since if Y is C2-contractible, *
*eD2 acts as 0 on
the N-spectrum Y . We therefore suppose Y is 1-contractible and prove Part (ii).
It suffices to consider the case X = G=H+ with H a cyclic subgroup of T . If *
*H = 1 the
result is obvious since both domain and codomain are 0, so we suppose H is non-*
*trivial. By
6.7 the only possible H-bad subgroups are 1 and D2. Now the argument of 6.5 app*
*lies to
show that ~
[G=H+; Y ]G -=! (1 - eD2)[G=H+; Y ]N
RATIONAL SO(3)-EQUIVARIANT COHOMOLOGY THEORIES. 11
is an isomorphism: simply apply the idempotent (1 - eD2) to all the N-equivaria*
*nt groups
in the proof.
7.The algebraic model of the category of (G; c)-spectra.
We proved in 6.1 that the algebraic model of (G; c)-spectra is a full subcate*
*gory of the al-
gebraic model of (N; c)-spectra, given in [3]. We have an algebraic model of (N*
*; c)-spectra,
and we construct a subcategory modelling (G; c)-spectra. In this section we rec*
*all the rele-
vant algebra, and in the next we complete the proof that we do have a model.
There are two models for rational (G; c)-spectra, as derived categories of ab*
*elian categories:
(G; c)-spectra ' D(As(G; c)) ' D(At(G; c)):
The standard abelian category As(G; c) has injective dimension 1, and the torsi*
*on abelian
category At(G; c) is of injective dimension 2. It is usually easiest to identif*
*y the model for
a (G; c)-spectrum in D(At(G; c)), at least providing its model has homology of *
*injective
dimension 1. This is then transported to the standard category, where calculat*
*ions are
sometimes easier.
To describe the categories, we should begin by describing the corresponding c*
*ategories
As(T ) and At(T ) for the subgroup T . This is constructed from simpler categor*
*ies, one for
each closed subgroup of T : for finite cyclic groups C the fibre category is th*
*e category of
Q[c]-modules with c of degree -2, and for T itself the fibre category is the ca*
*tegory of
graded Q-modules. There is some subtlety in how these are assembled, and it can*
* be viewed
as considering a category of sheaves over the topological category SfT of subgr*
*oups of T ,
although we will not emphasize this view here.
We need to use the discrete set FT of finite cyclic subgroups of T . On this *
*we consider
the sheaf R of rings with stalks Q[c] where c has degree -2. We need to conside*
*r the ring
R = map (FT; Q[c]) of global sections. For each subgroup H, we let eH 2 R den*
*ote the
idempotent with support H. If w : FT -! Z is a function, we write cw for the el*
*ement of R
with cw(H) = cw(H). Now consider the multiplicative set E generated by the univ*
*ersal Euler
classes e(V ) for the representations V of T with V T = 0. These are defined by*
* e(V ) = cv,
where v(H) = dimC(V H). In particular if V is a 1-dimensional representation wi*
*th kernel
of order n, we have e(V ) = cffi(n)where ffi(n)(H) = 1 if |H| divides n and 0 o*
*therwise.
Equivalently,
E = {cw | w : FT -! Z0 of finite support}:
L *
* Q
We let t*F= E-1R: as a graded vector space this is H Q in positive degrees an*
*d H Q in
degrees zero and below.
The objects of the standard model As(T ) are triples (N; fi; V ) where N is a*
*n R-module
(called the nub), V is a graded rational vector space (called the vertex) and f*
*i : N -! t*FV
is a morphism of R-modules (called the basing map) which becomes an isomorphism*
* when
E is inverted. When no confusion is likely, we simply say that N -! t*F V is an*
* object of
the standard abelian category. An object of As(T ) should be viewed as the modu*
*le N with
the additional structure of a trivialization of E-1N. A morphism (N; fi; V ) -!*
* (N0; fi0; V 0)
of objects is given by an R-map : N -! N0 and a Q-map OE : V - ! V 0compatible*
* under
the basing maps.
12 J.P.C.GREENLEES
We then form the derived category D(As(T )) by taking differential graded obj*
*ects of As(T )
and inverting homology isomorphisms. Thus homology gives a functor H* : D(As(T *
*)) -!
As(T ). Since the category As(T ) has injective dimension 1, homotopy types of*
* objects of
the derived category D(As(T )) are classified by their homology in As(T ), so t*
*hat homotopy
types correspond to isomorphism classes of objects of the abelian category As(T*
* ). In the
sheaf theoretic approach, N is the space of global sections of a sheaf on the s*
*pace of closed
subgroups T , the vertex V is the value of the sheaf at the subgroup T and the*
* fact that
the basing map fi : N -! t*F V is an isomorphism away from E is the manifestati*
*on of
the patching condition for sheaves together with the localization theorem for e*
*ach individual
finite subgroup.
The objects of the torsion abelian category At(T ) are triples (V; q; Q) wher*
*e V is a graded
rational vector space, Q is an E-torsion R-module and q : t*F V -! Q is a morp*
*hism
of R-modules. The condition on Q is equivalent to requiringL(i) that Q is the *
*sum of its
idempotent factors Q(H) = eH Q in the sense that Q = H Q(H) and (ii) that eac*
*h Q(H)
is a torsion Q[c]-module. When no confusion is likely, we simply say that t*F V*
* - ! Q is
an object of the torsion abelian category. In the sheaf theoretic approach, the*
* module Q(H)
is the cohomology of the structure sheaf with support at H. By contrast with th*
*e standard
abelian category, the torsion abelian category has injective dimension 2. Thus*
* not every
object X of the derived category D(At) is determined up to equivalence by its h*
*omology
H*(X) in the abelian category At(T ). We say that X is formal if it is determi*
*ned up to
isomorphism by its homology. It is not hard to check that X is formal if its ho*
*mology has
injective dimension 0 or 1 in At(T ). In general, if H*(X) = (t*F V -! Q), th*
*e object
X is equivalent to the fibre of a map (t*F V - ! 0) -! (t*F 0 -! Q) (in the der*
*ived
category) between objects in Atof injective dimension 1. This map is classified*
* by an element
of Ext(t*F V; Q), so that X is formal if the Ext group is zero in even degrees.*
* Thus X is
formal if both V and Q are in even degrees or if Q is injective in the sense th*
*at each Q(H)
is an injective Q[c]-module.
Notice that the action of N on T by conjugation gives an action of W on T -ma*
*ps between
N-spectra. It is shown in [3, 3.2] that the abelian categories for (N; c) are d*
*escribed by
As(N; c) = As(T )[W ] and At(N; c) = At(T )[W ]:
Thus the fibre category over a finite cyclic group C is the category of Q[c][W *
*]-modules, and
the fibre category over T itself is the category of graded Q[W ]-modules.
For (G; c)-spectra, As(G; c) is a full subcategory of As(N; c) and At(G; c) i*
*s a full subcat-
egory of At(N; c). The additional condition only involves the isotropy group 1.*
* Theorem 5.1
states that the forgetful map
(G; 1)-spectra -! (N; 1)-spectra
is full and faithful. A very special case of [3, 3.2] (together with [4, 4.4.1]*
*) shows that
(N; 1)-spectra ' D(tors-Q[c][W ]-mod):
However the composite is not essentially surjective.
To motivate the algebraic model of the image, we return to topology. Note tha*
*t H*(BG; Q) =
Q[c2] where c2 has degree -4. and that the restriction H*(BG; Q) -! H*(BT ; Q) *
*takes c2
to c2. Furthermore the conjugation action of W on T takes c to -c.
RATIONAL SO(3)-EQUIVARIANT COHOMOLOGY THEORIES. 13
Lemma 7.1. The functor rGN: Q[c2]-modules -! Q[c] [W ]-modules defined by
rGN(M) = Q[c] Q[c2]M
is an exact functor of abelian categories. It is also full and faithful, and t*
*akes c2-power
torsion modules to c-power torsion modules.
Proof: The functor is faithful because Q[c] is a free Q[c2]-module. It is full*
* because any
W -map f : rGN(M) -! rGN(M0) is rGN(fW ). The rest of the lemma follows.
Now At(G; c) is the full subcategory of objects in which e1Q lies in rGN(Q[c2*
*]-modules ),
and As(G; c) is the full subcategory of objects in which e1ker(fi : N - ! t*F V*
* ) lies in
rGN(Q[c2]-modules ).
Lemma 7.2. The categories As(G; c) and At(G; c) are abelian.
Theorem 7.3. There are equivalences of triangulated categories
(G; c)-spectra ' D(At(G; c)) ' D(As(G; c)):
Proof: It suffices to prove the first equivalence, since the second equivalence*
* is immediate
from the definitions of the algebraic categories.
We have shown that restriction displays (G; c)-spectra as a full triangulated*
* subcategory
of (N; c)-spectra. It remains to show that (N; c)-spectra which are restriction*
*s of G-spectra
have models in the subcategory D(At(G; c)) and that all objects in this subcate*
*gory occur.
The model for (N; c)-spectra is built up one subgroup at a time, and the subc*
*ategory is
defined by imposing a condition on the isotropy group 1. Accordingly, by workin*
*g through
the proof that D(At(N; c)) is equivalent to the category of (N; c)-spectra, we *
*may verify the
above two statements by looking at one subgroup at a time. Since this fibrewise*
* analysis is
interesting in itself, we devote Section 8 to it.
8. The fibres.
In this section we summarize the analysis of (G; H)-spectra for each subgroup*
* H. This is
not a necessary step if H is exceptional or dihedral (since it follows from Sec*
*tion 10 and the
theory of Mackey functors), but it is clearer to treat all groups.
Theorem 8.1. The categories of G-spectra with a single stable isotropy group ar*
*e described
as follows.
(1) If H is of finite index in its normalizer then
(G; H)-spectra ' D(Q[WG(H)]-modules ):
The graded Q[WG(H)]-module classifying a (G; H)-spectrum X is ss*(H X).
(2) If C is a non-trivial finite cyclic group
(G; C)-spectra ' D(tors-Q[c][W ]-mod);
where c has degree -2. The torsion Q[c][W ]-module classifying a (G; C)-*
*spectrum X
is ss*(E(N=C)+ ^T=C CX).
14 J.P.C.GREENLEES
(3) Free spectra are described by
(G; 1)-spectra ' D(tors-Q[c2]-mod)
where c2 has degree -4. The torsion Q[c2]-module classifying a (G; 1)-sp*
*ectrum X is
ss*(EG+ ^G X). Restriction is described by the commutative diagram
resGN
(G; 1)-spectra -! (N; 1)-spectra
'# #'
rGN
D(tors-Q[c2]-mod) -! D(tors-Q[c][W ]-mod):
The algebraic restriction functor is described by rGN(M) = Q[c] Q[c2]M, *
*using the
ring map Q[c2] -! Q[c] defined by taking c2 to c2, and letting W act as *
*the sign
representation on c.
Remark 8.2. The equivalences of categories in the theorem each state a triangul*
*ated cate-
gory Ho(C) is equivalent to the derived category D(A) of an abelian category A *
*of injective
dimension 0 or 1. The equivalence arises from an Adams spectral sequence based *
*on A, for
calculating maps in Ho(C): because of the injective dimension of A is 1, the *
*spectral
sequence collapses, and because of parity one may choose splittings giving an e*
*quivalence.
The proof is given in [4, 4.3.2]. Because of the low injective dimension, all o*
*bjects are classi-
fied by their A-valued homology, and this is the functor mentioned in the state*
*ment. Ideally
the equivalence would be constructed as a Quillen equivalence of model categori*
*es, and joint
work with B.E. Shipley now suggests this will be possible.
Proof: First consider the subgroups of finite index in their normalizer. We wil*
*l show in 11.4
and 11.5 that if X and Y are (G; H)-spectra then
~= H H W (H)
[X; Y ]G -! [ X; Y ] G :
Also, since X and Y are (G; H)-spectra, H X and H Y are free WG(H)-spectra.
Now if F is finite we may consider the descent spectral sequence
H*(F ; [A; B]*1) ) [EF+ ^ A; B]F*:
If we work over the rationals this collapses and converges to show
H0(F ; [A; B]1*) = [EF+ ^ A; B]F*:
This shows that if WG(H) is finite,
X 7-! ss*(H X)
defines a full and faithful functor
(G; H)-spectra -! D(Q[WG(H)]-modules ):
To see the functor is essentially surjective, note all objects of Q[WG(H)] are *
*retracts of sums
of copies of Q[WG(H)], so it suffices to show Q[WG(H)] is in the image. For thi*
*s we use the
fact that (G=H)H = WG(H). To obtain a spectrum with stable isotropy (H) we use*
* the
space G=H+ ^ "E[G H] where [G H] is the family of conjugates of proper subgroup*
*s of H.
It follows from 6.1 that if C is a cyclic group there is a full and faithful *
*functor
(G; C)-spectra -! (N; C)-spectra:
RATIONAL SO(3)-EQUIVARIANT COHOMOLOGY THEORIES. 15
A special case of [3, 3.2] (together with [4, 4.4.1]) shows that
(N; C)-spectra ' D(tors-Q[c][W ]-mod):
If C 6= 1 the composite functor is essentially surjective. Indeed, N=T+ ^ EN=C+*
* generates
(N=C; 1)-spectra, and this space is the C-fixed point space of G=T+ ^ E[G C]+ ^*
* "E[G C],
where [G C] is the family of subgroups of G conjugate to a subgroup of C.
Finally, we come to the isotropy group 1, where the composite is not essentia*
*lly surjec-
tive. We need only show that the image category is the image of the algebraic *
*functor
rGN. The G-spectrum G+ generates free G-spectra, so it suffices to observe tha*
*t its im-
age lies in the image of rGNand generates it. This is immediate since G+ corre*
*sponds to
ssT*(G+) = ss*(G+=T ) = ss*(CP+1).
Remark 8.3. For the proof of Theorem 7.3 we only need the fibres over H with H *
*a subgroup
of T . For these cases we do not need to appeal to Section 11. The appeal for H*
* = T can be
replaced by a reference to Theorem 6.1 and the N-equivariant result.
9. Mackey functors.
We now begin the analysis relevant to spectra with no stable isotropy groups *
*contained
in a maximal torus. It will turn out these are all products of Eilenberg-MacLan*
*e spectra, so
we start by analyzing the possible coefficients.
First recall that a Mackey functor is a contravariant additive functor
M : G-orbits- ! Q-modules ;
where G-orbitsis the full subcategory of G-spectra on the orbits G=H+. Before d*
*escribing
the model of (G; m)-spectra, we need to summarize the relevant properties of Ma*
*ckey func-
tors. This is the special case of the results of [2] in which the group is SO(3*
*). I am grateful
to L.G.Lewis for pointing out that in [2] the category of hO-sheaves must be re*
*placed by a
full subcategory in Theorems A and B to obtain correct statements. However in t*
*he present
paper, we use only the composite equivalence
Mackey functors' continuous Weyl-toral modules;
which is true as stated with the same proof. Alternatively it is easy to deal w*
*ith the special
case of SO(3) directly.
The main result states that the category of rational Mackey functors M is equ*
*ivalent to the
category of equivariant sheaves V on SfG equipped with restriction maps V (T ) *
*-! V (Cn)
for all n. More precisely, we should make SfG into a topological category with*
* one non-
identity morphism Cn -! T for each n 1, and ask that V be a sheaf on the cate*
*gory.
However, since we are only interested in sheaves supported on EqD0, where there*
* are no non-
identity morphisms, we do not need this extra precision. To be explicit, the st*
*alks of V may
be constructed from the Mackey functor M as follows. If H is totally isolated a*
*s a subgroup
of itself V (H) = eH M(H), where eH 2 A(H) is the idempotent of H supported at*
* H,
V (T ) = M(T ) and V (N) = lim eM(N), where the direct limit is over all id*
*empotents
! e(N)=1
e with e(N) = 1.
16 J.P.C.GREENLEES
Notice that for any Mackey functor M, an idemptent e 2 A(G) gives a new Macke*
*y functor
eM, defined by (eM)(H) = (eM)(G=H+) := M(eG=H+). In particular, the category of
rational Mackey functors splits:
G-Mackey ' (G; c)-Mackey x (G; d0)-Mackey x (G; e)-Mackey ;
where d0is the idempotent with support D0and e is the idempotent with support E*
*. In terms
of sheaves, we find (G; c)-Mackey functors correspond to equivariant sheaves su*
*pported on
the subcategory C: they are thus specified by W -modules V (T ), V (Cn) for all*
* n 1, where
the W -action on V (C1) is trivial, and W -maps V (T ) -! V (Cn).
The (G; e)-Mackey functors are easy to describe, since they are equivariant s*
*heaves over a
topological union of orbits. They are thus specified by giving a WG(E)-module V*
* (E) for each
E 2 E, and the category of (G; e)-Mackey functors is of projective and injectiv*
*e dimension
0 by Mashke's theorem.
Finally, (G; d0)-Mackey functors correspond to equivariant sheaves supported *
*on the space
D0. Since we need to perform calculations, it is useful to be more explicit. *
*The following
results are obtained by adapting Section 4 of [3]: we need only omit D2 and D4 *
*from D to
obtain D0. The stalks are V (N), and V (D) for finite dihedral groups D. By G-e*
*quivariance,
the stalks V (D) are specified by WG(D2n)-modules V (D2n) for each n 3.
Since WG(D2n) is of order 2 for n 3, we abbreviate it too by W , relying on *
*context to
determine which group of order 2 is intended. Now the orbit of D2n is isolated *
*for each n,
so the only piece of data required to recover the sheaf from its stalks is the *
*way a germ from
V (N) may be represented in some neighbourhood of G. Each germ is defined on D2*
*n for all
sufficiently large n, so we consider the end space
Y
V (D1 ) := lim V (D2n);
! ini
of sections of D \ {N} near infinity, and the behaviour of sections is describe*
*d by a W -map
oe : V (N) -! V (D1 ).
Summary 9.1. A (G; d0)-Mackey functor is given by a vector space V (N) together*
* with
W -modules V (D2n) for n 3 and a W -map oe : V (G) -! V (D1 ). This correspond*
*ence is
natural and gives an equivalence of categories.
It is then not hard to give a sufficient condition for a Mackey functor to be*
* injective, and
to conclude that all (G; d0)-Mackey functors have an injective resolution of le*
*ngth 1.
Lemma 9.2. A (G; d0)-Mackey functor is injective if the map oe is an epimorphis*
*m, and
hence the category of (G; d0)-Mackey functors is of injective dimension 1.
For completeness we record the situation for projective functors, although we*
* do not use
it elsewhere.
Remark 9.3. By contrast, the surjectivity of oe is not a sufficient condition f*
*or the corre-
sponding (G; d0)-Mackey functor to be projective. However all (G; d0)-Mackey fu*
*nctors admit
projective resolution of length 1.
RATIONAL SO(3)-EQUIVARIANT COHOMOLOGY THEORIES. 17
10. The ordinary part of the model.
In this section we discuss the torally isolated subgroups, which are associat*
*ed to the
idempotent m = 1 - c supported on subgroups not conjugate to a subgroup of T . *
*We show
that any (G; m)-spectrum splits as a product of Eilenberg-MacLane spectra. Inde*
*ed, we may
use the same proof as given in [5, Appendix A] for finite groups.
In the usual way, the nth H-equivariant stable homotopy group of a G-spectrum*
* X is
defined by ssHn(X) = [G=H+^Sn; X]G = [Sn; X]H , and these are assembled into th*
*e homotopy
Mackey functor ss_Gn(X) defined by by H 7-! ssHn(X). The Eilenberg-MacLane spec*
*trum HM
represents ordinary cohomology with coefficients in the Mackey functor M: it is*
* characterized
by the dimension axiom, which states that ss_G0(HM) = M and ss_Gn(HM) = 0 for n*
* 6= 0.
For the proof, we define a related cohomology theory. Given any injective rat*
*ional Mackey
functor I we may define a cohomology theory hI*G(.) by
hInG(X) = Hom (ss_Gn(X); I);
Evidently, if I is supported on the set of torally isolated subgroups, the coho*
*mology theory
vanishes on (G; c)-spectra.
Lemma 10.1. If I is an injective (G; m)-Mackey functor then hI is an Eilenberg-*
*MacLane
spectrum: hI = HI.
Proof: Since ss_G0(G=H+) = [.; G=H+]G is a free (or representable) functor on t*
*he orbit
category, it is clear that hI has the correct homotopy groups in degree 0. We m*
*ust calculate
hInG(G=H+) for each subgroup H, and show that it is zero if n 6= 0. We have alr*
*eady observed
that hI*G(.) is zero on (G; c)-spectra, so this deals with the case in which H *
*lies in C. This
leaves us to deal with the torally isolated groups
H 2 {G; N; A5; 4; A4} [ {D2n | n 2}:
We use the generic letter B for these groups.
We need to examine the functor ss_Gn(G=H+), which is made up from the groups *
*ssKn(G=H+).
Since I is a (G; m)-Mackey functor, the groups with K in C have no effect, and *
*we need only
consider torally isolated groups K. The case K = G gives the general pattern.
The tom Dieck splitting theorem for the G-space X states
M
ssGn(X) = ssn(EWG(B)+ ^WG(B)XB )
(B)
M
ssn(EWG(C)+ ^WG(C)L(C)XC ) ssn(EG+ ^W adXT );
(C)
where B runs through torally isolated groups and C through the finite cyclic gr*
*oups and
L(C) is the representation of WG(C) = N=C on the tangent space to G=C at the id*
*entity.
The summands for T and the finite cyclic groups do not contribute, since I is c*
*oncentrated
on torally isolated groups. The other Weyl groups WG(B) are all finite and henc*
*e over the
rationals we have M
mssGn(X) = m ssn(XB )WG(B):
(B)
We must now consider X = G=H+ for torally isolated groups H, and observe that (*
*G=H)B
is 0-dimensional in all cases. This is a simple verification, and the same ver*
*ification deals
with the other subgroups K.
18 J.P.C.GREENLEES
The result is obvious for H = G, so suppose H is a proper subgroup. Now, if *
*(G=H)B
were not finite there would be a one parameter family Bgt H, and hence H would *
*be of
higher dimension than B. This means we need only look at the special case H = *
*N and
B = D2n. If n 3 then the only elements conjugating B into H normalize Cn and t*
*herefore
lie in N. This leaves the case (G=N)D4. This has three points. Indeed, if Dg4 N*
* then there
is exactly one of the three non-identity elements s of D4 so that sg is the uni*
*que element !
of order 2 in T . If s = !, g normalizes ! and hence lies in N. If s is rotat*
*ion about the
x-axis g is a rotation moving the x-axis to the z-axis, and its axis thus lies *
*on one of the two
planes bisecting the angle between the planes x = 0 and z = 0. If s is rotation*
* about the
y-axis the analysis is similar.
Now if M is any (G; m)-Mackey functor, we may choose an injective resolution *
*0 -!
M - ! I - ! J - ! 0, and by definition of hI there is a map HM - ! hI lifting t*
*he
embedding M -! I. Thus by the lemma we have a cofibre sequence HM -! hI -! hJ,
and this gives a Universal Coefficient Theorem.
Corollary 10.2. For any (G; m)-Mackey functor M there is a short exact sequence
0 -! Ext(ss_Gn-1(X); M) -! HMnG(X) -! Hom (ss_Gn(X); M) -! 0:
Corollary 10.3. Any (G; m)-spectrum X splits as a product of Eilenberg-MacLane *
*spectra
Y
X -'! nHss_Gn(X):
n
Proof: By 10.2 we may lift the identity of ss_Gn(X) to give a map X -! nHss_Gn(*
*X) inducing
the identity of ss_Gn. Using these as components we obtain a map into the prod*
*uct. The
resulting map is an isomorphism in homotopy groups ssHnfor all n and H by const*
*ruction,
and thus a weak equivalence.
Note that since the product in the corollary is equivalent to a sum, this giv*
*es a short exact
sequence for calculating [X; Y ]G for any two (G; m)-spectra X and Y .
Corollary 10.4. For any (G; m)-spectra, X and Y there is a short exact sequence
0 -! Ext(ss_G*(X); ss_G*(Y )) -! [X; Y ]G -! Hom (ss_G*(X); ss_G*(Y )) -*
*! 0:
We may go a little further.
Corollary 10.5. The category of (G; m)-spectra is equivalent to the derived cat*
*egory of
graded (G; m)-Mackey functors:
(G; m)-spectra ' D((G; m)-Mackey ):
Proof: It remains to comment on the naturality of the splitting. First choose *
*a splitting
of each (G; m)-spectrum X as a product of Eilenberg-MacLane spectra. This deter*
*mines a
splitting of the short exact sequence of 10.4, and this allows us to define a f*
*unctor
(G; m)-spectra -! D((G; m)-Mackey )
to the derived category. The functoriality follows from the fact that the compo*
*site of two
elements of Ext is zero. It is easy to see the map is an equivalence of categor*
*ies.
RATIONAL SO(3)-EQUIVARIANT COHOMOLOGY THEORIES. 19
11. Reduction to the Weyl group.
The purpose of this section is to consider the geometric fixed point map
K : [X; Y ]G -! [K X; K Y ]WG(K);
and to show that under various hypotheses on the isotropy of X and Y it is an i*
*somorphism.
This could be the basic ingredient in our analysis, since for our group G = SO(*
*3), either
K = 1 and we may use the result of Section 5 or we may apply known results sinc*
*e WG(K) is
finite, a circle or O(2). Because restriction to N is so close to being an isom*
*orphism we have
chosen to avoid using these results in the proof of the main equivalence, but t*
*hey remain
conceptually helpful.
The basic result is straightforward obstruction theory.
Lemma 11.1. If K is normal in bK, and H Y ' * unless H K then
Kb K K WG(K)
[X; Y ] ~=[ X; Y ] :
Proof: By the isotropy hypothesis on Y , the inclusion Y - ! Y ^ "E[6 K] is an *
*equivalence,
where [6 K] is the family of subgroups not containing K. Now we quote [6, II.9].
Corollary 11.2. If H Y ' * unless H K, and X is built from cells G=H+ with H
NG(K) and so that if K H we have
(G=H)K = NG(K)=H;
then
[X; Y ]G ~=[K X; K Y ]WG(K):
Remark 11.3. Note that the condition on (G=H)K is slightly different to the req*
*uirement
that K is H-good, when we need (G=H)K = NG(H)=H.
Proof: It suffices to consider the case X = G=H+ with H NG(K). If K 6 H then b*
*oth
sides are zero, so we may suppose K H. Note that WH (K) WG(K) and
WG(K)=WH (K) = (NG(K)=K)=(NH (K)=K) = NG(K)=NH (K) = NG(K)=H
and since K H it follows that
NG(K)=H (G=H)K :
Now consider the diagram
[G=H+; Y ]G____//[K G=H+; K Y ]WG(K)______//[NG(K)=H+; K Y ]WG(K)
~=|| |~=|
fflffl|~= = |fflffl
[S0; Y ]H______//_[K S0; K Y ]WH(K)____//[WG(K)=WH (K)+; K Y ]WG(K)
The result follows since the inclusion
NG(K)=H -! (G=H)K
is an equivalence.
20 J.P.C.GREENLEES
We restate the condition H Y ' * unless H K as saying that the stable isotro*
*py groups
of Y all contain K.
The crudest application of the result is exactly analogous to the Hopf classi*
*fication theorem
for maps of an n-dimensional complex into an (n - 1)-connected space. This is a*
*dequate to
deal with subgroups K with WG(K) finite.
Corollary 11.4. If the stable isotropy groups of X are all contained in conjuga*
*tes of K and
the stable isotropy groups of Y all contain K then
[X; Y ]G ~=[K X; K Y ]WG(K):
Proof: It suffices to observe that such a spectrum X may be constructed using c*
*ells G=K+,
and cells G=H+ with H not containing K.
We now turn to the case of cyclic groups K = C; the case C = 1 is different a*
*nd was dealt
with in Section 5.
Corollary 11.5. If H Y ' * unless C H and X is built from cells G=H+ with H N
then if either
o |C| 3 or
o |C| = 2 and all cells of X are G=H+ with H topologically cyclic
then
[X; Y ]G ~=[CX; CY ]N=C:
Proof: We just need to verify that if C H N then (G=H)C = N=H. For gH to be f*
*ixed
by C we require Cg H N. Since N is the stabilizer of the z-axis, it suffices *
*to show that
either of the conditions imply g preserves the z-axis. However all elements of *
*N with order
3 are rotations about the z-axis. If |C| 3 then g must preserve the z axis an*
*d hence
lie in N; since C is normal in N, the condition is satisfied. However N contain*
*s elements of
order 2 with many other axes, so if |C| = 2 we must require that all non-identi*
*ty elements
of H preserve the z axis.
References
[1]J.P.C.Greenlees "A rational splitting theorem for the universal space for a*
*lmost free actions." Bull.
London Math. Soc. 28 (1996) 183-189.
[2]J.P.C.Greenlees "Rational Mackey functors for compact Lie groups I" Proc. L*
*ondon Math. Soc 76 (1998)
549-578
[3]J.P.C.Greenlees "Rational O(2)-equivariant cohomology theories." Fields Ins*
*titute Communications 19
(1998) 103-110
[4]J.P.C.Greenlees "Rational S1-equivariant cohomology theories" Mem. American*
* Math. Soc. 661 (1999),
xii + 289 pp
[5]J.P.C.Greenlees and J.P.May "Generalized Tate cohomology" Mem. American Mat*
*h. Soc. 543 vi + 178
pp.
[6]L.G.Lewis, J.P.May and M.Steinberger (with contributions by J.E.McClure) Eq*
*uivariant stable homo-
topy theory" Lecture notes in mathematics, 1213, Springer-Verlag, Berlin, ix*
* + 538pp
Department of Pure Mathematics, Hicks Building, Sheffield S3 7RH. UK.
E-mail address: j.greenlees@sheffield.ac.uk