HOMOTOPY LIE GROUPS
JESPER M. MOLLER
Abstract.Homotopy Lie groups, recently invented by W.G. Dwyer and C.W.
Wilkerson [13], represent the culmination of a long evolution. The basic*
* philos-
ophy behind the process was formulated almost 25 years ago by Rector [32*
*, 33]
in his vision of a homotopy theoretic incarnation of Lie group theory. W*
*hat
was then technically impossible has now become feasible thanks to modern
advances such as Miller's proof of the Sullivan conjecture [25] and Lann*
*es'
division functors [22]. Today, with Dwyer and Wilkerson's implementation*
* of
Rector's vision, the tantalizing classification theorem seems to be with*
*in grasp.
Supported by motivating examples and clarifying exercises, this guide *
*quick-
ly leads, without ignoring the context or the proof strategy, from class*
*ical finite
loop spaces to the important definitions and striking results of this ne*
*w theory.
1.Introduction
The aim of this report is to advertise the discovery by W.G. Dwyer and C.W.
Wilkerson of a remarkable class of spaces called homotopy Lie groups or p-compa*
*ct
groups. These purely homotopy theoretic objects capture the essence of the idea
of a Lie group. I shall here focus on [13] where Dwyer and Wilkerson introduce
homotopy Lie groups and prove (1.1) and (1.3). Subsequent developments [11, 12,
14], possibly leading to a classification theorem in the near future, will be b*
*riefly
described in the final section.
A p-compact group is a pointed topological space, BX, with all of its homo-
topy theory concentrated at the prime p, whose loop space X = BX satisfies a
cohomological finiteness condition. The p-compact r-torus BT = K(Zp; 2)r is an
example of a p-compact group.
A maximal torus for the p-compact group BX is a map Bi: BT ! BX of a
p-compact torus BT to BX satisfying an injectivity and a maximality condition.
Theorem 1.1. [13, 8.13, 9.4] Any p-compact group admits a maximal torus, unique
up to conjugacy.
The Weyl group, WT(X) , associated to the maximal torus Bi: BT ! BX is,
when X is connected, faithfully represented in the vector space H2(BT ; Qp) and
the algebra homomorphism induced by the WT(X) -invariant map Bi,
(1.2) H*(Bi; Qp):H*(BX; Qp) ! H*(BT ; Qp)WT(X)
takes the p-adic rational cohomology of BX into the invariant ring of WT(X) .
____________
1991 Mathematics Subject Classification. 55P35, 55R35.
Key words and phrases. Finite loop space, p-compact group, maximal torus, We*
*yl group.
1
2 J. M. MOLLER
Theorem 1.3. [13, 9.7] Let X be a connected p-compact group with maximal torus
T -! X. Then:
(1) T and X have the same rank.
(2) The Weyl group WT(X) is faithfully represented as a reflection group in
the Qp-vector space H2(BT ; Qp).
(3) The homomorphism ( 1.2) is an isomorphism.
The precise definition of a p-compact group, which is motivated by the concept
of a finite loop space, follows shortly.
1.4. Finite loop spaces. Suppose G is a compact Lie group. Take a free, con-
tractible G-space EG and define BG = EG=G to be the orbit space. Then the
associated fibre sequence
EG -!BG -!G -!EG -!BG
contains a homotopy equivalence BG -!G.
This phenomenon is embedded in the general concept of a finite loop space.
Definition 1.5.A finite loop space is a connected, pointed space BX such that
X = BX is homotopy equivalent to a finite CW-complex.
Note that X _ by definition _ is the loop space of BX. It is customary, though
ambiguous, to refer to the finite loop space BX by its underlying space X and t*
*hen
call BX the classifying space of X.
We have already seen that compact Lie groups are finite loop spaces. The clas-
sifying space of SU (2), for instance, is the infinite quaternionic projective *
*space
B SU(2) = HP 1. However, the class of finite loop spaces is much larger. A stri*
*king
example was provided by Rector [31] who found an uncountable family of homo-
topically distinct finite loop spaces BX with X homotopy equivalent to SU(2). In
other words, the homotopy type SU (2) supports uncountable many distinct loop
space structures.
Rector's example destroyed all hopes of a classification theorem in the spiri*
*t of
compact Lie groups _ as long as one sticks to integral spaces, that is. The sit*
*uation
looks brighter in the category of Fp-local spaces.
1.6. Notation. In the following, p denotes a fixed prime number, Fp the field
with p elements, Zp the ring of p-adic integers, and Qp = Zp Q the field of p-a*
*dic
numbers.
H*(-) denotes singular cohomology with Fp-coefficients, H*(-; Fp), while
H*(-; Qp) denotes H*(-; Zp) Q (and not singular cohomology with Qp-
coefficients).
A space K is Fp-finite if H*(K) is finite dimensional over Fp. A map A -!B is
an Fp-equivalence if it induces an isomorphism H*(B) -!H*(A) on H*(-).
HOMOTOPY LIE GROUPS 3
1.7. Fp-local spaces. A space K is Fp-local if any Fp-equivalence A -!B induces
a homotopy equivalence map (B; K) -!map (A; K) of mapping spaces.
Fp-local spaces exist _ the classifying space K(Fp; i) for the functor Hi(-) *
*is
an obvious example. In fact, any space can be made Fp-local in a minimal and
functorial way.
Theorem 1.8. (Bousfield [4, 3.2]) The exists a functor K Kp, of the (ho-
motopy) category of CW-complexes into itself, with a natural transformation
jK :K ! Kp such that jK is an Fp-equivalence and Kp is Fp-local.
Note these categorical consequences of the definition and the theorem:
o Any Fp-equivalence between Fp-local spaces is a homotopy equivalence.
o A map is an Fp-equivalence if and only if its Fp-localization is a homot*
*opy
equivalence.
o K is Fp-local if and only if jK :K ! Kp is a homotopy equivalence.
If K is nilpotent or is connected and either H1(K; Fp) = 0 or ss1(K) is finit*
*e, then
[5, VI.5.3, VII.3.2, VII.5.1] [13, x11] the Bousfield localization, Kp, coincid*
*es with
the (perhaps more familiar) Bousfield-Kan localization, (Fp)1 K. In particular *
*[5,
VI.5.2], ssi(Kp) ~=ssi(K) Zp when K is connected, pointed and nilpotent with
finitely generated abelian homotopy groups. (This is far from true without the
nilpotency hypothesis (2.2).)
1.9. Homotopy Lie groups. By design, actually inescapably [7], the uncount-
ably many finite loop spaces BX of Rector's example all Fp-localize to the stan*
*dard
(B SU(2))p. This observation indicates that "Fp-local finite loop spaces" are b*
*etter
behaved than integral loop spaces. The problem, however, is that this term is a*
*b-
surd as finite complexes are unlikely to be Fp-local. The solution proposed in *
*[13]
is to replace the topological finiteness criterion in (1.5) by a cohomological *
*one.
Definition 1.10.[13, 2.2] A p-compact group is an Fp-local space BX such that
X = BX is Fp-finite.
Again, it is customary to use X, by definition the loop space of BX, when
referring to the p-compact group BX, the classifying space of X.
2. Examples of p-compact groups
Let G be any compact Lie group whose component group ss0(G) is a p-group.
Define BG^ = (BG)p. Then G^is a p-compact group with H*(BG^) = H*(BG),
ss0(G^) = ss0(G), and ssi(G^) = ssi(G) Zp for all i 1 [13, x11].
This example includes all finite p-groups such as the trivial group {1} and t*
*he
cyclic p-groups Z=pn; n 1.
2.1. Toral groups. When applied to an r-torus S = SO(2)r, the above construc-
tion produces a p-compact r-torus T = ^S. The classifying space BT = K(Zp; 2)r
is an Eilenberg-MacLane space with homotopy in dimension two and H*(BT ) =
Fp[t1; : :;:tr] is polynomial on r generators of degree two.
Alternatively, BT = (BT )p, where T = (Z=p1 )r is a p-discrete r-torus.
More generally, a p-compact toral group P is a p-compact group with BP =
(BP )p, where P , a p-discrete toral group, is an extension of a p-discrete tor*
*us T
4 J. M. MOLLER
by a finite p-group ss. Note that the Fp-localized sequence BT -! BP -! Bss is *
*[5,
II.5.1] a fibration sequence as ss necessarily acts nilpotently on Hi(BT ; Fp).
Since map (BP; BX) ' map(BP ; BX) for any p-compact group X and P is the
union of an ascending chain of finite p-groups, results for finite p-groups can*
* often,
by discrete approximation [13, x6], be extended to p-compact toral groups.
2.2. Exotic examples. Call a connected p-compact group exotic if it is not of
the form ^Gfor any connected compact Lie group G. Sullivan spheres and, more
generally, (many) Clark-Ewing p-compact groups are exotic.
Proposition 2.3.(Sullivan) Assume that n > 2 is an integer dividing p-1. Then
the Fp-local sphere (S2n-1)p is a p-compact group.
The construction goes as follows: The cyclic group Z=n acts on T = Z=p1 as
Z=n < Aut(T ) ~=Z*pwhen n|(p - 1). Define BX = (BN )p where N = To Z=n is
the semidirect product. The computation
H*(BX) = H*(BN ) = H*(BT )Z=n = Fp[t]Z=n = Fp[tn];
which uses (1.7) and the fact that n is prime to p, shows that the mod p cohomo*
*logy
of BX is polynomial on one generator of degree 2n. Thus the Fp-local space BX
is (2n - 1)-connected [5, I.6.1] and its loop space X is (2n - 2)-connected with
H*(X) abstractly isomorphic to H*(S2n-1). The Hurewicz theorem tells us that
this abstract isomorphism is realizable by an Fp-equivalence S2n-1 -!X, i.e. (1*
*.7)
by a homotopy equivalence (S2n-1)p -!Xp = X.
Clark and Ewing [6] observed that applicability of Sullivan's construction is*
* not
restricted to the rank one case. Let T be a p-discrete r-torus and W < Aut(T ) *
*~=
GL r(Zp) a finite group of order prime to p acting on T. Put BX = (BN )p where
N = To W . The invariant ring
H*(BX) = H*(BN ) = H*(BT )W = Fp[t1; : :;:tr]W
is, essentially by the Shephard-Todd theorem [2, 7.2.1] [21, x23], a finitely g*
*enerated
polynomial algebra if and only if W is a reflection group in GL r(Qp). If this*
* is
the case, H*(X) is an exterior algebra on finitely many odd degree generators.
In particular, X is Fp-finite and BX a p-compact group. These Clark-Ewing p-
compact groups are fairly well understood [8].
The Clark-Ewing p-compact group associated to any non-Coxeter group from the
list [6] of irreducible p-adic reflection groups is (7.6) exotic. This scheme p*
*roduces
many exotics if p is odd but none for p = 2 as the only non-Coxeter 2-adic refl*
*ection
group, W (DI(4)), number 24 on the list and of rank 3, has even order.
To come up with an exotic 2-compact group, a much more sophisticated approach
is required. In their landmark paper [18], Jackowski and McClure showed how to
decompose BG, for any compact Lie group G, as a generalized pushout of classify*
*ing
spaces of subgroups (proper subgroups if the center of G is trivial). Dwyer and
Wilkerson realized that a similar decompositon applies in the case of p-compact
groups [11, x8] and that this could be used in the construction of an exotic 2-
compact group.
HOMOTOPY LIE GROUPS 5
Theorem 2.4. [12] There exists a connected 2-compact group DI(4) such that
H*(B DI(4); F2) is isomorphic (as an algebra over the Steenrod algebra) to the *
*rank
4 mod 2 Dickson algebra H*(B(F2)4; F2)GL4(F2)and H*(B DI(4); Q2) is isomorphic
to the invariant ring of W (DI(4)).
Assuming that such a space exists, it is possible [10] to read off from its c*
*ohomol-
ogy a finite diagram that looks like the cohomological image of a diagram of sp*
*aces.
Some effort is required to verify that the picture seen in cohomology actually *
*is
realizable on the level of spaces. The generalized pushout of this diagram is *
*the
exotic 2-compact group.
Since G1 := SO(3) = DI(2) and G2 = DI(3), it is tempting to put G3 = DI(4).
No 2-compact group deserves the name DI(n) with n 5 [19, 35].
2.5. Cohomological invariants. If the space K is Fp-finite, also H*(K; Qp) is
finite dimensional over Qp, and the Euler characteristic
X X
O(K) = (-1)idimFpHi(K) = (-1)idimQp Hi(K; Qp)
and the cohomological dimension
cd(K) = max{i | Hi(K) 6= 0}
are defined [13, 4.3, 6.13].
For a connected p-compact group X, in particular, H*(X; Qp) is a connected
finite dimensional Hopf algebra so, by Borel [3] or Milnor-Moore [26], H*(X; Qp*
*)=
E(x1; : :;:xr) is an exterior algebra on finitely many generators of odd degree*
*, |xi| =
2di- 1, and H*(BX; Qp) = Qp[y1; : :;:yr] is a polynomial algebra on generators
of even degree, |yi| = 1 + |xi|; 1 i r. The number, r = rk(X), of generators
is [13, 5.9] the rank of X. The cohomologicalPdimension of X is [14, 3.8] given*
* by
cd(X) = max{i | Hi(X; Qp) 6= 0} = ri=1(2di- 1). For instance, cd(G^) = dimG,
rk(G^) is the rank of G, a p-compact r-torus has rank r, cd(P ) = rk(P ) if (an*
*d only
if) P is a p-compact toral group, rk(S2n-1) = 1 and cd(S2n-1) = 2n - 1, while
rk(DI(4)) = 3 and [2, Appendix A] cd(DI(4)) = 27.
Exercise 2.6.The trivial p-compact group has Euler characteristic O({1}) = 1.
The empty space has Euler characteristic O(;) = 0. For a connected p-compact
group X, X is trivial, O(X) 6= 0 , rk(X) = 0 [13, 5.10].
3. Morphisms
A p-compact group morphism f :X ! Y is a based map Bf :BX ! BY be-
tween the classifying spaces. The trivial morphism 0: X ! Y is the constant
map B0: BX ! BY , and the identity morphism 1: X ! X is the identity map
B1: BX ! BX .
Note the fibration sequence
(3.1) X -f!Y -! Y=f -! BX -Bf-!BY
where Y=f, or Y=X when f is understood, denotes the homotopy fibre of Bf.
Two morphisms f; g :X ! Yare conjugate if the maps Bf; Bg :BX ! BY are
freely homotopic and Rep(X; Y ) = ss0map (BX; BY ) = [BX; BY ] denotes the set
of conjugacy classes of morphisms of X to Y .
6 J. M. MOLLER
3.2. Monomorphisms, epimorphisms, and isomorphisms. The morphism
f :X ! Y is a monomorphism if Y=X is Fp-finite, an epimorphism if Y=X is the
classifying space of some p-compact group, and an isomorphism if Y=X is con-
tractible.
Example 3.3. {1} -! X is a monomorphism with X={1} = X, X -! {1} is
an epimorphism with {1}=X = BX, 1: X ! X is an isomorphism with X=X =
{1}, and the diagonal : X ! Xn is a monomorphism since Xn=X is homotopy
equivalent to Xn-1. There exists [12, 1.8] a monomorphism of S"pin(7)to DI(4).
These definitions are motivated by
Example 3.4. Let f :G ! H be a monomorphism (an epimorphism) of com-
pact Lie groups. The homotopy fibre of the induced map Bf :BG ! BH is
H=f(G) (B(kerf)) so the corresponding p-compact group morphism f^:^G! ^H
is a monomorphism (an epimorphism). (Not all morphisms between ^Gand ^Hare
induced from homomorphisms between G and H [16].)
A diagram X -! Y -! Z of p-compact group morphisms is a short exact sequence
if BX -! BY -! BZ is a fibration sequence. Any p-compact group sits in a short
exact sequence of the form X0 -!X -! ss0(X) where X0 is the identity component *
*of
X; the identity component of a p-compact toral group, for instance, is a p-comp*
*act
torus (2.1).
Exercise 3.5.Let f :X ! Y and g :Y ! Z be morphisms.
(1) If f and g are monomorphisms, then g O f is a monomorphism.
(2) If X is a p-compact toral group and g O f a monomorphism, then f is a
monomorphism.
(3) Assume that X -f!Y -g!Z is a short exact sequence. Show that f is a
monomorphism and g an epimorphism. Show also that if X is a p-compact
r-torus and Z a p-compact s-torus, then Y is a p-compact (r + s)-torus.
To be fair, part (2) of this exercise, requiring the theory of kernels [13, 7*
*.1-7.3],
is highly nontrivial. (The condition on X can be removed [13, 9.11].)
An inspection of the Serre spectral sequence for the left segment of (3.1) yi*
*elds
Proposition 3.6.[13, 6.14] If f :X ! Y is a monomorphism, then cd(Y ) =
cd(X) + cd(Y=X).
3.7. Nontrivial elements. The existence of nontrivial elements in nontrivial p-
compact groups represents the first and decisive step in constructing the maxim*
*al
torus.
Theorem 3.8. [13, 5.4, 5.5, 7.2, 7.3] Let X be a nontrivial p-compact group.
(1) There exists a monomorphism Z=p -!X.
(2) If X is connected, there exists a monomorphism S -! X from a p-compact
1-torus S to X.
HOMOTOPY LIE GROUPS 7
Note that (2) implies (1): In case the identity component X0 is nontrivial, u*
*se
(2) to get (3.5) a monomorphism Z=p -!S -! X0 -!X. Otherwise, (1) reduces to
obstruction theory. (A sketch of the proof of (3.8) will be presented in x6.)
Analogously, any nontrivial, connected compact Lie group contains a copy of
SO(2) (in its maximal torus).
Exercise 3.9.Use (3.8) and Lannes theory [22] to show that BX is Fp-finite only
if X is trivial. Next show that a p-compact group morphism which is both a
monomorphism and an epimorphism, is an isomorphism.
4. Homotopy Fixed Point Spaces
Let ss be a finite p-group and K a space. A ss-space with underlying space K *
*is
a fibration Khss-!Bss over Bss with fibre K. A ss-map is a map uhss:Khss! Lhss
over Bss.
The homotopy orbit space is the total space, Khss, and the homotopy fixed poi*
*nt
space, Khss, is the space of sections (which may very well be empty). These spa*
*ces
are related by the evaluation map Bss x Khss-! Khss.
For brevity, a ss-space will often be denoted by its underlying space and a s*
*s-map
by its restriction to the underlying spaces.
Example 4.1. The trivial ss-space with underlying space K is the trivial fibrat*
*ion
K x Bss -! Bss with homotopy orbit space Khss= Bss x K and homotopy fixed
point space Khss= map(Bss; K).
The homotopy fixed point construction Khssis functorial in both variables:
o For any ss-map u: K ! L , composition with uhss:Khss! Lhssdetermines
a map uhss:Khss! Lhss.
o For any subgroup < ss, any ss-space is also a -space. The inclusion
: ! ss induces a map Kh :Kh ! Khss over B: Bss ! B and a map
Kh :Khss! Kh of homotopy fixed point spaces.
The homotopy orbit space and the homotopy fixed point space are homotopy in-
variant constructions in that any ss-map u: K ! L, which is an ordinary (nonequ*
*iv-
ariant) homotopy equivalence, induces homotopy equivalences uhss:Khss! Lhss
and uhss:Khss! Lhss.
4.2. Exactness. Let U denote the (ordinary, nonequivariant) homotopy fibre of
a ss-map u: K ! L (where L is assumed to be connected), or, equivalently, the
homotopy fibre of uhss:Khss! Lhss. The pullback diagram
Uhss____//Khss
| |u
| | hss
fflffl| fflffl|
Bss__l__//_Lhss
shows that any homotopy fixed point l 2 Lhssmakes U into a ss-space such that
Proposition 4.3.[13, 10.6] U -! K -! L is a fibration sequence of ss-maps be-
tween ss-spaces and Uhss-! Khss-! Lhssis a fibration sequence of homotopy fixed
point spaces (where l 2 Lhssserves as base point).
8 J. M. MOLLER
4.4. Exponential Laws. The exponential law in its simple form reads
(4.5) map (Khss; A) = map(K; A)hss;
and in slightly more general form,
(4.6) Khss= (Kh )h(ss=)
as Khss= (Kh )h(ss=)for any normal subgroup C ss.
The standing assumption made here that ss be finite is not essential; ss can *
*(and
will) be replaced by a p-discrete toral group or even a p-compact toral group.
5.Centralizers
Let P be a p-compact toral group, Y any p-compact group, and g :P ! Y a
morphism of P to Y .
The centralizer of g, CY (g), or CY (P ) when g is understood, is the loop sp*
*ace of
BCY (g) = map (BP; BY )Bg, the mapping space component containing Bg. Note
the evaluation map BCY (g) x BP -! BY . Base point evaluation, BCY (g) -!BY ,
in particular, provides the first nontrivial example of a monomorphism.
Theorem 5.1. [13, 5.1, 5.2, 6.1] CY (g) is a p-compact group and CY (g) -!Y is*
* a
monomorphism.
The difficulty here is to show that CY (g) and Y=CY (g) are Fp-finite spaces.*
* (It
is unknown if this remains true with P replaced by a general p-compact group).
5.2. Central maps. The morphism g :P ! Y is said to be central if
(1) CY (g) -!Y is an isomorphism, or,
(2) g extends to a morphism Y x P -! Y which is the identity on Y .
These two conditions are equivalent as the adjoint of a morphism as in (2) is an
inverse to the evaluation monomorphism in (1).
Example 5.3. [15] [7, 2.5] [34, 9.6] [11, 12.5] Let CG (f) be the centralizer o*
*f a
homomorphism f :ss ! Gof a finite p-group ss into a compact Lie group G whose
component group is a finite p-group. Then ss0(CG (f)) is a p-group [17, A.4] and
there is an isomorphism
C"G(f)-! CG^^f
which is adjoint to the Fp-localization of the map BCG (f) x Bss -! BG induced
by the homomorphism CG (f) x ss -! G. Thus ^f:ss ! ^Gis a central morphism of
p-compact groups if f :ss ! Gis central as a homomorphism of Lie groups.
Here are two more examples of central morphisms.
Theorem 5.4. [13, 5.3, 6.1] The constant morphism 0: P ! Y is central.
This first example is an immediate consequence of the Sullivan conjecture as
proved by H. Miller [25].
In contrast to the very deep Theorem 5.4, nothing more than elementary ob-
struction theory is needed for the second example of a central morphism.
Lemma 5.5. The identity map 1: S ! S of a p-compact torus S is central.
HOMOTOPY LIE GROUPS 9
In other words, S is abelian. (See x8 for more on central maps and abelian
p-compact groups). In fact, any morphism into S is central.
Consider a morphism g :S ! Ydefined on a p-compact torus. Identifying S with
the centralizer of the identity morphism, composition of maps provides a morphi*
*sm
CY (g) x S -! CY (g). The restriction to the second factor is a central factori*
*zation
g0:S ! CY (g)of g through its own centralizer.
Lemma 5.6. [13, 8.2, 8.3] Suppose that g :S ! Y is a monomorphism of a p-
compact torus S to Y . Then there exists a short exact sequence
0
S -g!CY (g) -!CY (g)=g0
0
of p-compact groups such that S -g!CY (g) -!Y is conjugate to g.
Note that (5.6) asserts the existence of a classifying space B(CX (g)=g0) for*
* the
homogeneous space CX (g)=g0.
Exercise 5.7.Any monomorphism S -! S is an isomorphism.
6. Algebraic Smith theory
Suppose that ss is a finite p-group and that the p-compact group classifying
spaces BX and BY are ss-spaces. Let f :X ! Y be a monomorphism such that
Bf :BX ! BY is a ss-map. Choose a base point y 2 (BY )hssand equip Y=X
with the corresponding ss-space structure (4.3) such that Y=X -! BX -! BY is
a fibration sequence of ss-maps and (Y=X)hss-! (BX)hss-! (BY )hssa fibration
sequence of homotopy fixed point spaces.
Algebraic Smith theory, based on work by J. Lannes and his collaborators and
concerned with the cohomological properties, in particular the Euler characteri*
*stic
(2.5), of the fibre (Y=X)hss, can be summarized as follows.
Theorem 6.1. [13, 4.5, 4.6, 5.7] [9] [24] Under the above assumptions the foll*
*owing
hold:
(1) (Y=X)hssis Fp-finite.
(2) O((Y=X)hss) = O(Y=X) mod p.
(3) O((Y=X)hss) = (Y=X; ss) if ss is cyclic.
The Lefschetz number of (6.1(3)) is the alternating sum
1X
(Y=X; ss) = (-1)itraceHi(; Qp)
i=0
where Hi(; Qp) is the automorphism of Hi(Y=X; Qp) induced by any generator
of the cyclic group ss.
The analogous Euler characteristic formulas were known to be true in classical
Smith theory dealing with fixed point spaces for (reasonable) group actions on *
*finite
complexes.
I refrain from commenting on the proof of (6.1) but refer to [23] for more de*
*tailed
information.
A particularly advantageous situation arises when the finite p-group ss can be
replaced by a p-discrete torus (2.1) T.
10 J. M. MOLLER
Corollary 6.2.[13, 4.7, 5.7] [9] Suppose that Bf :BX ! BY is a T-map. Then:
(1) O((Y=X)hA) = O(Y=X) for any finite subgroup A < T.
(2) (Y=X)hT 6= ; if O(Y=X) 6= 0.
It is unknown if (1) also holds for infinite subgroups, such as T itself.
As a demonstration of the power of Smith theory, I now sketch the proof of (3*
*.8).
Let X be a connected, nontrivial p-compact group so that (2.6) rk(X) > 0.
Consider the map
(6.3) map (B; BX): map (BZ=pn+1; BX) ! map(BZ=pn; BX)
induced by the inclusion : Z=pn ! Z=pn+1 , n 0. Viewing BX as a trivial
Z=pn+1-space and BZ=pn = (Z=p)hZ=pn+1as the total space of a p-fold covering
map of BZ=pn+1, we get (4.1, 4.5)
n+1
map (BZ=pn+1; BX) = (BX)hZ=p
n+1 p hZ=pn+1
map (BZ=pn; BX) = map(Z=p; BX)hZ=p = (BX )
and map (B; BX) = (B)hZ=pn+1reveals itself as induced by the p-fold diagonal
(3.3) B: BX ! map(Z=p; BX) = BXp , a Z=pn+1-map with fibre Xp=X.
By (6.1), the homotopy fibre (Xp=X)hZ=pn+1of (6.3) is Fp-finite and the remar*
*k-
able, but straightforward, Lefschetz number calculation [13, 5.11],
n+1 p n+1 rk(X)
O((Xp=X)hZ=p ) = (X =X; Z=p ) = p ;
shows (2.6) that it is nonempty and noncontractible.
If all maps of BZ=p to BX were inessential, then (5.4) map (B; BX) would
be a homotopy equivalence of the total space map (BZ=p; BX) = BCX (0) to the
base space BX of (6.3) with n = 0 and the homotopy fibre would be contractible
_ but it is not, so there must exist an essential map Bf1: BZ=p ! BX , i.e. [1*
*3,
x7] a monomorphism f1: Z=p ! X. Since the homotopy fibre over Bf1 of (6.3)
with n = 1 is nonempty, f1 extends (up to conjugacy) to a monomorphism [13,
x7] f2: Z=p2 ! X. Proceeding inductively, we obtain a morphism f1 :Z=p1 ! X
that restricts to a monomorphism on Z=pn for all n 1. The Fp-localization of
Bf1 is (2.1) a monomorphism f :S ! X of a p-compact 1-torus S to X.
This proves (3.8). Also (5.1) is a quick consequence of (6.1).
7.Maximal tori and Weyl groups
Let X be any p-compact group. The maximal torus of X is constructed by an
inductive procedure.
If X(= CX ({1})={1}) is not homotopically discrete, it is (3.8) the target of*
* a
monomorphism S1 -!X defined on a p-compact 1-torus S1. This monomorphism
factors through its own centralizer (5.6) to give a short exact sequence
S1 -!CX (S1) -!CX (S1)=S1
of p-compact groups.
HOMOTOPY LIE GROUPS 11
If CX (S1)=S1 is not homotopically discrete, it is (3.8) the target of a mono*
*mor-
phism S2=S1 -!CX (S1)=S1 defined on a p-compact 1-torus S2=S1. Pull back along
this monomorphism induces a commutative diagram of p-compact group morphisms
S1_______//S2________//S2=S1
|| | |
|| | |
|| fflffl| fflffl|
S1_____//CX (S1)___//_CX (S1)=S1
where S2 is (3.5) a p-compact 2-torus and the middle arrow a monomorphism
(CX (S1)=S2 ' CX(S1)=S1_S2=S1is Fp-finite). Thus X is the target of a monomorph*
*ism
(3.5) S2 -!CX (S1) -!X defined on a p-compact 2-torus.
By dimension considerations (3.6), this inductive procedure eventually stops *
*at
a maximal torus for X where
Definition 7.1.[13, 8.8, 8.9] A maximal torus is a monomorphism i: T ! X of a
p-compact torus T to X such that CX (T )=T is a homotopically discrete p-compact
group.
We have thus established the existence part of (1.1).
Let i: T ! X be a maximal torus such that Bi: BT ! BX is a fi-
bration. The Weyl space WT(X) is the topological monoid of self-
maps of BT over BX. As a space, WT(X) is the fibre over B of
map(BT; B): map (BT; BT ) ! map(BT; BX) , i.e. (5.5)
a a
(7.2) WT(X) = (X=T )hT = CX (i)=CT(w) = CX (T )=T
w w
where the disjoint union indexed by all w 2 Rep(T; T ), necessarily (3.5, 5.7) *
*central
automorphisms, with i O w conjugate to i. The right hand side shows that the We*
*yl
space is homotopically discrete.
Definition 7.3.[13, 9.6] The Weyl group WT(X) is the component group (!)
ss0WT(X) of the Weyl space.
The homotopy fixed point space (X=T )hT from (7.2), by discrete approximation
(2.1) homotopy equivalent [13, 6.1, 6.7] to (X=T )hA for some finite subgroup A*
* < T,
is (6.1) Fp-finite and the computation (6.2)
(7.4) O(X=T ) = O((X=T )hA) = O((X=T )hT) = |WT(X) |
shows that the Euler characteristic of the homogeneous space X=T equals the ord*
*er
of the Weyl group; in particular, O(X=T ) > 0.
Another application of (6.2) now yields the uniqueness part of (1.1): Suppose
that i1: T1 ! X and i2: T2 ! X are maximal tori. The fact that (X=T2)hT1 =
(X=T2)hT1 6= ; 6= (X=T1)hT2 = (X=T1)hT2 means that there exist morphisms
u: T1 ! T2and v :T2 ! T1, necessarily isomorphisms (3.5, 5.7), such that i1 is
conjugate to i2 O u and i2 to i1 O v.
Approaching the key results of [13], we now specialize to connected p-compact
groups.
12 J. M. MOLLER
Proposition 7.5.[13, 9.1] Let X be a connected p-compact group and T -! X a
maximal torus. Then the morphism T -! CX (T ) is an isomorphism.
Again, the proof uses Smith theory.
With CX (T )=T = {1}, (7.2) shows that the monoid morphism WT(X) -!
Rep(T; T ) = [BT; BT ] is injective, i.e. that WT(X) is faithfully represented*
* in
H2(BT ; Qp) := H2(BT ; Zp) Q. The WT(X) -invariant map Bi: BT ! BX in-
duces an algebra map
H*(Bi; Qp):H*(BX; Qp) ! H*(BT ; Qp)WT(X)
into the invariant ring of this faithful representation. By means of a transfer
map, generalizing the Becker-Gottlieb transfer [1] to fibrations with Fp-finite*
* fibres
and ingeniously constructed [13, 9.13] via the Kan-Thurston theorem [20], Dwyer
and Wilkerson show that H*(Bi; Qp)is injective. The extension H*(BX; Qp)
H*(BT ; Qp)is finite since the fibre X=T is Fp-finite so these two rings have i*
*den-
tical Krull dimensions, i.e. rk(T ) = rk(X) (1.3.(1)). The culmination of [13]*
* is
(1.3.(2)) asserting that H*(BX; Qp) is isomorphic to the invariant ring of the *
*Weyl
group. The fact that the invariant ring is polynomial (2.5) implies, by the cl*
*as-
sical Shephard-Todd theorem [2, 7.2.1], that (1.3.(3)) WT(X) is represented as*
* a
reflection group in the vector space H2(BT ; Qp). Thus WT(X) must be isomorphic
to a product of irreducible p-adic reflection groups from the Clark-Ewing list *
*[2,
7.1] and H*(BX; Qp) must be isomorphic to a tensor product of the corresponding
graded invariant rings.
Example 7.6. Suppose that G is a compact Lie group with ss0(G) a p-group. Then
any Lie theoretic maximal torus T -! G induces a maximal torus ^T-!G^ of the p-
compact group ^G. The associated Weyl groups are isomorphic. The Weyl group of a
p-compact toral group P is ss0(P ). The Weyl group of the Sullivan sphere (S2n-*
*1)p
is Z=n and the Weyl group of the Clark-Ewing p-compact group (B(T oW ))p is
W . The Weyl group of DI(4) is W (DI(4)), abstractly isomorphic to the product *
*of
a cyclic group of order two and the simple group of order 168.
Exercise 7.7.Modify the above construction of the maximal torus for a p-compact
group to obtain an unconventional construction of the maximal torus for a compa*
*ct
Lie group [23] [13, 1.2].
While the p-adic rational cohomology H*(BX; Qp) is under control, it is quite
another matter with the cohomology algebra H*(BX) with coefficients in Fp. The
difference between the two situations is that H*(BX; Qp) embeds into the poly-
nomial ring H*(BT ; Qp)while H*(BX) embeds into H*(BNp(T )) where BNp(T ),
the p-normalizer of the maximal torus [13, 9.8], is the Borel construction for *
*the
action of a Sylow p-subgroup of the Weyl group on BT . As a result, H*(BX) need
not be polynomial but can actually be surprisingly complicated [36]. However,
Dwyer and Wilkerson are able to settle the perhaps most basic structural questi*
*on.
Theorem 7.8. [13, 2.3] H*(BX) is a finitely generated Fp-algebra for any p-
compact group X.
Given (7.8) it is but a small step to verify [13, 1.1] the old conjecture that
H*(BX) is a finitely generated Fp-algebra for any finite loop space X.
HOMOTOPY LIE GROUPS 13
8. Classification
The classification scheme for p-compact groups, not yet completed, exhibits a
pronounced analogy to Lie theory.
8.1. Centers. A p-compact group is abelian if its identity map is central. By
(5.5), p-compact tori are abelian.
Theorem 8.2. [11, 1.1] [29, 3.1] A p-compact group is abelian if and only if i*
*t is
isomorphic to a product of a finite abelian p-group and a p-compact torus.
If Z -! X is a central monomorphism, then [11, 5.1] [29, 3.5] Z is abelian.
Theorem 8.3. [11, 1.2] [29, 4.4] For any p-compact group X there exists a cent*
*ral
monomorphism Z(X) -! X such that any central monomorphism into X factors,
in an essentially unique way, through Z(X).
The terminal central monomorphism, esentially unique, of (8.3) is the center *
*of
X.
The (discrete approximation to the) center can be defined as the group of ele-
ments in (the discrete approximation to) CX (T ) that are central in X. Another
candidate to the center title is the centralizer of the identity morphism. Fort*
*unately,
there is no discrepancy.
Theorem 8.4. [11, 1.3] The map BZ(X) -! map(BX; BX)B1, corresponding to
the isomorphism CX (Z(X)) -!X, is a homotopy equivalence.
The highly nontrivial proof of (8.4) involves decomposing BX as a generalized
pushout _ a technique also applied in the proof of (2.4).
Example 8.5. Let G be a connected compact Lie group with Lie theoretic center
Z(G). Then the maps
(BZ(G))p -!map (BG^; BG^)B1- BZ(G^)
adjoint to the Fp-localization of BZ(G) x BG -! BG and BZ(G^) x BG^ -!BG^,
respectively, are homotopy equivalences [11, 1.4, 12.1] (8.4). The center of t*
*he
p-compact group ^E6is cyclic of order 3 if p = 3 and trivial otherwise.
For a connected p-compact group X [29, 5.2],
(8.6) rk(Z(X)) = dimQp(ss1(X) Qp) = dimQpH2(BT ; Qp)WT(X)
so the center is finite if and only if the fundamental group is. The quotient *
*p-
compact group [13, 8.3] X=Z(X) has trivial center [11, 6.3] [29, 4.6].
The classification of connected p-compact groups essentially reduces to the s*
*im-
ply connected case by the following result.
Theorem 8.7. [29, 5.4] For any connected p-compact group X, there exists a sho*
*rt
exact sequence
A -!Y x S -! X
where A is a finite abelian p-group, A -!Y x S -pr1-!Y is a central monomorphism
into the simply connected p-compact group Y , and S is a p-compact torus.
14 J. M. MOLLER
In (8.7), Y is the universal covering p-compact group of X, A is the torsion
subgroup of ss1(X), and S = Z(X)0 is the identity component of the center.
8.8. Semisimplicity. Call a connected p-compact group with maximal torus T -!
X simple if the faithful representation of the Weyl group WT(X) in H2(BT ; Qp)*
* is
irreducible. Sullivan spheres and Clark-Ewing p-compact groups (2.2) are simple
by design. G^is simple for any connected compact simple Lie group G.
For any connected p-compact group, the WT(X) -representation H2(BT ; Qp)
splits as a direct sum
H2(BT ; Qp) = M1 . . .Mn
of irreducible representations. Provided the center Z(X) = 0 is trivial, all Mi*
* are
nontrivial (8.6) and [14, 1.5] this splitting of Qp[WT(X) ]-modules descends t*
*o a
splitting of Zp[WT(X) ]-modules,
H2(BT ; Zp) = L1 . . .Ln
where Li= H2(BT ; Zp) \ Mi. The splitting criterion [14, 1.4], guaranteeing the*
* re-
alizability as a splitting of X of any splitting of the Zp[WT(X) ]-module H2(B*
*T ; Zp),
now leads to the main result on semisimplicity.
Theorem 8.9. [14, 1.3] [30] Any connected p-compact group with trivial center *
*is
isomorphic to a product of simple p-compact groups.
For any simply connected p-compact group Y , Z(Y ) is finite, and the decompo-
sition of the center free quotient Y=Z(Y ) into simple factors lifts to a decom*
*position
of Y into simple factors [14, 1.6].
The decompositions of (8.7) and (8.9) are also useful for the classification *
*of
endomorphisms of p-compact groups [28] [27].
The final stage, still unresolved, of the classification scheme has to addres*
*s exis-
tence and uniqueness of simple p-compact groups. At the core of this problem, t*
*wo
obstruction groups loom.
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E-mail address: moller@math.ku.dk