Vector bundles over classifying spaces of compact Lie groups
Transcript Of Vector bundles over classifying spaces of compact Lie groups
Acta Math., 176 (1996), 109-143 (~) 1996by Institut Mittag-Leffler.All rights reserved
Vector bundles over classifying spaces of compact Lie groups
by
STEFAN JACKOWSKI
and
Uniwersytet Warszawski Warszaw, Poland
BOB OLIVER
Universitd Paris Nord Villetaneuse, France
In this paper, we describe the vector bundles over the classifying space BG of a compact
Lie group G, up to stabilization by bundles coming from linear representations of G. In
particular, the Grothendieck group of vector bundles over BG is expressed in terms of
the representation rings of certain subgroups of G. Define Vect(X), for any space X, and Rep(G), for any group G, to be the abelian
monoids of isomorphism classes of complex vector bundles over X and complex finitedimensional G-representations, respectively; with addition in both cases defined by direct sum. Let
(~G:Rep(G) -~ Vect(BG)
be the homomorphism defined by sending a complex G-representation V to the vector
bundle (EG•
associated to the universal principal bundle EG~BG. This map
is already known to be bijective when G is a finite p-group [DZ] or when G is p-toral
[Nb]. (A group G is p-toral if its identity component Go is a torus and G/Go is a finite
p-group.) But in general, c~Gis neither surjective nor injective, and Vect(BG) can in fact
even be uncountable. The situation does, however, become much simpler after passing
to Grothendieck groups.
For each p-toral subgroup P of G, consider the composite
Vect(BG) restr)Vect(BP) ~ 1 Rep(P) C R(P),
where R(P) is the complex representation ring of G. These maps define a homomorphism re: Vect(BG) ~ Rp (G) ~f li___mR(P),
P
where the inverse limit is taken over all p-toral subgroups of G (for all primes p) with respect to inclusion and conjugation of subgroups.
Now let K(X), for any space X, denote the Grothendieck group of the monoid Vect(X). Our main result (Theorem 1.8 below) is the following:
The first author was partly supportedby PolishScientificGrant 2P30101007.
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S. JACKOWSKI AND B. OLIVER
THEOREM. For any compact Lie group G, ra extends to an isomorphism of groups
fa:K(BG) ~-, R~,(G).
Furthermore, any vector bundle over BG is a summand of a bundle ( E G x a V ) ~ B G for some G-representation V; and thus K(BG) can be obtained from Vect(BG) by inverting only those vector bundles coming from G-representations.
In contrast to the situation for bundles over finite-dimensional spaces, any subbundle of a trivial bundle over BG is itself trivial. For finite G, this follows from the version of the Sullivan conjecture proven by Miller [Mi, Theorem A], since any summand of a trivial bundle is classified by a map into a finite-dimensional Grassmannian. When dim(G)>0, it follows, via a somewhat more complicated argument, from [FM, Theorem 3.1].
When G is connected, then R~,(G)mR(G), and the theorem implies that an: R(G)--* K(BG) is an isomorphism.
When X is compact or finite-dimensional, K(X) is by definition equal to the Ktheory ring K(X). This is usually extended to a representable functor K(X), defined for an arbitrary space X, by setting K(x)d-e--f[X, Z x BU]. Here, U denotes the infinite increasing union of the unitary groups U(n). The obvious natural transformation
~x:K(X)--*K(X)
need not be an isomorphism. In fact, the geometrically defined functor K(-) can behave very differently from K ( - ) . For example, K ( - ) is not exact, and it does not satisfy Bott periodicity in general (see the discussion after Theorem 1.1). This helps to explain why K ( - ) is more difficult to compute than K ( - ) .
The K-theory of classifying spaces of compact Lie groups has been computed by Atiyah and Segal [AS]. Their completion theorem says that the composite
R(G) aa K(BG) ~a K(BG)
extends to an isomorphism aA n: R(G) A -~:-~K(BG), where R(G)^ is the completion of the representation ring with respect to its augmentation ideal. We thus have the following commutative diagram
ool o+ R(G) ~G , R(G)^
K(SG),
where ha denotes the completion homomorphism.
VECTOR BUNDLES OVER CLASSIFYING SPACES OF COMPACT LIE GROUPS
111
One consequence of the above theorem is that ~BG is a monomorphism (Corollary 1.9). Its image can in fact be described internally, using the exterior power opera-
tions on K(BG). Adams, in [Ad], defined and studied the subgroup FF(BG)C_K(BG) generated by the "formally finite-dimensional elements"; i.e., those elements xEK(BG) such that Ak(x)=Ofor k sufficiently large. Our results, when combined with his, imply that FF(BG)=Im(13Ba).
The Atiyah-Segal completion theorem also describes the groups K-i(BG) for i>0; i.e., the homotopy groups of the mapping space map(BG, Z • BU). Since ]__[n~__0BU(n)
is a topological monoid and commutative up to homotopy, the space of maps from X into it (the "topological monoid of vector bundles over X") is also a homotopy commutative topological monoid. Thus, we can consider its topological group completion ~ c ( X ) , where 7r0(~C(X))~K(X). When X is a finite complex, ~C(X) has the homo-
topy type of map(X, Z • BU). In Proposition 2.4 below, we show that when G is finite, the connected components of ~ c (BG) have the same homotopy type as the components of map(BG, Z • BU). In contrast, even when G is a (nontrivial) torus, then the components of ~ c (BG) are quite different from those of map(BG, Z • BU) (see Proposition 2.5),
and in fact their homotopy groups are nonvanishing in odd degrees.
The above discussion has focused on the case of complex bundles, but all of these results (except for those in Proposition 2.5) are also shown to hold for real bundles.
This paper grew out of our earlier efforts to understand maps between the classi-
fying spaces of compact Lie groups. The set of n-dimensional vector bundles over BG corresponds to that of homotopy classes of maps from BG to BU(n). Thus the starting
point for our computation of K(BG) are the theorems of Dwyer-Zabrodsky and Not-
bohm which describe up to p-completion the mapping space map(BP, BL) for a p-total
group P and an arbitrary compact Lie group L (Theorem 1.1 below). A decomposition
of BG at any prime p as a homotopy direct limit of classifying spaces of p-toral sub-
groups of G (Theorem 1.2) provides a tool for passing to more general groups. The key new element in the proof of the main theorem is provided by the vanishing of certain higher derived functors of inverse limits, which in turn depends on the equivariant Bott periodicity theorem.
In the course of the proof of vanishing higher limits, we also show the following extension of Smith theory (Proposition 3.3). If X is a finite-dimensional G-complex with
finitely many orbit types and all isotropy subgroups p-total, then X H is also Fp-acyclic, not only for HCG a p-total subgroup (as follows from Smith theory), but also whenever
H is a subgroup of any p-toral subgroup of G.
The main results about K(BG) and KO(BG) axe shown in w and those about the
spaces ~C(BG) and ~R(BG) in w Also, two other algebraic descriptions of Rp(G) are
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S. JACKOWSKI AND B. OLIVER
given in w (Proposition 1.12 and the following discussion). The vanishing theorem for higher inverse limits needed in the first two sections is shown in w
Both authors would like to thank the Mittag-Leftier Institute for its hospitality while the idea of this work emerged and was carried out. The first author also thanks Universit~ Paris Nord, and the Sonderforschungsbereich 170 in Gbttingen, for their hospitality during later stages of the work. We would also like to thank Haynes Miller, Max Karoubi, and Charles Thomas for their very helpful suggestions; and Ib Madsen for his many
comments as editor of Acta on the exposition.
Notation. All complex (real) representations of G are assumed to be equipped with
G-invariant hermitian (inner) product. For any such representation V, we let Aut(V) denote the group of unitary (orthogonal) automorphisms of the vector space V, and
Qv: G--*Aut(V) the homomorphism induced by the action. Also, for any HC_G,AutH(V)
denotes the subgroup of H-equivariant automorphisms. It will be convenient to state some of the results simultaneously for real and complex
vector bundles, or for orthogonal and unitary groups. In such situations, if F = C or R,
we write U(n,F) for U(n) or O(n), respectively. Similarly, KF(-)=K(-) or KO(-) and RF(-)=R(-) or RO(-). And VectV(X) denotes the monoid of F-vector bundles
over a space X, K F ( X ) (=K(X) or KO(X)) its Grothendieck group, and RepF(G) the monoid of F-representations of the group G.
Throughout the paper, Yp denotes the p-completion of a space Y in the sense of Bousfield and Kan [BK].
1. K(BG) for a compact Lie group G
Throughout this section, G denotes a fixed compact Lie group. Rather than working di-
rectly with vector bundles over BG, we work with their classifying maps BG-~BU(n, F) (F=C or R), via the isomorphism of monoids
co
VectF(BG)~-H [BG,BU(n, F)].
n~O
Maps BG---~BU(n,F) are studied using our general strategy for studying maps between
classifying spaces, as outlined in [JMO2, w The first part of this section thus consists mostly of results already shown elsewhere, and needed in the proof of the main theorem (Theorem 1.8). The main exception to this is Proposition 1.5, where we prove the vanishing of the higher inverse limits which occur as obstructions, using also the results in w
VECTOR BUNDLES OVER CLASSIFYING SPACES OF COMPACT LIE GROUPS
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The starting point for understanding vector bundles over BG is the following theorem, which in particular implies that K(BP)~-R(P) for any p-toral group P. In fact, the theorem says that the underlying monoids are isomorphic in this case.
THEOREM 1.1 (Dwyer-Zabrodsky and Notbohm). Set F = C or R. For any prime p and any p-toral group P, the homomorphism of monoids
aF: RepF(p) ~
oo
VectF(Bp) ~- I I [BP, BU(n, F)],
rt~O
which sends a representation V to the vector bundle (E P • p V )+B P (or to the map BOy), is an isomorphism. Also, for any P-representation V over F, which is odd-dimensional if F = R , the homomorphism P x A u t p ( V ) (Qv,incl)Aut(V) induces (by adjointness) a homotopy equivalence
B Autp(V)p -~>map(BP, B Aut(V)~')Bov.
Proof. These are special cases of the following theorems of Dwyer and Zabrodsky [DZ] (when P is a finite p-group) and Notbohm [Nb] (in the general case). For any compact Lie group L, the map
Hom(P, L)/Inn(L) ~
[BP, BL]
(1)
is a bijection. If L is connected, then for any 0: P--*L, the homomorphism
PXCL(o(P)) (e,ind)L
induces a homotopy equivalence
BCL(O(P))p ~->map(BP, (SL)p)s Q.
(2)
Point (2) follows easily from results in [DZ] and [Nb], and is shown explicitly in [JMO,
Theorem 3.2 (iii)].
Theorem 1.1 is just the special case of (1) and (2) when L=Aut(V)~-U(n) or
O(n). Note in particular that if F = R and n=dim(V) is odd, then Aut(V)~-O(n) ~-
SO(n) x {=kI}.
[]
Theorem 1.1 provides some simple examples of the exotic behavior of the functor K ( - ) . For example, if Cp denotes the cyclic subgroup of order p, for any prime p, then the sequence
K(BS1/BCp) P-~ K ( B S ' ) restr K(BCp)
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is not exact. More precisely, if {,~IBS1 (for nEZ) denotes the line bundle with Chern class n times some fixed generator of H 2(BS1), then [{1]- [{p+l] lies in the kernel of the above restriction map, but not in the image of K(BS1/BCp). One can also show, using Theorem 1.1 again (and Propositions 2.3 and 2.5 below), that for any prime p and any finite p-group P,
K(BP) ~-R(P)- Zr (where r =rk(R(P))), ~:(p2 (BP+)) = ~:((S 2x B P ) / B P ) ~- Z,
and K ( S 2x B P ) / K ( B P ) ~-Z x (~p)r-1.
In particular, these groups are pairwise nonisomorphic (if P # 1), and so Bott periodicity
fails for K(-).
To pass from p-toral groups to an arbitrary compact Lie group G, we use a decompo-
sition of BG, at each prime p, as a homotopy direct limit of classifying spaces of p-toral
subgroups of G. This decomposition is indexed by a certain orbit category TIp(G). The
objects in Tip(G) are the orbits G / P such that (1) PC_G is p-total, (2) N ( P ) / P is finite,
and (3) there is no normal p-subgroup Ir
The morphisms in Tip(G) are
the G-maps between orbits. This is a discrete category, and is in fact equivalent to a
finite category (see Lemma 1.7 below). It is also a "directed" or "EI" category, in that all
endomorphisms of Tip(G) are isomorphisms, and hence there are morphisms in at most
one direction between any pair of nonisomorphic objects. The importance of Tip(G) lies
in the following theorem.
THEOREM 1.2. For any prime p, the map
qG,p: hocolim (EG/P) --+BG,
G/P6T~v (G)
induced by the projection EG--+BG, is an Fp-homology equivalence. In particular, for any connected complex Y, qa,p induces a homotopy equivalence
(--~
map(SG, Y~) e)map( hocolim (EG/P), Yp).
G/PE'R.v(G )
Proof. The first statement is shown in [JMO, Theorem 1.4]; and the second then
follows from [BK, Proposition II.2.8].
[]
Theorem 1.2 will be directly useful only to describe maps from BG to the BU(n, F)p; i.e., after p-completion. The next proposition will allow us to combine such maps, to get maps to BU(n, F) itself.
VECTOR BUNDLES OVER CLASSIFYING SPACES OF COMPACT LIE GROUPS 115
PROPOSITION 1.3. Let T C G be a maximal torus of G, and set w = I N ( T ) / T 1. Then the following square is a pullback if F=C, or if F = R and n is odd:
[BG, BU(n, F)] restr 1
[BT, BU(n, F)]
9 1-l l [Ba, BU(n, F);]
restr 1 > I'Ipl~o[BT, B U ( n , F)p].
Proof. This is a consequence of the arithmetic pullback square for the space
BU(n, F). For details, see [JMO3, Proposition 1.2].
[]
In view of Theorem 1.2, we are faced with the problem of comparing maps defined on a homotopy direct limit with maps defined on its pieces. This will be studied via the obstruction theory described in the next proposition.
This obstruction theory is based on a skeletal decomposition of the homotopy colimit which generalizes the classical construction of obstructions related to the skeletal decomposition if the source space is a polyhedron (or CW-complex). For each n ) 0 , the "n-skeleton" of hocolim(F) is defined by setting
h~
( H F(c~
\ k = 0 \aO'-*...-'-~ck
where one divides out by face and degeneracy relations. The space hocolim(F) itself is just the case n=co. Note in particular that hocolim(~ is the disjoint union of
the F(c), taken over all ceOb(C), and that hocolim(1)(F) is the union of the mapping
cylinders of all maps induced by morphisms in C. The following result describes the obstructions we will need to consider.
PROPOSITION 1.4. Fix a discrete category C, and a (covariant) functor F: C--*Top.
Let Y be any other space, and fix maps fc: F(c)--*Y (for all cEOb(C)) whose homo-
topy classes define an element ]=([fc])~ecers
(i.e., life extends to a map
hocolim(1)(F) --~Y ) . Set
an (c) = ~-n(map(F(c), Y), f~)
for all ceOb(C). Then given a map fn: hocolim(n)(F)---~Y (any n ) l ) which extends ], the obstruction to constructing a map on hocolim(n+l) (F) which extends
f= [hocolim(=-1)(F)
C
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S. JACKOWSKI AND B. OLIVER
lies in li_mn+l(an). Also, given two maps f,f':hocolimc(F)--*Y and a homotopy Fn defined on hocolim) (n)(F) (any n)O), the obstruction to constructing a homotopy on hocolim(~+1) (F) x I which extends
Fn[ (hocolira(n-l) (F) • I)
C
lies in lie_.__nln+l(O/n+l).Both of these obstructions are natural in Y.
Proof. This is shown by Wojtkowiak in [Wo]. Note in particular that for an to be
well defined as a functor from C to groups or abelian groups, one must first choose a map
fl: hocolim 0) (F)--*Y which can be extended to hocolim(2)(F). This applies to all of the
above situations except where one wants to extend a map fl from hocolim(t) (F) to the
2-skeleton; and in this case the obstruction set lim2(al) is defined for any functor
from C to the category of groups with morphisms given by conjugacy classes of homo-
morphisms.
[]
When applying Proposition 1.4, we will need to deal with the higher limits of homotopy groups of mapping spaces
map( EG / P, B Aut(V)p) s o , ~--B Autp(V)p
(Theorem 1.1), where Autp(V) is a product of unitary, orthogonal, and symplectic groups. Since the higher homotopy groups of these spaces are unknown, we instead stabilize, by taking limits over all VERepF(G). Such limits can be made more precise by taking them over some sequence V1C_C1_/2c V3c_... of G-representations (with G-invariant inner or hermitian product), such that each finite-dimensional G-representation is contained in Vk for sufficiently large k. We can of course assume that each Vk is odddimensional (in order to apply Theorem 1.1 or Proposition 1.3). Alternatively, one can choose a "universal" G-representation U (with G-invariant inner product), i.e., a representation which contains infinitely many copies of each irreducible G-representation, and then take the limit over all finite-dimensional subrepresentations of U.
PROPOSITION 1.5. For each i>0, let HF:T~p(G)---~Zp-rrtod be the functor defined by setting
HE(G/P)= lim ~ri(map(EG/P, B Aut(V)p),Bpv).
VERepF(G)
Then
VECTOR BUNDLES OVER CLASSIFYING SPACES OF COMPACT LIE GROUPS
117
as functors on T~p(G), and
li+___mj II~ -- 0
(2)
~e~(G)
for all i,j>0.
Proof. By Corollary 3.6, fi__mJnp(a)(Zp•
for all i,j>O. So (2) follows
immediately once point (1) has been shown.
Fix a G-representation V over F, such that dim(V) is odd if F = R . By Theorem 1.1,
for any p-toral subgroup PCG, the group homomorphism
P • Autp(Y) (~v,incl)) Aut(V)
is, after taking classifying spaces, adjoint to a homotopy equivalence
cp,y :S Autp (Y)p - ~ map(BP, B Aut(Y)p )Sev .
So for each i>0, cp,y induces an isomorphism
Cip, v . 9Zp|
Autp(Y)) ~ 7ri(map(BP, B Aut(Y)p), BQv).
By construction, the cip,_ are natural with respect to inclusions of representations V C W (where this means taking direct sums with the basepoints of the corresponding spaces in
the orthogonal complement of V). Hence we can take the limit over all VCRepF(G) to
get an isomorphism
cip: Zp|
v
B Autp(Y))) -~ fi__,m(lri(map(B P, S Aut(V)), BOy)) = HE(G/P).
V
For each P and V, Autp(V)~-AutG(G/P•
the group of G-bundle auto-
morphisms covering the identity on G/P. Thus, for each i>0,
Iri(B Autp (Y)) ~ 7ri_l(AutG (G/P x Y l G/P))
can be identified with the set of isomorphism classes of G-vector bundles over
Ei(G/p+ ) ~- (DiA(G/ p+ ))Us~_IA(G/p+) (DiA(G/P+ ))
with a fixed identification of V with the fiber over the base point. Hence
Yl__,m(Tri(BAutp(V))) ~ K-FG(Ei(G/P+ )) = KF~i(G/P);
V
and this together with c~ defines the isomorphism KF~i(G/P)~-HF(G/P).
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It remains to show that this isomorphism is natural with respect to morphisms in T~p(G). We do this by replacing Cp,y, for each V, by a homotopy equivalent map which is clearly functorial in G/P. Regard EG and BAut(V) as the nerves of topological categories CG and BAut(V): CG has as objects the elements of G and has a unique morphism between each pair of objects; while BAut(V) has one object, and a morphism for each element of Aut(V). Let gzv:EGxGBAut(V)--*BAut(V) (where G acts on B Aut(V) via 0y and conjugation) be the map induced by the functor $G x B Aut(V)---~ BAut(Y) which sends each morphism ([g--~h], a) to Qy(g).a.Qy(h) -1.
For any G / P in T~p(G), let
ev: G / P x B Auta(G/Px V ~G/P)p -* B Aut (V)p
be the evaluation map: for each gPEG/P, ev(gP,-) is induced by restriction to the fiber over gP. Define
CP,V = (r
~(Id x ev): E G / P x B Autv ( G / P x V ~ G/P)p = EGxG(G/P x B Autv(G/Px Y ~G/P)~) -~ B Aut (V)p;
and let
C'p,y: B Autc ( G / P x Y l G/P)~ --*map(EG/P, B Aut (Y)~)
be its adjoint map. The restriction of Cp,u to B P x B Autp (V) is just the map induced by
(Yy, incl) and multiplication. So cP' ,V is homotopy equivalent to the map cp,v constructed
earlier, and is natural in G / P by construction.
[]
In general, of course, direct and inverse limits cannot be switched. The following lemma describes one case where this can be done.
LEMMA 1.6. Fix a finite category C and a directed category 7), and let M: d• be any functor to abelian groups. Then for any i>~O,
li~__mi(libra(M)) ----li___~m(li~__m(i M)) 9
C D
D C
Proof. For any functor M':C--*Ab, the higher limits li_mc(M' ) are the homology groups of a cochain complex
0 ,H/,(c)--Hi'(c,), H
C~
CO--~CI
Co --~CI ---*C2
(cf. [BK, XI.6.2] or [02, Lemma 2]). In particular, this applies when M ' = M ( - , d) for any d in 7), and when Ml=li_m d M ( - , d). Since d is a finite category, all of the products
Vector bundles over classifying spaces of compact Lie groups
by
STEFAN JACKOWSKI
and
Uniwersytet Warszawski Warszaw, Poland
BOB OLIVER
Universitd Paris Nord Villetaneuse, France
In this paper, we describe the vector bundles over the classifying space BG of a compact
Lie group G, up to stabilization by bundles coming from linear representations of G. In
particular, the Grothendieck group of vector bundles over BG is expressed in terms of
the representation rings of certain subgroups of G. Define Vect(X), for any space X, and Rep(G), for any group G, to be the abelian
monoids of isomorphism classes of complex vector bundles over X and complex finitedimensional G-representations, respectively; with addition in both cases defined by direct sum. Let
(~G:Rep(G) -~ Vect(BG)
be the homomorphism defined by sending a complex G-representation V to the vector
bundle (EG•
associated to the universal principal bundle EG~BG. This map
is already known to be bijective when G is a finite p-group [DZ] or when G is p-toral
[Nb]. (A group G is p-toral if its identity component Go is a torus and G/Go is a finite
p-group.) But in general, c~Gis neither surjective nor injective, and Vect(BG) can in fact
even be uncountable. The situation does, however, become much simpler after passing
to Grothendieck groups.
For each p-toral subgroup P of G, consider the composite
Vect(BG) restr)Vect(BP) ~ 1 Rep(P) C R(P),
where R(P) is the complex representation ring of G. These maps define a homomorphism re: Vect(BG) ~ Rp (G) ~f li___mR(P),
P
where the inverse limit is taken over all p-toral subgroups of G (for all primes p) with respect to inclusion and conjugation of subgroups.
Now let K(X), for any space X, denote the Grothendieck group of the monoid Vect(X). Our main result (Theorem 1.8 below) is the following:
The first author was partly supportedby PolishScientificGrant 2P30101007.
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S. JACKOWSKI AND B. OLIVER
THEOREM. For any compact Lie group G, ra extends to an isomorphism of groups
fa:K(BG) ~-, R~,(G).
Furthermore, any vector bundle over BG is a summand of a bundle ( E G x a V ) ~ B G for some G-representation V; and thus K(BG) can be obtained from Vect(BG) by inverting only those vector bundles coming from G-representations.
In contrast to the situation for bundles over finite-dimensional spaces, any subbundle of a trivial bundle over BG is itself trivial. For finite G, this follows from the version of the Sullivan conjecture proven by Miller [Mi, Theorem A], since any summand of a trivial bundle is classified by a map into a finite-dimensional Grassmannian. When dim(G)>0, it follows, via a somewhat more complicated argument, from [FM, Theorem 3.1].
When G is connected, then R~,(G)mR(G), and the theorem implies that an: R(G)--* K(BG) is an isomorphism.
When X is compact or finite-dimensional, K(X) is by definition equal to the Ktheory ring K(X). This is usually extended to a representable functor K(X), defined for an arbitrary space X, by setting K(x)d-e--f[X, Z x BU]. Here, U denotes the infinite increasing union of the unitary groups U(n). The obvious natural transformation
~x:K(X)--*K(X)
need not be an isomorphism. In fact, the geometrically defined functor K(-) can behave very differently from K ( - ) . For example, K ( - ) is not exact, and it does not satisfy Bott periodicity in general (see the discussion after Theorem 1.1). This helps to explain why K ( - ) is more difficult to compute than K ( - ) .
The K-theory of classifying spaces of compact Lie groups has been computed by Atiyah and Segal [AS]. Their completion theorem says that the composite
R(G) aa K(BG) ~a K(BG)
extends to an isomorphism aA n: R(G) A -~:-~K(BG), where R(G)^ is the completion of the representation ring with respect to its augmentation ideal. We thus have the following commutative diagram
ool o+ R(G) ~G , R(G)^
K(SG),
where ha denotes the completion homomorphism.
VECTOR BUNDLES OVER CLASSIFYING SPACES OF COMPACT LIE GROUPS
111
One consequence of the above theorem is that ~BG is a monomorphism (Corollary 1.9). Its image can in fact be described internally, using the exterior power opera-
tions on K(BG). Adams, in [Ad], defined and studied the subgroup FF(BG)C_K(BG) generated by the "formally finite-dimensional elements"; i.e., those elements xEK(BG) such that Ak(x)=Ofor k sufficiently large. Our results, when combined with his, imply that FF(BG)=Im(13Ba).
The Atiyah-Segal completion theorem also describes the groups K-i(BG) for i>0; i.e., the homotopy groups of the mapping space map(BG, Z • BU). Since ]__[n~__0BU(n)
is a topological monoid and commutative up to homotopy, the space of maps from X into it (the "topological monoid of vector bundles over X") is also a homotopy commutative topological monoid. Thus, we can consider its topological group completion ~ c ( X ) , where 7r0(~C(X))~K(X). When X is a finite complex, ~C(X) has the homo-
topy type of map(X, Z • BU). In Proposition 2.4 below, we show that when G is finite, the connected components of ~ c (BG) have the same homotopy type as the components of map(BG, Z • BU). In contrast, even when G is a (nontrivial) torus, then the components of ~ c (BG) are quite different from those of map(BG, Z • BU) (see Proposition 2.5),
and in fact their homotopy groups are nonvanishing in odd degrees.
The above discussion has focused on the case of complex bundles, but all of these results (except for those in Proposition 2.5) are also shown to hold for real bundles.
This paper grew out of our earlier efforts to understand maps between the classi-
fying spaces of compact Lie groups. The set of n-dimensional vector bundles over BG corresponds to that of homotopy classes of maps from BG to BU(n). Thus the starting
point for our computation of K(BG) are the theorems of Dwyer-Zabrodsky and Not-
bohm which describe up to p-completion the mapping space map(BP, BL) for a p-total
group P and an arbitrary compact Lie group L (Theorem 1.1 below). A decomposition
of BG at any prime p as a homotopy direct limit of classifying spaces of p-toral sub-
groups of G (Theorem 1.2) provides a tool for passing to more general groups. The key new element in the proof of the main theorem is provided by the vanishing of certain higher derived functors of inverse limits, which in turn depends on the equivariant Bott periodicity theorem.
In the course of the proof of vanishing higher limits, we also show the following extension of Smith theory (Proposition 3.3). If X is a finite-dimensional G-complex with
finitely many orbit types and all isotropy subgroups p-total, then X H is also Fp-acyclic, not only for HCG a p-total subgroup (as follows from Smith theory), but also whenever
H is a subgroup of any p-toral subgroup of G.
The main results about K(BG) and KO(BG) axe shown in w and those about the
spaces ~C(BG) and ~R(BG) in w Also, two other algebraic descriptions of Rp(G) are
112
S. JACKOWSKI AND B. OLIVER
given in w (Proposition 1.12 and the following discussion). The vanishing theorem for higher inverse limits needed in the first two sections is shown in w
Both authors would like to thank the Mittag-Leftier Institute for its hospitality while the idea of this work emerged and was carried out. The first author also thanks Universit~ Paris Nord, and the Sonderforschungsbereich 170 in Gbttingen, for their hospitality during later stages of the work. We would also like to thank Haynes Miller, Max Karoubi, and Charles Thomas for their very helpful suggestions; and Ib Madsen for his many
comments as editor of Acta on the exposition.
Notation. All complex (real) representations of G are assumed to be equipped with
G-invariant hermitian (inner) product. For any such representation V, we let Aut(V) denote the group of unitary (orthogonal) automorphisms of the vector space V, and
Qv: G--*Aut(V) the homomorphism induced by the action. Also, for any HC_G,AutH(V)
denotes the subgroup of H-equivariant automorphisms. It will be convenient to state some of the results simultaneously for real and complex
vector bundles, or for orthogonal and unitary groups. In such situations, if F = C or R,
we write U(n,F) for U(n) or O(n), respectively. Similarly, KF(-)=K(-) or KO(-) and RF(-)=R(-) or RO(-). And VectV(X) denotes the monoid of F-vector bundles
over a space X, K F ( X ) (=K(X) or KO(X)) its Grothendieck group, and RepF(G) the monoid of F-representations of the group G.
Throughout the paper, Yp denotes the p-completion of a space Y in the sense of Bousfield and Kan [BK].
1. K(BG) for a compact Lie group G
Throughout this section, G denotes a fixed compact Lie group. Rather than working di-
rectly with vector bundles over BG, we work with their classifying maps BG-~BU(n, F) (F=C or R), via the isomorphism of monoids
co
VectF(BG)~-H [BG,BU(n, F)].
n~O
Maps BG---~BU(n,F) are studied using our general strategy for studying maps between
classifying spaces, as outlined in [JMO2, w The first part of this section thus consists mostly of results already shown elsewhere, and needed in the proof of the main theorem (Theorem 1.8). The main exception to this is Proposition 1.5, where we prove the vanishing of the higher inverse limits which occur as obstructions, using also the results in w
VECTOR BUNDLES OVER CLASSIFYING SPACES OF COMPACT LIE GROUPS
113
The starting point for understanding vector bundles over BG is the following theorem, which in particular implies that K(BP)~-R(P) for any p-toral group P. In fact, the theorem says that the underlying monoids are isomorphic in this case.
THEOREM 1.1 (Dwyer-Zabrodsky and Notbohm). Set F = C or R. For any prime p and any p-toral group P, the homomorphism of monoids
aF: RepF(p) ~
oo
VectF(Bp) ~- I I [BP, BU(n, F)],
rt~O
which sends a representation V to the vector bundle (E P • p V )+B P (or to the map BOy), is an isomorphism. Also, for any P-representation V over F, which is odd-dimensional if F = R , the homomorphism P x A u t p ( V ) (Qv,incl)Aut(V) induces (by adjointness) a homotopy equivalence
B Autp(V)p -~>map(BP, B Aut(V)~')Bov.
Proof. These are special cases of the following theorems of Dwyer and Zabrodsky [DZ] (when P is a finite p-group) and Notbohm [Nb] (in the general case). For any compact Lie group L, the map
Hom(P, L)/Inn(L) ~
[BP, BL]
(1)
is a bijection. If L is connected, then for any 0: P--*L, the homomorphism
PXCL(o(P)) (e,ind)L
induces a homotopy equivalence
BCL(O(P))p ~->map(BP, (SL)p)s Q.
(2)
Point (2) follows easily from results in [DZ] and [Nb], and is shown explicitly in [JMO,
Theorem 3.2 (iii)].
Theorem 1.1 is just the special case of (1) and (2) when L=Aut(V)~-U(n) or
O(n). Note in particular that if F = R and n=dim(V) is odd, then Aut(V)~-O(n) ~-
SO(n) x {=kI}.
[]
Theorem 1.1 provides some simple examples of the exotic behavior of the functor K ( - ) . For example, if Cp denotes the cyclic subgroup of order p, for any prime p, then the sequence
K(BS1/BCp) P-~ K ( B S ' ) restr K(BCp)
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S. JACKOWSKI AND B. OLIVER
is not exact. More precisely, if {,~IBS1 (for nEZ) denotes the line bundle with Chern class n times some fixed generator of H 2(BS1), then [{1]- [{p+l] lies in the kernel of the above restriction map, but not in the image of K(BS1/BCp). One can also show, using Theorem 1.1 again (and Propositions 2.3 and 2.5 below), that for any prime p and any finite p-group P,
K(BP) ~-R(P)- Zr (where r =rk(R(P))), ~:(p2 (BP+)) = ~:((S 2x B P ) / B P ) ~- Z,
and K ( S 2x B P ) / K ( B P ) ~-Z x (~p)r-1.
In particular, these groups are pairwise nonisomorphic (if P # 1), and so Bott periodicity
fails for K(-).
To pass from p-toral groups to an arbitrary compact Lie group G, we use a decompo-
sition of BG, at each prime p, as a homotopy direct limit of classifying spaces of p-toral
subgroups of G. This decomposition is indexed by a certain orbit category TIp(G). The
objects in Tip(G) are the orbits G / P such that (1) PC_G is p-total, (2) N ( P ) / P is finite,
and (3) there is no normal p-subgroup Ir
The morphisms in Tip(G) are
the G-maps between orbits. This is a discrete category, and is in fact equivalent to a
finite category (see Lemma 1.7 below). It is also a "directed" or "EI" category, in that all
endomorphisms of Tip(G) are isomorphisms, and hence there are morphisms in at most
one direction between any pair of nonisomorphic objects. The importance of Tip(G) lies
in the following theorem.
THEOREM 1.2. For any prime p, the map
qG,p: hocolim (EG/P) --+BG,
G/P6T~v (G)
induced by the projection EG--+BG, is an Fp-homology equivalence. In particular, for any connected complex Y, qa,p induces a homotopy equivalence
(--~
map(SG, Y~) e)map( hocolim (EG/P), Yp).
G/PE'R.v(G )
Proof. The first statement is shown in [JMO, Theorem 1.4]; and the second then
follows from [BK, Proposition II.2.8].
[]
Theorem 1.2 will be directly useful only to describe maps from BG to the BU(n, F)p; i.e., after p-completion. The next proposition will allow us to combine such maps, to get maps to BU(n, F) itself.
VECTOR BUNDLES OVER CLASSIFYING SPACES OF COMPACT LIE GROUPS 115
PROPOSITION 1.3. Let T C G be a maximal torus of G, and set w = I N ( T ) / T 1. Then the following square is a pullback if F=C, or if F = R and n is odd:
[BG, BU(n, F)] restr 1
[BT, BU(n, F)]
9 1-l l [Ba, BU(n, F);]
restr 1 > I'Ipl~o[BT, B U ( n , F)p].
Proof. This is a consequence of the arithmetic pullback square for the space
BU(n, F). For details, see [JMO3, Proposition 1.2].
[]
In view of Theorem 1.2, we are faced with the problem of comparing maps defined on a homotopy direct limit with maps defined on its pieces. This will be studied via the obstruction theory described in the next proposition.
This obstruction theory is based on a skeletal decomposition of the homotopy colimit which generalizes the classical construction of obstructions related to the skeletal decomposition if the source space is a polyhedron (or CW-complex). For each n ) 0 , the "n-skeleton" of hocolim(F) is defined by setting
h~
( H F(c~
\ k = 0 \aO'-*...-'-~ck
where one divides out by face and degeneracy relations. The space hocolim(F) itself is just the case n=co. Note in particular that hocolim(~ is the disjoint union of
the F(c), taken over all ceOb(C), and that hocolim(1)(F) is the union of the mapping
cylinders of all maps induced by morphisms in C. The following result describes the obstructions we will need to consider.
PROPOSITION 1.4. Fix a discrete category C, and a (covariant) functor F: C--*Top.
Let Y be any other space, and fix maps fc: F(c)--*Y (for all cEOb(C)) whose homo-
topy classes define an element ]=([fc])~ecers
(i.e., life extends to a map
hocolim(1)(F) --~Y ) . Set
an (c) = ~-n(map(F(c), Y), f~)
for all ceOb(C). Then given a map fn: hocolim(n)(F)---~Y (any n ) l ) which extends ], the obstruction to constructing a map on hocolim(n+l) (F) which extends
f= [hocolim(=-1)(F)
C
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S. JACKOWSKI AND B. OLIVER
lies in li_mn+l(an). Also, given two maps f,f':hocolimc(F)--*Y and a homotopy Fn defined on hocolim) (n)(F) (any n)O), the obstruction to constructing a homotopy on hocolim(~+1) (F) x I which extends
Fn[ (hocolira(n-l) (F) • I)
C
lies in lie_.__nln+l(O/n+l).Both of these obstructions are natural in Y.
Proof. This is shown by Wojtkowiak in [Wo]. Note in particular that for an to be
well defined as a functor from C to groups or abelian groups, one must first choose a map
fl: hocolim 0) (F)--*Y which can be extended to hocolim(2)(F). This applies to all of the
above situations except where one wants to extend a map fl from hocolim(t) (F) to the
2-skeleton; and in this case the obstruction set lim2(al) is defined for any functor
from C to the category of groups with morphisms given by conjugacy classes of homo-
morphisms.
[]
When applying Proposition 1.4, we will need to deal with the higher limits of homotopy groups of mapping spaces
map( EG / P, B Aut(V)p) s o , ~--B Autp(V)p
(Theorem 1.1), where Autp(V) is a product of unitary, orthogonal, and symplectic groups. Since the higher homotopy groups of these spaces are unknown, we instead stabilize, by taking limits over all VERepF(G). Such limits can be made more precise by taking them over some sequence V1C_C1_/2c V3c_... of G-representations (with G-invariant inner or hermitian product), such that each finite-dimensional G-representation is contained in Vk for sufficiently large k. We can of course assume that each Vk is odddimensional (in order to apply Theorem 1.1 or Proposition 1.3). Alternatively, one can choose a "universal" G-representation U (with G-invariant inner product), i.e., a representation which contains infinitely many copies of each irreducible G-representation, and then take the limit over all finite-dimensional subrepresentations of U.
PROPOSITION 1.5. For each i>0, let HF:T~p(G)---~Zp-rrtod be the functor defined by setting
HE(G/P)= lim ~ri(map(EG/P, B Aut(V)p),Bpv).
VERepF(G)
Then
VECTOR BUNDLES OVER CLASSIFYING SPACES OF COMPACT LIE GROUPS
117
as functors on T~p(G), and
li+___mj II~ -- 0
(2)
~e~(G)
for all i,j>0.
Proof. By Corollary 3.6, fi__mJnp(a)(Zp•
for all i,j>O. So (2) follows
immediately once point (1) has been shown.
Fix a G-representation V over F, such that dim(V) is odd if F = R . By Theorem 1.1,
for any p-toral subgroup PCG, the group homomorphism
P • Autp(Y) (~v,incl)) Aut(V)
is, after taking classifying spaces, adjoint to a homotopy equivalence
cp,y :S Autp (Y)p - ~ map(BP, B Aut(Y)p )Sev .
So for each i>0, cp,y induces an isomorphism
Cip, v . 9Zp|
Autp(Y)) ~ 7ri(map(BP, B Aut(Y)p), BQv).
By construction, the cip,_ are natural with respect to inclusions of representations V C W (where this means taking direct sums with the basepoints of the corresponding spaces in
the orthogonal complement of V). Hence we can take the limit over all VCRepF(G) to
get an isomorphism
cip: Zp|
v
B Autp(Y))) -~ fi__,m(lri(map(B P, S Aut(V)), BOy)) = HE(G/P).
V
For each P and V, Autp(V)~-AutG(G/P•
the group of G-bundle auto-
morphisms covering the identity on G/P. Thus, for each i>0,
Iri(B Autp (Y)) ~ 7ri_l(AutG (G/P x Y l G/P))
can be identified with the set of isomorphism classes of G-vector bundles over
Ei(G/p+ ) ~- (DiA(G/ p+ ))Us~_IA(G/p+) (DiA(G/P+ ))
with a fixed identification of V with the fiber over the base point. Hence
Yl__,m(Tri(BAutp(V))) ~ K-FG(Ei(G/P+ )) = KF~i(G/P);
V
and this together with c~ defines the isomorphism KF~i(G/P)~-HF(G/P).
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S. JACKOWSKI AND B. OLIVER
It remains to show that this isomorphism is natural with respect to morphisms in T~p(G). We do this by replacing Cp,y, for each V, by a homotopy equivalent map which is clearly functorial in G/P. Regard EG and BAut(V) as the nerves of topological categories CG and BAut(V): CG has as objects the elements of G and has a unique morphism between each pair of objects; while BAut(V) has one object, and a morphism for each element of Aut(V). Let gzv:EGxGBAut(V)--*BAut(V) (where G acts on B Aut(V) via 0y and conjugation) be the map induced by the functor $G x B Aut(V)---~ BAut(Y) which sends each morphism ([g--~h], a) to Qy(g).a.Qy(h) -1.
For any G / P in T~p(G), let
ev: G / P x B Auta(G/Px V ~G/P)p -* B Aut (V)p
be the evaluation map: for each gPEG/P, ev(gP,-) is induced by restriction to the fiber over gP. Define
CP,V = (r
~(Id x ev): E G / P x B Autv ( G / P x V ~ G/P)p = EGxG(G/P x B Autv(G/Px Y ~G/P)~) -~ B Aut (V)p;
and let
C'p,y: B Autc ( G / P x Y l G/P)~ --*map(EG/P, B Aut (Y)~)
be its adjoint map. The restriction of Cp,u to B P x B Autp (V) is just the map induced by
(Yy, incl) and multiplication. So cP' ,V is homotopy equivalent to the map cp,v constructed
earlier, and is natural in G / P by construction.
[]
In general, of course, direct and inverse limits cannot be switched. The following lemma describes one case where this can be done.
LEMMA 1.6. Fix a finite category C and a directed category 7), and let M: d• be any functor to abelian groups. Then for any i>~O,
li~__mi(libra(M)) ----li___~m(li~__m(i M)) 9
C D
D C
Proof. For any functor M':C--*Ab, the higher limits li_mc(M' ) are the homology groups of a cochain complex
0 ,H/,(c)--Hi'(c,), H
C~
CO--~CI
Co --~CI ---*C2
(cf. [BK, XI.6.2] or [02, Lemma 2]). In particular, this applies when M ' = M ( - , d) for any d in 7), and when Ml=li_m d M ( - , d). Since d is a finite category, all of the products