ICM 1998, Berlin, Aug. 18–27

Abstracts of Plenary and Invited Lectures

Section:

6. Topology

1991 MS Classification: Bob Oliver, Département de Mathématiques Université Paris-Nord, Villetaneuse, France Vector bundles over classifying spaces

A vector bundle over a space X can be thought of as a collection of (complex) vector spaces, one for each point in X, which are combined in one topological space. For any X, let Vect(X) denote the set of isomorphism classes of vector bundles over X, made into a commutative monoid by the operation of direct sum. Define 𝕂(X) to be the Grothendieck group of Vect(X); i. e., the abelian group of all formal differences (x − x′) for x, x′ ∈ Vect(X)

(with the obvious relations). When X is compact, then K(X) {def \atop =} 𝕂(X), the K-theory of X, defines a cohomology theory; and in particular has nice properties such as forming exact sequences and Bott periodicity which make it (relatively) easy to calculate these groups. When X is non-compact, and in particular when dim(X) = ∞, then this is no longer the case, and the groups 𝕂(X) no longer define a cohomology theory. So instead, one must define K(X) via classifying spaces. In general, for any topological group G, Milnor constructed a classifying space BG for G, with the property that for any space X, bundles over X with “structure group” G are in one-to-one correspondence with the set [X, BG] of homotopy classes of maps X → BG. In particular, if U denotes the infinite unitary group, then one defines K(X) = [X, ℤ × BU] for arbitrary X, and this coincides with the above definition when X is compact. There is a natural homomorphism β_{X} : 𝕂(X) → K(X) for each X, which is an isomorphism when X is compact, but not in general. In fact, the geometrically defined functor 𝕂(−) can behave very differently from K(−): it is not a cohomology theory, and is in general much more difficult to compute than K(−). If G is a compact Lie group and BG is its classifying space, then 𝕂(BG) and K(BG) can both be studied by comparison with the representation ring R(G) of G. Let

α_{G} : R(G) ⟶ 𝕂(BG)

be the homomorphism defined by sending a complex G-representation V to its Borel construction (EG×_{G}V), regarded as a vector bundle over BG. Alternatively, if one thinks of a representation (with hermitian product) as a homomorphism ρ : G → U(n), then α_{G} sends ρ to the vector bundle classified by Bρ : BG → BU(n). The completion theorem of Atiyah and Segal says that

β_{BG} ∘ α_{G} : R(G) ⟶ K(BG)

extends to an isomorphism α̂_{G} : R(G) ⟶^{≅} K(BG), where R(G) = \underleftarrow{lim}_{n} (R(G)/I^{n}) is completion with respect to the augmentation ideal I =

Ker[R(G) → ℤ]. The new result which I want to describe, proven in joint work with Stefan Jackowski, is a description of the Grothendieck group 𝕂(BG) itself, also in terms of representations. The starting point for doing this was to consider the case where G is p-toral; i. e., an extension of a torus by a finite p-group. By theorems of Dwyer–Zabrodsky (when G is a finite p-group) and of Notbohm (when G is p-toral), α_{G} : R(G) → 𝕂(BG) is an isomorphism in this case. Hence, for an arbitrary compact Lie group G, there are homomorphisms

𝕂(BG) \underrightarrow{restr} 𝕂(BP) ⟶_{≅}^{α_{P}^{−1}} R(P),

for each p-toral subgroup P ⊆ G; and these combine to define a homomorphism

r_{G} : Vect(BG) ⟶ R_{P}(G) {def \atop =} \underleftarrow{lim}_{P} R(P).

Here, the inverse limit is taken over all p-toral subgroups of G (for all primes p) with respect to inclusion and conjugation of subgroups. Our main theorem is that r_{G} is an isomorphism; i. e., that 𝕂(BG) ≅ R_{P}(G), for any compact Lie group G. This is proven by first studying Vect_{n}(BG) ≅ [BG, BU(n)] for each n, using techniques developed in joint work with Jim McClure, based on a decomposition of BG as a homotopy direct limit of spaces BP for p-toral P ⊆ G. These sets Vect_{n}(BG) are in themselves quite complicated in general (they contain non-vanishing higher derived functors of inverse limits, and can even be uncountable); but all of the “difficult” parts of Vect(BG) vanish when one takes the Grothendieck construction. Among the consequences of this theorem are that β_{BG} is always injective, that α_{G} is surjective if G is finite or connected; and that Coker(α_{G}) always has finite exponent. In fact, 𝕂(BG) ≅ R(G) (α_{G} is an isomorphism) whenever if π_{0}(G) (the group of connected components of G) has the property that all of its elements have prime power order. As one example, let G be a finite group, and let r_{p} (for any prime p) denote the number of conjugacy classes of elements of p-power order. Then 𝕂(BG) ≅ ℤ × ∏_{p}(ℤ)^{r_{p}−1}, while in contrast K(BG) ≅ ℤ × ∏_{p}(ℤ̂_{p})^{r_{p}−1}.