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Vector bundles over classifying spaces of compact Lie groups. (English) Zbl 0896.55003

Let \(F\) be \(\mathbf R\) or \(\mathbf C\). For any topological space \(X\), let \(\mathbf K F(X)\) denote the Grothendieck group of the abelian monoid \({\text{Vect}^F}(X)\) of isomorphism classes of \(F\)-vector bundles over \(X\); the addition in \({\text{Vect}^F}(X)\) is defined by direct sum. Fix an arbitrary compact Lie group \(G\); let \(BG\) denote its classifying space.
In their main theorem, the authors establish an isomorphism between the groups \(\mathbf K F(BG)\) and \(RF_{\mathcal P}(G) := {{\varprojlim}_P} RF(P)\). Here the inverse limit is taken over all \(p\)-toral subgroups \(P\) of \(G\), for all prime numbers \(p\), with respect to inclusion and conjugation of subgroups (a group \(P\) is \(p\)-toral if its identity component \(P_0\) is a torus and \(P/{P_0}\) is a finite \(p\)-group), and \(RF(P)\) is the \(F\)-representation ring of \(P\). In addition to this, the main theorem shows that for any \(F\)-vector bundle \(\xi\) over \(BG\) there exists a \(G\)-representation \(V\) such that \(\xi\) can be expressed as a summand of the vector bundle \(EG {\times _G} V\) associated to the universal principal bundle \(EG\) over \(BG\); hence \(\mathbf K F(BG)\) can be obtained from \({\text{Vect}^F}(BG)\) by inverting only those vector bundles coming from \(G\)-representations.
The starting point in the proof of the main theorem is a description, up to \(p\)-completion, of the mapping space \(\text{map}(BP, BL)\) for a \(p\)-toral group \(P\) and an arbitrary compact Lie group \(L\). This description is derived from theorems of W. Dwyer and A. Zabrodsky [Algebraic topology, Proc. Symp., Barcelona/Spain 1986, Lect. Notes Math. 1298, 106-119 (1987; Zbl 0646.55007)], and D. Notbohm [Math. Z. 207, No. 1, 153-168 (1991; Zbl 0731.55011)]. Then the authors pass to more general groups using a decomposition of \(BG\) at any prime \(p\) as a homotopy direct limit of classifying spaces of \(p\)-toral subgroups of \(G\).
If \(F=\mathbf C\) and \(X\) is a compact space or a finite-dimensional CW-complex, then \(\mathbf K(X)\) is equal to the complex \(K\)-theory ring \(K(X) := [X, \mathbf Z \times BU]\) (where \(U\) is the union of \(U(i)\), \(i\geq 1\), with the inductive topology). But the authors show that for other spaces the functor \(\mathbf K(-)\) can behave very differently from \(\mathbf K(-)\): for example, Bott periodicity can fail for \(\mathbf K(-)\).
Among other results, the authors also compare (the homotopy types of) the mapping space \(\text{map}(X, \mathbf Z \times BU)\) and \({{\mathfrak K}^{\mathbf C}} (X)\); the latter is the topological group completion of the space of maps from \(X\) into the disjoint union of \(BU(n)\), \(n \geq 0\). For instance, they show that for \(G\) finite the connected components of \({{\mathfrak K}^{\mathbf C}}(BG)\) have the same homotopy type as the components of \(\text{map}(BG, \mathbf Z \times BU)\), but this is not true for \(G\) a nontrivial torus.

MSC:

55N15 Topological \(K\)-theory
57R22 Topology of vector bundles and fiber bundles
55P60 Localization and completion in homotopy theory
55N91 Equivariant homology and cohomology in algebraic topology
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
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