nLab
universal equivariant PU(H)-bundle

Contents

Context

Bundles

Representation theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

under construction

Contents

Idea

In equivariant generalization of how the universal principal bundle with structure group the projective unitary group PU(ℋ) on a separable Hilbert space \mathcal{H} classifies the 3-twist of twisted KU-cohomology theory, so the the universal equivariant principal bundle with this structure group PU(ℋ) serves to classify the 3-twists of equivariant KU-cohomology theory.

Details

For finite equivariance group GG and in specialization of the Murayama-Shimakawa construction, the base space of the universal GG-equivariant PU()PU(\mathcal{H})-principal bundle is

(1)PU()=TopGrpds(G×GG,PU()*)GActions(TopSpaces), \mathcal{B} PU(\mathcal{H}) \;\; = \;\; TopGrpds \big( G \times G \rightrightarrows G ,\, PU(\mathcal{H}) \rightrightarrows \ast \big) \;\;\; \in \; \in G Actions(TopSpaces) \,,

where GG acts by right multiplication on the arguments.

This space is considered in BEJU 2014, Thm. 3.5 (without reference to Murayama & Shimakawa 1995) as the equivariant classifying space for the 3-twist of twisted equivariant K-theory.

For subgroups HGH \subset G the HH-fixed locus of this space is the Borel construction

(PU()) H=Grps(H,PU()) adPU() \big( \mathcal{B} PU(\mathcal{H}) \big)^H \;\; = \;\; Grps\big(H, PU(\mathcal{H})\big) \sslash_{\!ad} PU(\mathcal{H})

of the adjoint action of PU(ℋ) on the space of group homomorphisms from HH.

Properties

Equivariant homotopy groups

Proposition

For HGH \subset G any subgroup, the higher homotopy groups of the HH-fixed locus of the equivariant classifying space (1) are concentrated on the integers in degree 3 and the Pontrjagin dual of KK in degree 1:

π k>0((PU()) H)={0 | k4 | k=3 0 | k=2 Grps(H,S 1) | k=1 \pi_{k \gt 0} \Big( \big( \mathcal{B} PU(\mathcal{H}) \big)^H \Big) \;\; = \;\; \left\{ \begin{array}{cll} 0 &\vert& k \geq 4 \\ \mathbb{Z} &\vert& k = 3 \\ 0 &\vert& k = 2 \\ Grps(H,S^1) &\vert& k = 1 \end{array} \right.

(BEJU 2014, around Cor. 1.11)

References

Last revised on September 15, 2021 at 10:32:46. See the history of this page for a list of all contributions to it.