# nLab universal equivariant PU(H)-bundle

Contents

### Context

#### Bundles

bundles

fiber bundles in physics

## Constructions

#### Representation theory

representation theory

geometric representation theory

## Theorems

#### Homotopy theory

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 $G$ and in specialization of the Murayama-Shimakawa construction, the base space of the universal $G$-equivariant $PU(\mathcal{H})$-principal bundle is

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

where $G$ 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 $H \subset G$ the $H$-fixed locus of this space is the Borel construction

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

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

## Properties

### Equivariant homotopy groups

###### Proposition

For $H \subset G$ any subgroup, the higher homotopy groups of the $H$-fixed locus of the equivariant classifying space (1) are concentrated on the integers in degree 3 and the Pontrjagin dual of $K$ in degree 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.