nLab universal equivariant PU(H)-bundle

under construction

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)

Related concepts

• $U(\mathcal{H})$

References

• Noé Bárcenas, Jesús Espinoza, Michael Joachim, Bernardo Uribe, Universal twist in Equivariant K-theory for proper and discrete actions, Proceedings of the London Mathematical Society, Volume 108, Issue 5 (2013) (arXiv:1202.1880, doi:10.1112/plms/pdt061)

• Bernardo Uribe, Wolfgang Lück, Section 15 of: Equivariant principal bundles and their classifying spaces, Algebr. Geom. Topol. 14 (2014) 1925-1995 (arXiv:1304.4862, doi:10.2140/agt.2014.14.1925)

• Noé Bárcenas, Jesús Espinoza, Bernardo Uribe, Mario Velasquez, Section 3.2 of: Segal’s spectral sequence in twisted equivariant K-theory for proper and discrete actions, Proceedings of the Edinburgh Mathematical Society 61 1 (2018) (arXiv:1307.1003, doi:10.1017/S0013091517000281)

• Jesus Espinoza, Bernardo Uribe, Topological properties of spaces of projective unitary representations, Rev. Acad. Colombiana Cienc. Exact. Fís. Natur. 40 (2016), no. 155, 337-352 (arXiv:1511.06785, scielo:S0370-39082016000200013, doi:10.18257/raccefyn.317)