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under construction
In equivariant generalization of how the universal principal bundle with structure group the projective unitary group PU(ℋ) on a separable Hilbert space 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.
For finite equivariance group and in specialization of the Murayama-Shimakawa construction, the base space of the universal -equivariant -principal bundle is
where 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 the -fixed locus of this space is the Borel construction
of the adjoint action of PU(ℋ) on the space of group homomorphisms from .
For any subgroup, the higher homotopy groups of the -fixed locus of the equivariant classifying space (1) are concentrated on the integers in degree 3 and the Pontrjagin dual of in degree 1:
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)
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