Let be an infinite discrete group and let be a classifying space for proper actions of . Every –equivariant vector bundle over gives rise to a compatible collection of representations of the finite subgroups of . We give the first examples of groups with a cocompact classifying space for proper actions admitting a compatible collection of representations of the finite subgroups of that does not come from a –equivariant (virtual) vector bundle over . This implies that the Atiyah–Hirzebruch spectral sequence computing the –equivariant topological –theory of has nonzero differentials. On the other hand, we show that for right-angled Coxeter groups this spectral sequence always collapses at the second page and compute the –theory of the classifying space of a right-angled Coxeter group.
"Equivariant vector bundles over classifying spaces for proper actions." Algebr. Geom. Topol. 17 (1) 131 - 156, 2017. https://doi.org/10.2140/agt.2017.17.131