We prove a Brown Representability Theorem in the context of spaces over a category. We discuss two applications to the representability of equivariant cohomology theories, with emphasis on Bredon cohomology with local coefficients

We give formulas for the Whitehead groups and the rational K-theory groups of the (integral group ring of the) Hilbert modular group in terms of their maximal finite subgroups.

We enumerate the regular and chiral polyhedra (in the sense of Grünbaum’s skeletal approach) whose vertex and edge sets are a subset of those of the primitive cubic lattice, the face-centred cubic lattice, or the body-centred cubic lattice

Let F be a (possibly non-simple) hypergraph and let forb(m; F) denote the maximum number of edges a simple hypergraph with m vertices can have if it doesn't contain F as a subhypergraph. A conjecture of Anstee and Sali predicts the asymptotic behaviour of forb(m; F) for fixed F. In this paper we prove that even finding this predicted asymptotic behaviour is an NP-hard problem, meaning that if the Anstee-Sali conjecture were true, finding the asymptotics of forb(m; F) would be NP-hard.

We use a spectral sequence to compute twisted equivariant K–theory groups for the classifying space of proper actions of discrete groups. We study a form of Poincaré duality for twisted equivariant K–theory studied by Echterhoff, Emerson and Kim in the context of the Baum–Connes conjecture with coefficients and verify it for the group Sl3(Z).