We give a simple universal property of the multiplicative structure on the Thom spectrum of an n‐fold loop map, obtained as a special case of a characterization of the algebra structure on the colimit of a lax scriptO‐monoidal functor. This allows us to relate Thom spectra to En‐algebras of a given characteristic in the sense of Szymik. As applications, we recover the Hopkins–Mahowald theorem realizing Hdouble-struckFp and HZ as Thom spectra, and compute the topological Hochschild homology and the cotangent complex of various Thom spectra.

For each of the groups G = O(2), SU (2), U (2), we compute the integral and F2-cohomology rings of BcomG (the classifying space for commutativity of G), the action of the Steenrod algebra on the mod 2 cohomology, the homotopy type of EcomG (the homotopy fiber of the inclusion BcomG → BG), and some low-dimensional homotopy groups of BcomG.

We derive a rigorous classification of topologically stable Fermi surfaces of noninteracting, discrete translation-invariant systems from electronic band theory, adiabatic evolution and their topological interpretations. For systems on an infinite crystal it is shown that there can only be topologically unstable Fermi surfaces. For systems on a halfspace and with a gapped bulk, our derivation naturally yields a K -theory classification. Given the d − 1-dimensional surface Brillouin zone X s of a d-dimensional half-space, our result implies that different classes of globally stable Fermi surfaces belong in K −1 (X s ) for systems with only discrete translation-invariance. This result has a chiral anomaly inflow interpretation, as it reduces to the spectral flow for d = 2. Through equivariant homotopy methods we extend these results for symmetry classes AI, AII, C and D and discuss their corresponding anomaly inflow interpretation.

We publicise a proof of the Jordan Curve Theorem which relates it to the Phragmen-Brouwer Property, and whose proof uses the van Kampen theorem for the fundamental groupoid on a set of base points. * www.bangor.ac.uk/r.brown.

We describe the connected components of the space $\text {Hom}(\Gamma ,SU(2))$ of homomorphisms for a discrete nilpotent group $\Gamma$ . The connected components arising from homomorphisms with non-abelian image turn out to be homeomorphic to $\mathbb {RP}^{3}$ . We give explicit calculations when $\Gamma$ is a finitely generated free nilpotent group. In the second part of the paper, we study the filtration $B_{\text {com}} SU(2)=B(2,SU(2))\subset \cdots \subset B(q,SU(2))\subset \cdots$ of the classifying space $BSU(2)$ (introduced by Adem, Cohen and Torres-Giese), showing that for every $q\geq 2$ , the inclusions induce a homology isomorphism with coefficients over a ring in which 2 is invertible. Most of the computations are done for $SO(3)$ and $U(2)$ as well.