Topological phases for free fermions in systems with crystal symmetry are classified by the topology of the valence band viewed as a vector bundle over the Brillouin zone. Additional symmetries, such as crystal symmetries which act non-trivially on the Brillouin zone, or time-reversal symmetry, endow the vector bundle with extra structure. These vector bundles are classified by a suitable version of K-theory. While relatively easy to define, these K-theory groups are notoriously hard to compute in explicit examples. In this paper we describe in detail how one can compute these K-theory groups starting with a decomposition of the Brillouin zone in terms of simple submanifolds on which the symmetries act nicely. The main mathematical tool is the Atiyah-Hirzebruch spectral sequence associated to such a decomposition, which will not only yield the explicit result for several crystal symmetries, but also sheds light on the origin of the topological invariants. This extends results that have appeared in the literature so far. We also describe examples in which this approach fails to directly yield a conclusive answer, and discuss various open problems and directions for future research.

We give a complete classification of topological field theories with reflection structure and spinstatistics in one and two spacetime dimensions. Our answers can be naturally expressed in terms of an internal fermionic symmetry group G which is different from the spacetime structure group. Fermionic groups encode symmetries of systems with fermions and time reversing symmetries. We show that 1-dimensional topological field theories with reflection structure and spin-statistics are classified by finite dimensional hermitian representations of G. In spacetime dimension two we give a classification in terms strongly G-graded stellar Frobenius algebras. Our proofs are based on the cobordism hypothesis. Along the way, we develop some useful tools for the computation of homotopy fixed points of 2-group actions on bicategories.

The tenfold way provides a strong organizing principle for invertible topological phases of matter. Mathematically, it is intimately connected with K-theory via the fact that there exist exactly ten Morita classes of simple real superalgebras. This connection is physically unsurprising, since weakly interacting topological phases are classified by K-theory. We argue that when strong interactions are present, care has to be taken when formulating the exact ten symmetry groups present in the tenfold way table. We study this phenomenon in the example of class D by providing two possible mathematical interpretations of a class D symmetry. These two interpretations of class D result in Morita equivalent but different symmetry groups. As K-theory cannot distinguish Morita-equivalent protecting symmetry groups, the two approaches lead to the same classification of topological phases on the weakly interacting side. However, we show that these two different symmetry groups yield different interacting classifications in spacetime dimension 2+1. We use the approach to interacting topological phases using bordism groups, reducing the relevant classification problem to a spectral sequence computation.

We show that two Pin-structures on a surface differ by a diffeomorphism of the surface if and only if they are cobordant (for comparison, the analogous fact has already been shown for Spin-structures). We give a construction that shows that this does not extend to dimensions greater than two. In addition, we count the number of Pin-structures on a surface in a given cobordism class.