# Topological phases and K-theory

### Abstract

Topological K-theory has shown up in the classification of symmetry-protected topological phases of free fermions. This started with the work of Kitaev, and was later continued by Freed and Moore, who modified K-theory to twisted K-theory in order to fit the physics. In more recent years, mathematically rigorous tools have been developed for the computation of this twisted K-theory. We present an introductory approach to both the mathematical and physical side of the subject, notably without assuming a prior knowledge of K-theory. The main results of Freed and Moore are summarised for the symmorphic case, and the basic cohomological tools for the computation of these K-groups are outlined. These tools are then applied to specific calculations. We classify all possible one-dimensional topological phases in all ten Altland–Zirnbauer classes, under a few assumptions. We do this mainly by using the Mayer–Vietoris exact sequence, and we find that for one-dimensional systems this method coincides with the Atiyah–Hirzebruch spectral sequence approach. We also treat the three-dimensional space group $F222$ in class A, i.e., topological phases protected by crystal symmetries only. This was already done by Gomi et al., but not in much detail; we provide a detailed version of this calculation. Using the Atiyah–Hirzebruch spectral sequence, we find the same result; in particular, we find a $\mathbf{Z}/2$-invariant.

This is my Bachelor’s thesis, which was supervised by Hessel Posthuma and Jan de Boer.

There are some errata for this thesis (mostly typos):

• On page 2, the phrase “cases were rigorous mathematical arguments” should read “cases where rigorous …”.
• In the proof of Proposition 4.5, the last sentence is false: $\operatorname{Aut}(\mathbf{Z}^d)$ is not a finite group. The proof can be corrected by embedding $P$ in $O(\mathbf{Z}^d)$ instead, which is a finite group.
• In the introduction to §5.3, the phrase " on the space of equivariant compact Hausdorff spaces" should read “on the category of …”.
• On page 77, $\mathcal{H}_i^-$ and $\mathcal{H}_i^+$ should be replaced by $\mathcal{H}_\bullet^-$ and $\mathcal{H}_\bullet^+$, respectively.
• On page 80, “Chapters 1” should read “Chapter 1”.
• In Proposition 7.3, the assignment $[E]\mapsto [E_H]$ should read $[E]\mapsto [E_{1\cdot H}]$.
• On page 111, there is a stray bracket ‘]’.