Stable homotopy theory of invertible gapped quantum spin systems I: Kitaev's $\mathrm{\Omega }$-spectrum

We provide a mathematical realization of a conjecture by Kitaev, on the basis of the operator-algebraic formulation of infinite quantum spin systems. Our main results are threefold. First, we construct an $\mathrm{\Omega }$-spectrum ${IP}_{\ast }$ whose homotopy groups are isomorphic to the smooth homotopy group of invertible gapped quantum systems on Euclidean spaces. Second, we develop a model for the homology theory associated with the $\mathrm{\Omega }$-spectrum ${IP}_{\ast }$, describing it in terms of the space of quantum systems placed on an arbitrary subspace of a Euclidean space. This involves introducing the concept of localization flow, a semi-infinite path of quantum systems with decaying interaction range, inspired by Yu’s localization C*-algebra in coarse index theory. Third, we incorporate spatial symmetries given by a crystallographic group $\mathrm{\Gamma }$ and define the $\mathrm{\Omega }$-spectrum ${IP}_{\ast }^{\mathrm{\Gamma }}$ of $\mathrm{\Gamma }$-invariant invertible phases. We propose a strategy for computing the homotopy group ${\pi }_n({IP}_d^{\mathrm{\Gamma }})$ that uses the Davis–Lück assembly map and its description by invertible gapped localization flow. In particular, we show that the assembly map is split injective, and hence ${\pi }_n({IP}_d^{\mathrm{\Gamma }})$ contains a computable direct summand.