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

We provide a mathematical realization of a proposal by Kitaev, on the basis of the operator-algebraic formulation of infinite quantum spin systems. Our main results are threefold. First, we construct an $\Omega$-spectrum $\mathit{IP}_*$ 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 $\Omega$-spectrum $\mathit{IP}_*$, 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 $\Gamma $ and define the $\Omega$-spectrum $\mathit{IP}_*^\Gamma$ of $\Gamma$-invariant invertible phases. We propose a strategy for computing the homotopy group $\pi_n(\mathit{IP}_d^\Gamma )$ that uses the Davis--L"{u}ck assembly map and its description by invertible gapped localization flow. In particular, we show that the assembly map is split injective, and hence $\pi_n(\mathit{IP}_d^\Gamma)$ contains a computable direct summand.