In the talk I report on recent joint work with Beaudry, Hermele, Moreno, Qi and Spiegel, where a homotopy theoretic framework for studying state spaces of quantum lattice spin systems has been introduced using the language of C*-algebraic quantum mechanics. First some old and new results about the state space of the quasi-local algebra of a quantum lattice spin system when endowed with either the natural metric topology or the weak* topology will be presented. Switching to the algebraic topological side we then determine the homotopy groups of the unitary group of a UHF algebra and then show that the pure state space of any UHF algebra is simply connected. We finally indicate how these and related results may lead to a framework for constructing Kitaev's loop-spectrum of bosonic invertible gapped phases of matter.