In the talk, an algebraic topological framework for studying state spaces of  quantum lattice spin systems is presented, using the language of algebraic quantum mechanics. We first provide 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. Switching to the algebraic topological side we then determine the homotopy groups of the unitary group of a UHF algebra and show that the pure state space of any UHF algebra is simply connected. We finally indicate how these and related results may lead to an approach for constructing Kitaev's loop-spectrum of bosonic invertible gapped phases of matter. The talk is on joint work with A. Beaudry, M. Hermele, J. Moreno, M. Qi and D. Spiegel.