We construct a topological space \(\mathcal {B}\) consisting of translation invariant injective matrix product states (MPS) of all physical and bond dimensions and show that it has the weak homotopy type \(K(\mathbb {Z}, 2) \times K(\mathbb {Z}, 3)\) . The implication is that the phase of a family of such states parametrized by a space X is completely determined by two invariants: a class in \(H^2(X;\mathbb {Z})\) corresponding to the Chern number per unit cell and a class in \(H^3(X;\mathbb {Z})\) , the so-called Kapustin–Spodyneiko (KS) number. The space \(\mathcal {B}\) is defined as the quotient of a contractible space \(\mathcal {E}\) of MPS tensors by an equivalence relation describing gauge transformations of the tensors. We prove that the projection map \(p:\mathcal {E}\rightarrow \mathcal {B}\) is a quasifibration, and this allows us to determine the weak homotopy type of \(\mathcal {B}\) . As an example, we review the Chern number pump—a family of MPS parametrized by \(S^3\) —and prove that it generates \(\pi _3(\mathcal {B})\) .
