Topology from Fermions

We show that there can be an odd number of fermionic zero modes in certain string-like, non-topological, bosonic configurations while the number of zero modes in the vacuum vanishes. We use the index $n_L-n_R$, where $n_L$ and $n_R$ are the number of left- and right-moving zero modes, to argue that such non-topological bosonic field configurations cannot evolve continuously into the vacuum and that they lie in a distinct topological sector. This topology is not given by the homotopy groups of the bosonic vacuum manifold; instead it is a property of the fermionic sector of the theory.