The manifold hypothesis suggests that word vectors live on a submanifold within their ambient vector space. We argue that we should, more accurately, expect them to live on a <i>pinched</i> manifold: a singular quotient of a manifold obtained by identifying some of its points. The identified, singular points correspond to polysemous words, i.e. words with multiple meanings. Our point of view suggests that monosemous and polysemous words can be distinguished based on the topology of their neighbourhoods. We present two kinds of empirical evidence to support this point of view: (1) We introduce a topological measure of polysemy based on persistent homology that correlates well with the actual number of meanings of a word. (2) We propose a simple, topologically motivated solution to the SemEval-2010 task on <i>Word Sense Induction & Disambiguation</i> that produces competitive results.