Definition

Let $M$ be a manifold (the hypothetical space of meanings). A pinching of $M$ is the quotient space obtained by choosing a finite collection of finite point-sets $S_1,\dots,S_k \subset M$ and identifying each $S_i$ to a single point. The result $W = M/\{S_1,\dots,S_k\}$ is a pinched manifold: it agrees with $M$ (is a genuine $n$-manifold, locally homeomorphic to an open ball) at every point except the $k$ new singular points, where the neighbourhood instead looks like $|S_i|$ open balls glued together at their centres.

Singular points are distinguished from ordinary ones by a purely topological test: the punctured neighbourhood (the neighbourhood minus the point itself) of an ordinary point is connected, while the punctured neighbourhood of a singular point obtained by gluing $k>1$ points has $k$ connected components. In dimension $\geq 2$ this criterion is exact; the unpunctured neighbourhoods of singular and ordinary points are not distinguishable by the usual invariants (both are contractible).

Intuition

Take a manifold of “meanings” $M$ — an idealised space in which every distinct sense of every word has its own point. A real word-vector embedding does not represent meanings; it represents words, and a polysemous word (e.g. “mole”: a skin blemish, a spy, a burrowing mammal, a unit of substance) must have all of its distinct meanings mapped to the same vector. That many-to-one identification is exactly the pinching construction: the word space $W$ is what you get by gluing together the several points of $M$ that share a word, and the glued points are singular — geometrically indistinguishable from an ordinary point in isolation, but topologically different from their neighbours, since a small ball around “mole” really looks like several half-formed neighbourhoods (birthmark-senses, spy-senses, mammal-senses) fused at a point.

Relative to manifolds and persistent-homology-profile

This is a sharper, falsifiable alternative to the plain manifold hypothesis for word/embedding spaces, not a measurement built on top of it: the claim is about the shape of the space itself (a specific kind of non-manifold, a singular quotient), not a scalar or profile computed from it. Persistent homology (see Persistent-homology / Betti profile across depth) is used here purely as the detection instrument — a degree-zero persistence diagram of a word’s punctured neighbourhood counts connected components, which is exactly the invariant that separates singular from ordinary points — not as the structural claim in its own right. This mirrors the general distinction elsewhere in this map between a shape (Circle, Torus, …) and the machinery used to detect it.

Key evidence

A topological measure of polysemy built from exactly this singular-point criterion correlates with real word-sense counts in a self-trained fastText embedding. Jakubowski, Gasić & Zibrowius (2020) formalize topological polysemy $\mathrm{TPS}_n(w)$: normalize a word $w$‘s $n$ nearest neighbours onto the unit sphere around $w$, compute the degree-zero persistence diagram of that punctured neighbourhood, and take its Wasserstein norm — the more disconnected the punctured neighbourhood, the more “singular” $w$‘s point looks, the higher the score. Fit on a 65M-token SemEval-2010 corpus with a from-scratch fastText model (no lexicographic supervision), $\mathrm{TPS}_{50}$ correlates with the SemEval word-sense-induction gold standard at $r=0.424$ ($p<10^{-3}$) and with WordNet synset counts across 62,049 words, while showing no correlation with raw word frequency (ruling out a frequency confound). A simple clustering solution to the SemEval-2010 task built on the same intuition (cluster punctured neighbourhoods; assign instances to the nearest cluster) is competitive with far more complex contemporary systems. See jakubowski-etal-2020-topology-of-word-embeddings.

Key papers

• Jakubowski, Gasić & Zibrowius (2020). Topology of Word Embeddings: Singularities Reflect Polysemy. arXiv:2011.09413 — origin of the pinched-manifold picture and the TPS measure above.