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Kontsevich's Asphericity Conjecture for Strata of Differentials

A conjecture attributed to Kontsevich holds that strata of quadratic differentials are aspherical, that is orbifold $K(\pi,1)$ spaces. False: when there are at least four zeros or poles, no connected component of a genus-one stratum is an orbifold $K(\pi,1)$, giving infinitely many counterexamples, along with counterexamples for associated stability spaces.

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geometry-topologyJun 23, 2026Significance 25/100Registry: unreviewed

Kontsevich's Asphericity Conjecture for Strata of Differentials

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A conjecture attributed to Kontsevich holds that strata of quadratic differentials are aspherical, that is orbifold $K(\pi,1)$ spaces. False: when there are at least four zeros or poles, no connected component of a genus-one stratum is an orbifold $K(\pi,1)$, giving infinitely many counterexamples, along with counterexamples for associated stability spaces.

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A conjecture attributed to Kontsevich holds that strata of quadratic differentials are aspherical, that is orbifold $K(\pi,1)$ spaces. False: when there are at least four zeros or poles, no connected component of a genus-one stratum is an orbifold $K(\pi,1)$, giving infinitely many counterexamples, along with counterexamples for associated stability spaces.

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