Celebratio Mathematica — Freedman

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Works connected to William H. Meeks, III

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article M. Freedman and W. H. Meeks, III: “Une obstruction élémentaire à l’existence d’une action continue de groupe dans une variété,” C. R. Acad. Sci. Paris Sér. A 286 : 4 (1978), pp. 195–198. MR 0515866 Zbl 0373.57021

It is natural to think that most manifolds have no symmetry, and therefore do not admit any nontrivial compact Lie group action. However it has been hard to find such manifolds: The first example was constructed by P. E. Conner, F. Raymond and P. Weinberger [1972]. Even the problem of excluding the effective actions of certain compact Lie groups has proved to be difficult [Atiyah and Hirzebruch 1970; Hirzebruch 1975; Yau 1977].

We prove that if a smooth manifold \( M \) admits an effective action of the circle \( \mathbb{S}^1 \), then its de Rham cohomology has certain properties from which one can deduce geometric properties of \( M \) that do not appear on its cohomology alone. For example, if \( M \) is a compact \( n \)-manifold, not homotopy equivalent to the \( n \)-sphere \( \mathbb{S}^n \), then the connected sum \( M\mathbin{\#}\mathbb{T}^n \) of \( M \) with the \( n \)-torus \( \mathbb{T}^n \) admits no effective action of the circle \( \mathbb{S}^1 \).”