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Transactions of the American Mathematical Society

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Circle actions on rational homology manifolds and deformations of rational homotopy types


Author: Martin Raussen
Journal: Trans. Amer. Math. Soc. 347 (1995), 137-153
MSC: Primary 57S10; Secondary 55P62, 57S17
DOI: https://doi.org/10.1090/S0002-9947-1995-1273540-6
MathSciNet review: 1273540
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Abstract: The aim of this paper is to follow up the program set in [LR85, Rau92], i.e., to show the existence of nontrivial group actions ("symmetries") on certain classes of manifolds. More specifically, given a manifold $ X$ with submanifold $ F$, I would like to construct nontrivial actions of cyclic groups on $ X$ with $ F$ as fixed point set. Of course, this is not always possible, and a list of necessary conditions for the existence of an action of the circle group $ T = {S^1}$ on $ X$ with fixed point set $ F$ was established in [Rau92]. In this paper, I assume that the rational homotopy types of $ F$ and $ X$ are related by a deformation in the sense of [A1178] between their (Sullivan) models as graded differential algebras (cf. [Sul77, Hal83]). Under certain additional assumptions, it is then possible to construct a rational homotopy description of a $ T$-action on the complement $ X\backslash F$ that fits together with a given $ T$-bundle action on the normal bundle of $ F$ in $ X$. In a subsequent paper [Rau94], I plan to show how to realize this $ T$-action on an actual manifold $ Y$ rationally homotopy equivalent to $ X$ with fixed point set $ F$ and how to "propagate" all but finitely many of the restricted cyclic group actions to $ X$ itself.


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DOI: https://doi.org/10.1090/S0002-9947-1995-1273540-6
Keywords: Symmetry, circle group, rational homotopy type, deformation
Article copyright: © Copyright 1995 American Mathematical Society