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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Circle actions on rational homology manifolds and deformations of rational homotopy types
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by Martin Raussen PDF
Trans. Amer. Math. Soc. 347 (1995), 137-153 Request permission

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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Additional Information
  • © Copyright 1995 American Mathematical Society
  • 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