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 | References | Similar Articles | Additional Information

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 with submanifold , I would like to construct nontrivial actions of cyclic groups on with 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 on with fixed point set was established in [Rau92]. In this paper, I assume that the rational homotopy types of and 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 -action on the complement that fits together with a given -bundle action on the normal bundle of in . In a subsequent paper [Rau94], I plan to show how to realize this -action on an actual manifold rationally homotopy equivalent to with fixed point set and how to "propagate" all but finitely many of the restricted cyclic group actions to itself.

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Additional Information

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