Abstract
This is the first of a series of papers with the aim to follow up the program initiated in [12], i.e., to investigate which classes of manifolds are symmetric in the sense that they allow some non-trivial group action. The general approach for constructing cyclic group actions on a given manifold M is to start with an action of the circle group T=S 1 on some manifold in the rational homotopy type of M and to “propagate” most of the restricted cyclic group actions to M itself. In [12], we only considered free actions, whereas the focus of this and the following paper [15] is on the non-free case.
This note is primarily concerned with a closer look at T-actions on Poincaré complexes and in particular with the interpretation of the Borel localization theorem in the language of deformations of algebras [14,2]. In the subsequent paper [15], we will show how to realize an abstractly given deformation in the rational homotopy category by a circle action on a manifold of the given rational homotopy type.
The first two sections of this paper should be of independent interest, whereas the short final ones are mainly to be considered as background material for [15].
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
C. Allday, Torus actions on a cohomology product of three odd spheres, Trans. Amer. Math. Soc. 203, 343–358(1975)
C. Allday, On the rational homotopy of fixed point sets of torus actions, Topology 17, 95–100(1978)
J. Barge, Structures différentiables sur les types d'homotopie rationelle simplement connexes, Ann.scient.Éc.Norm.Sup., 4e série, 9, 469–501(1976)
A. Borel et al.: Seminar on Transformation Groups. Ann. of Math. Studies 46, Princeton, N.J.: Princeton University Press 1961
A.K.Bousfield, V.K.A.M.Gugenheim, On PL De Rham theory and rational homotopy type, Memoirs of the Amer. Math. Soc. 179, 1976
G.E. Bredon, The cohomology ring structure of a fixed point set, Ann. of Math. 80, 524–537(1964)
G.E. Bredon, Introduction to compact transformation groups, New York: Academic Press 1972
T. Chang, T. Skjelbred, The topological Schur lemma and related results, Ann. of Math. 100, 307–321(1974)
S.Halperin, Lectures on minimal models, Mémoire de la Soc. Math. de France 9/10, 1983
W.Y. Hsiang, Cohomology Theory of Topological Transformation Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete 85, Berlin, Heidelberg, New York: Springer-Verlag 1975
T. Kawasaki, Cohomology of twisted projective spaces and lens complexes, Math. Ann. 206, 243–248(1973)
P. Löffler, M. Raußen, Symmetrien von Mannigfaltigkeiten und rationale Homotopietheorie, Math.Ann. 272, 549–576(1985)
J.W. Milnor, J.D. Stasheff, Characteristic Classes, Ann. of Math. Studies 76, Princeton, N.J.: Princeton University Press 1974
V. Puppe, Cohomology of fixed point sets and deformations of algebras, manuscripta math. 23, 343–354(1978)
M.Raußen, Symmetries on manifolds via rational homotopy theory, in preparation.
D. Sullivan, Infinitesimal computations in topology, Publ. IHES 47, 269–331(1977)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1992 Springer-Verlag
About this paper
Cite this paper
Raußen, M. (1992). Rational cohomology and homotopy of spaces with circle action. In: Aguadé, J., Castellet, M., Cohen, F.R. (eds) Algebraic Topology Homotopy and Group Cohomology. Lecture Notes in Mathematics, vol 1509. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0087521
Download citation
DOI: https://doi.org/10.1007/BFb0087521
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-55195-9
Online ISBN: 978-3-540-46772-4
eBook Packages: Springer Book Archive