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

HTML articles powered by AMS MathViewer

- 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.## References

- Christopher Allday,
*On the rational homotopy of fixed point sets of torus actions*, Topology**17**(1978), no. 1, 95–100. MR**501036**, DOI 10.1016/0040-9383(78)90015-0 - C. Allday and V. Puppe,
*Cohomological methods in transformation groups*, Cambridge Studies in Advanced Mathematics, vol. 32, Cambridge University Press, Cambridge, 1993. MR**1236839**, DOI 10.1017/CBO9780511526275 - A. K. Bousfield and V. K. A. M. Gugenheim,
*On $\textrm {PL}$ de Rham theory and rational homotopy type*, Mem. Amer. Math. Soc.**8**(1976), no. 179, ix+94. MR**425956**, DOI 10.1090/memo/0179 - Murray Gerstenhaber,
*On the deformation of rings and algebras*, Ann. of Math. (2)**79**(1964), 59–103. MR**171807**, DOI 10.2307/1970484 - Phillip A. Griffiths and John W. Morgan,
*Rational homotopy theory and differential forms*, Progress in Mathematics, vol. 16, Birkhäuser, Boston, Mass., 1981. MR**641551** - S. Halperin,
*Lectures on minimal models*, Mém. Soc. Math. France (N.S.)**9-10**(1983), 261. MR**736299** - Peter Löffler and Martin Raußen,
*Symmetrien von Mannigfaltigkeiten und rationale Homotopietheorie*, Math. Ann.**271**(1985), no. 4, 549–576 (German). MR**790115**, DOI 10.1007/BF01456134 - Timothy James Miller,
*On the formality of $(k-1)$-connected compact manifolds of dimension less than or equal to $4k-2$*, Illinois J. Math.**23**(1979), no. 2, 253–258. MR**528561** - Martin Raußen,
*Rational cohomology and homotopy of spaces with circle action*, Algebraic topology (San Feliu de Guíxols, 1990) Lecture Notes in Math., vol. 1509, Springer, Berlin, 1992, pp. 313–325. MR**1185981**, DOI 10.1007/BFb0087521
— - Dennis Sullivan,
*Infinitesimal computations in topology*, Inst. Hautes Études Sci. Publ. Math.**47**(1977), 269–331 (1978). MR**646078**, DOI 10.1007/BF02684341

*Symmetries on manifolds via rational homotopy theory*, in preparation.

## 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