Topological obstructions to certain Lie group actions on manifolds

Author:
Pisheng Ding

Journal:
Trans. Amer. Math. Soc. **358** (2006), 3937-3967

MSC (2000):
Primary 57S15, 58J20; Secondary 57R91, 57R20

Published electronically:
August 1, 2005

MathSciNet review:
2219004

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given a smooth closed -manifold , this article studies the extent to which certain numbers of the form are determined by the fixed-point set , where classifies the universal cover of , , is a polynomial in the Pontrjagin classes of , and is in the subalgebra of generated by . When , various vanishing theorems follow, giving obstructions to certain fixed-point-free actions. For example, if a fixed-point-free -action extends to an action by some semisimple compact Lie group , then . Similar vanishing results are obtained for spin manifolds admitting certain -actions.

**1.**Michael Atiyah and Friedrich Hirzebruch,*Spin-manifolds and group actions*, Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, pp. 18–28. MR**0278334****2.**M. F. Atiyah and I. M. Singer,*The index of elliptic operators. III*, Ann. of Math. (2)**87**(1968), 546–604. MR**0236952****3.**William Browder and Wu Chung Hsiang,*𝐺-actions and the fundamental group*, Invent. Math.**65**(1981/82), no. 3, 411–424. MR**643560**, 10.1007/BF01396626**4.**P. E. Conner and Frank Raymond,*Actions of compact Lie groups on aspherical manifolds*, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), Markham, Chicago, Ill., 1970, pp. 227–264. MR**0271958****5.**Fuquan Fang and Xiaochun Rong,*Fixed point free circle actions and finiteness theorems*, Commun. Contemp. Math.**2**(2000), no. 1, 75–86. MR**1753140**, 10.1142/S0219199700000062**6.**Victor W. Guillemin and Shlomo Sternberg,*Supersymmetry and equivariant de Rham theory*, Mathematics Past and Present, Springer-Verlag, Berlin, 1999. With an appendix containing two reprints by Henri Cartan [ MR0042426 (13,107e); MR0042427 (13,107f)]. MR**1689252****7.**Wu-yi Hsiang,*Cohomology theory of topological transformation groups*, Springer-Verlag, New York-Heidelberg, 1975. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 85. MR**0423384****8.**R. K. Lashof, J. P. May, and G. B. Segal,*Equivariant bundles with abelian structural group*, Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982) Contemp. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1983, pp. 167–176. MR**711050**, 10.1090/conm/019/711050**9.**H. Blaine Lawson Jr. and Shing Tung Yau,*Scalar curvature, non-abelian group actions, and the degree of symmetry of exotic spheres*, Comment. Math. Helv.**49**(1974), 232–244. MR**0358841****10.**John McCleary,*A user’s guide to spectral sequences*, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 58, Cambridge University Press, Cambridge, 2001. MR**1793722****11.**Jonathan Rosenberg and Shmuel Weinberger,*Higher 𝐺-indices and applications*, Ann. Sci. École Norm. Sup. (4)**21**(1988), no. 4, 479–495. MR**982331****12.**Shing Tung Yau,*Remarks on the group of isometries of a Riemannian manifold*, Topology**16**(1977), no. 3, 239–247. MR**0448379**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
57S15,
58J20,
57R91,
57R20

Retrieve articles in all journals with MSC (2000): 57S15, 58J20, 57R91, 57R20

Additional Information

**Pisheng Ding**

Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405

Email:
pding@indiana.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-05-03788-8

Keywords:
Group actions,
circle actions,
characteristic numbers,
index theory,
$G$-signature theorem,
geometric topology

Received by editor(s):
October 9, 2003

Received by editor(s) in revised form:
June 9, 2004

Published electronically:
August 1, 2005

Additional Notes:
The author thanks his Ph.D. advisor, Sylvain Cappell, for pointing out the general direction along which the results of this article are developed. The author is grateful to the referee for many constructive suggestions.

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.