Topological obstructions to certain Lie group actions on manifolds
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Abstract:
Given a smooth closed $S^{1}$-manifold $M$, this article studies the extent to which certain numbers of the form $\left ( f^{\ast }\left ( x\right ) \cdot P\cdot C\right ) \left [ M\right ]$ are determined by the fixed-point set $M^{S^{1}}$, where $f:M\rightarrow K\left ( \pi _{1}\left ( M\right ), 1\right )$ classifies the universal cover of $M$, $x\in H^{\ast }\left ( \pi _{1}\left ( M\right ) ;\mathbb {Q}\right )$, $P$ is a polynomial in the Pontrjagin classes of $M$, and $C$ is in the subalgebra of $H^{\ast }\left ( M;\mathbb {Q}\right )$ generated by $H^{2}\left ( M;\mathbb {Q}\right )$. When $M^{S^{1}}=\varnothing$, various vanishing theorems follow, giving obstructions to certain fixed-point-free actions. For example, if a fixed-point-free $S^{1}$-action extends to an action by some semisimple compact Lie group $G$, then $\left ( f^{\ast }(x)\cdot P\cdot C\right ) [M]=0$. Similar vanishing results are obtained for spin manifolds admitting certain $S^{1}$-actions.References
- 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
- M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR 236952, DOI 10.2307/1970717
- William Browder and Wu Chung Hsiang, $G$-actions and the fundamental group, Invent. Math. 65 (1981/82), no. 3, 411–424. MR 643560, DOI 10.1007/BF01396626
- 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
- Fuquan Fang and Xiaochun Rong, Fixed point free circle actions and finiteness theorems, Commun. Contemp. Math. 2 (2000), no. 1, 75–86. MR 1753140, DOI 10.1142/S0219199700000062
- 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, DOI 10.1007/978-3-662-03992-2
- Wu-yi Hsiang, Cohomology theory of topological transformation groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 85, Springer-Verlag, New York-Heidelberg, 1975. MR 0423384
- 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, DOI 10.1090/conm/019/711050
- 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 358841, DOI 10.1007/BF02566731
- 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
- Jonathan Rosenberg and Shmuel Weinberger, Higher $G$-indices and applications, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 4, 479–495. MR 982331
- Shing Tung Yau, Remarks on the group of isometries of a Riemannian manifold, Topology 16 (1977), no. 3, 239–247. MR 448379, DOI 10.1016/0040-9383(77)90004-0
Additional Information
- Pisheng Ding
- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
- Email: pding@indiana.edu
- 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.
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 358 (2006), 3937-3967
- MSC (2000): Primary 57S15, 58J20; Secondary 57R91, 57R20
- DOI: https://doi.org/10.1090/S0002-9947-05-03788-8
- MathSciNet review: 2219004