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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
DOI: https://doi.org/10.1090/S0002-9947-05-03788-8
Published electronically: August 1, 2005
MathSciNet review: 2219004
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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.


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Additional Information

Pisheng Ding
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: pding@indiana.edu

DOI: https://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.

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