Topological obstructions to certain Lie group actions on manifolds
Trans. Amer. Math. Soc. 358 (2006), 3937-3967
Primary 57S15, 58J20; Secondary 57R91, 57R20
August 1, 2005
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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.
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Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Received by editor(s):
October 9, 2003
Received by editor(s) in revised form:
June 9, 2004
August 1, 2005
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.
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