Circle action and some vanishing results on manifolds

Details Circle action and some vanishing results on manifolds Circle action and some vanishing results on manifolds

2010-12-07

arxiv.org/abs/1012.1507v2

International Journal of Mathematics, 22 (2011), 1603-1610

Mathematics

Algebraic Topology

7 pages, typos corrected and minors modified

Scientific paper

Kawakubo and Uchida showed that, if a closed oriented $4k$-dimensional manifold $M$ admits a semi-free circle action such that the dimension of the fixed point set is less than $2k$, then the signature of $M$ vanishes. In this note, by using $G$-signature theorem and the rigidity of the signature operator, we generalize this result to more general circle actions. Combining the same idea with the remarkable Witten-Taubes-Bott rigidity theorem, we explore more vanishing results on spin manifolds admitting such circle actions. Our results are closely related to some earlier results of Conner-Floyd, Landweber-Stong and Hirzebruch-Slodowy.

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