Circle action and some vanishing results on manifolds

TL;DR

This paper generalizes vanishing signature results for manifolds with circle actions, using the G-signature theorem and rigidity theorems, extending earlier findings to broader classes of manifolds.

Contribution

It introduces new vanishing theorems for signatures of manifolds with circle actions by leveraging advanced rigidity techniques, broadening previous results.

Findings

Vanishing of the signature for certain circle actions on manifolds.

Extension of previous vanishing results using G-signature and rigidity theorems.

Connections to classical results by Conner-Floyd, Landweber-Stong, and Hirzebruch-Slodowy.

Abstract

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.