A result on the \(\widehat A\) and elliptic genera on non-spin manifolds with circle actions. (English) Zbl 1013.58012
Summary: We prove the vanishing of the \(\widehat A\)-genus of compact smooth manifolds with finite second homotopy group and endowed with smooth \(S^1\) actions. These manifolds are not necessarily spin, hence, this vanishing extends that of Atiyah and Hirzebruch on spin manifolds with \(S^1\) actions.
The proof is accomplished by proving a rigidity theorem under circle actions of the elliptic genus on these manifolds.
The proof is accomplished by proving a rigidity theorem under circle actions of the elliptic genus on these manifolds.
Keywords:
non-spin manifolds; vanishing of the \(\widehat A\)-genus; compact smooth manifolds; rigidity theorem under circle actions; elliptic genusReferences:
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