Addendum to "The $\hat A$-genus of $S^1$-manifolds with finite second homotopy group"

TL;DR

This paper provides an explicit construction of $S^1$-manifolds with finite second homotopy group and non-zero $\hat A$-genus using equivariant surgeries, addressing previous limitations in the methodology.

Contribution

It introduces a new explicit equivariant surgery technique to construct $S^1$-manifolds with desired topological and geometric properties.

Findings

Constructed examples of $S^1$-manifolds with finite second homotopy group and non-zero $\hat A$-genus.

Demonstrated the effectiveness of explicit equivariant surgeries in this context.

Resolved limitations of previous equivariant surgery lemmas.

Abstract

In C. R. Math. Acad. Sci. Paris 348 (2010) pp. 283--285 (arXiv:0811.0840) we constructed examples of $S^1$-manifolds with finite second homotopy group and non-vanishing $\hat A$-genus. The reasoning was based on an equivariant surgery lemma which only holds under additional assumptions. To remedy the situation we give a construction using explicit equivariant surgeries.