A note on the $\hat A$-genus for $\pi_2$-finite manifolds with $S^1$-symmetry

TL;DR

This paper constructs examples of $S^1$-manifolds with finite $\pi_2$ and non-zero $\hat A$-genus, contributing to the understanding of positive quaternionic Kähler manifolds.

Contribution

It provides explicit examples of $S^1$-manifolds with finite second homotopy group and non-vanishing $\hat A$-genus, linking to quaternionic Kähler geometry.

Findings

Examples of $S^1$-manifolds with finite $\pi_2$ and non-zero $\hat A$-genus are constructed.

The results relate to the classification of positive quaternionic Kähler manifolds.

The $\hat A$-genus can be non-zero in these specific topological settings.

Abstract

We construct examples of $S^1$-manifolds with finite second homotopy group and non-vanishing $\hat A$-genus. This is related to the classification of positive quaternionic Kaehler manifolds.