Characteristic classes, elliptic operators and compact group actions (Q1355391)

From the Introduction: ``Our aim is to introduce characteristic classes of vector bundles, to relate them to the indices of elliptic operators and the Lefschetz theorems for such operators, and finally to show how to use these to prove the nonexistence of nontrivial actions preserving a spin structure for compact connected Lie groups. Specifically, we will show how to prove the following [for a complete proof, see \textit{J. L. Heitsch} and \textit{C. Lazarov}, Mich. Math. J. 38, 285-297 (1991; Zbl 0726.57021)]. Theorem 5.2. Let \(F\) be an oriented foliation of a compact oriented manifold \(M\) and assume that \(F\) admits a spin structure. If a compact connected Lie group acts nontrivially on \(M\) as a group of isometries taking each leaf of \(F\) to itself and preserving the spin structure on \(F\), then the \(\widehat{\mathcal A}\) genus of \(F\) is zero''.