\(\text{Spin}^c\)-manifolds with \(\text{Pin}(2)\)-action
type: Item
instance of: scholarly article
title: \(\text{Spin}^c\)-manifolds with \(\text{Pin}(2)\)-action
zbMATH Open document ID: 0963.19002
author: Anand Dessai
published in: Mathematische Annalen
publication date: 2001-06-18T00:00:00Z
review text: It is a classical result of Atiyah and Hirzebruch that the \(\widehat A\)-genus (the index of the Dirac operator) on a Spin-manifold with a non-trivial \(S^1\)-action vanishes. The author proves here that the Witten genus (heuristically the index of the ``Dirac operator'' on the free loop space of the manifold) vanishes for \(\text{Spin}^c\)-manifolds with ``nice'' \(\text{Pin}(2)\)-action provided the first Chern class and first Pontrjagin class are torsion. The main tool is a vanishing theorem for certain indices of \(\text{Spin}^c\)-Dirac operators which the author proved in [\textit{A. Dessai}, ``Rigidity theorems for \(\text{Spin}^c\)-manifolds'', Topology 39, 239-258 (2000; Zbl 0944.58019)]. Applications are given in the cases when the manifold is a homotopy complex projective space and when it is of dimension four.
reviewed by: U. Tillmann
Mathematics Subject Classification ID: 19L47
Mathematics Subject Classification ID: 55R91
Mathematics Subject Classification ID: 19J35
zbMATH DE Number: 1395776
zbMATH Keywords: elliptic genus
zbMATH Keywords: Dirac operators
zbMATH Keywords: vanishing theorem
zbMATH Keywords: Witten genus
MaRDI profile type: Publication
full work available at URL: s002080050327
OpenAlex ID: W2031288677
DOI: 10.1007/S002080050327
description: scientific article; zbMATH DE number 1395776