Elliptic genera, involutions, and homogeneous spin manifolds
type: Item
instance of: scholarly article
title: Elliptic genera, involutions, and homogeneous spin manifolds
zbMATH Open document ID: 0712.57010
DOI: 10.1007/BF00147351
published in: Geometriae Dedicata
publication date: 1990-01-01T00:00:00Z
review text: The authors study the normalized elliptic genus \(\Phi (X)=\phi (X)/\epsilon^{k/2}\) for 4k-dimensional homogeneous spin manifolds X, and show that they are constant (zero for odd k) modular functions. When \(\Phi\) (X) is a constant it equals the signature of X. For example, on quaternionic projective spaces \(P_ k({\mathbb{H}})\) one finds by this method that \(\Phi (P_ k({\mathbb{H}}))=1\) for k even and vanishes for k odd. The authors derive a general formula for sign(G/H), \(G\supset H\) compact Lie groups, and determine its value in some cases by making use of the theory of involutions on compact Lie groups. The basic tool is a reduction formula relating \(\Phi\) (X) to the value of \(\Phi\) on the self- intersection of the fixed point set of an involution g on X. Under appropriate hypotheses the formula takes the simple form \(\Phi (X)=\Phi (X^ g\circ X^ g)\).
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 57R19
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 58J26
zbMATH DE Number: 4171962
zbMATH Keywords: signature of homogeneous spaces
zbMATH Keywords: normalized elliptic genus
zbMATH Keywords: 4k- dimensional homogeneous spin manifolds
zbMATH Keywords: modular functions
zbMATH Keywords: quaternionic projective spaces
zbMATH Keywords: involutions on compact Lie groups
zbMATH Keywords: self-intersection of the fixed point set of an involution
author: Friedrich Hirzebruch
author: Peter Slodowy
reviewed by: Peter Landweber
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00147351
OpenAlex ID: W2044460972
description: scientific article; zbMATH DE number 4171962