On the topology of double coset manifolds
type: Item
instance of: scholarly article
title: On the topology of double coset manifolds
zbMATH Open document ID: 0793.57019
DOI: 10.1007/BF01459492
published in: Mathematische Annalen
publication date: 1994-04-26T00:00:00Z
full work available at URL: https://eudml.org/doc/165122
review text: Given a compact Lie group \(G\) and two closed subgroups \(K\) and \(H\) of \(G\) such that \(K \times H\) operates freely on \(G\) by \(g \cdot (k,h)=k^{-1} gh\), we can form the quotient \(K \backslash G/H\), which is a compact manifold called a double coset manifold. The objective of the present paper is to show that the topology of double coset manifolds is in many respects as easy to handle as that of homogeneous spaces. In particular, there is an explicit description of the tangent bundle. Since, by the known collapsing results for the Eilenberg-Moore spectral sequence, the cohomology of \(K \backslash G/H\) is readily computable, formulas for the characteristic classes can be deduced, which generalize those of Borel-Hirzebruch for homogeneous spaces. For example, rational characteristic classes vanish in dimensions larger than \(\dim (K \backslash G/H)-(rk G-rk K-rk H)\). Finally, there is a formula for the Euler characteristics of \(K \backslash G/H\).
Mathematics Subject Classification ID: 57T15
Mathematics Subject Classification ID: 57T35
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 55R40
zbMATH DE Number: 537203
zbMATH Keywords: free action
zbMATH Keywords: double coset manifold
zbMATH Keywords: tangent bundle
zbMATH Keywords: cohomology
zbMATH Keywords: characteristic classes
zbMATH Keywords: rational characteristic classes
zbMATH Keywords: Euler characteristics
author: Wilhelm Singhof
reviewed by: Wilhelm Singhof
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 537203