Hermitian structures on \(A_{\infty{}}\) ring spaces
type: Item
instance of: scholarly article
title: Hermitian structures on \(A_{\infty{}}\) ring spaces
zbMATH Open document ID: 0768.55009
DOI: 10.1007/BF00997380
author: Zbigniew Fiedorowicz
author: Roland Schwänzl
author: Rainer Vogt
published in: \(K\)-Theory
publication date: 1993-05-16T00:00:00Z
review text: Let \(M\) be a compact manifold. Rationally, one can get geometric information about \(M\) from the Hermitian \(K\)-theory of the group ring \(\mathbb{Z}[\pi_ 1M]\) of the fundamental group \(\pi_ 1M\). This paper is the first of a series aimed at analogous results with only the prime 2 inverted. For this purpose the group ring \(\mathbb{Z}[\pi_ 1M]\) is replaced by the free infinite loop space \(Q(G_ +)\) on the loop group \(G\) of \(M\). This is an \(A_ \infty\) ring space with structures analogous to an involution and a central unit. The authors show that, after inverting 2, these structures can be replaced by genuine involutions with good properties. This produces what the authors call a Hermitian \(A_ \infty\) ring, which can be used as input for the machinery that they develop in later papers.
Mathematics Subject Classification ID: 55P47
Mathematics Subject Classification ID: 19G38
zbMATH DE Number: 166867
zbMATH Keywords: Hermitian \(K\)-theory
zbMATH Keywords: group ring of the fundamental group
zbMATH Keywords: \(A_ \infty\) ring space
zbMATH Keywords: Hermitian \(A_ \infty\) ring
zbMATH Keywords: free infinite loop space
reviewed by: Richard John Steiner
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00997380
OpenAlex ID: W2318780166
description: scientific article; zbMATH DE number 166867