Surgery on the Novikov complex
type: Item
instance of: scholarly article
title: Surgery on the Novikov complex
zbMATH Open document ID: 0861.57042
DOI: 10.1007/BF00533216
published in: \(K\)-Theory
publication date: 1997-05-12T00:00:00Z
review text: The main result of the paper is the Theorem: Let \(M\) be a closed connected manifold with \(\dim(M)\geq 6\) and fundamental group \(G=\pi_1(M)\). Let \(\xi: G\to S^1\) be a group homomorphism whose kernel is finitely presented. Let \(D_*\) be a finite-dimensional nonnegative finitely generated based free chain complex over the Novikov ring \(\Lambda^-_\xi\). Suppose that \(D_*\) is simple homotopy equivalent to \(C^\Delta_*(\widetilde M)\otimes_\Lambda\Lambda^-\) and that \(D_i=0\) for \(i\leq 1\) and \(i\geq\dim(M)-1\). Let \(N\) be an integer. Then there is a Morse map \(f:M\to S^1\) which belongs to \(\xi\) and a perfect gradient field for \(f\) such that \(C(f,v)\) is \(N\)-equivalent to \(D_*\). Here are some explanations. Let \(\widehat{\widehat\Lambda}\) be the Abelian group of (arbitrary) formal sums \(\lambda=\sum_{g\in G}\lambda_g\cdot g\). The support \(\text{supp}(\lambda)\) of \(\lambda\) is the subset of \(G\) consisting of those elements \(g\in G\) for which \(\lambda_g\) is non-zero. Let \(\Lambda^-_\xi\) be the subset \(\widehat{\widehat\Lambda}\) consisting of elements \(\lambda\) with the property that for each \(c\in\mathbb{Z}\) the set of \(\text{supp}(\lambda)\cap\xi^{-1}([c,+\infty[)\) is finite. This set \(\lambda^-_\xi\) has an obvious ring structure and contains the integral group ring \(\Lambda\) of \(G\) as subring. Given an integer \(N\), two elements \(a\), \(b\) of \(\Lambda^-_\xi\) are called \(N\)-equivalent, if \(\text{supp}(a-b)\) is contained in \(\xi^{-1}(]-\infty,-N])\). Two finite-dimensional non-negative finitely generated based free chain complexes \(D_*\) and \(E_*\) over the Novikov ring \(\Lambda^-_\xi\) are called \(N\)-equivalent if there is a one-to-one-correspondence of their bases in each dimension and the entries of the matrices describing the \(p\)th differential are \(N\)-equivalent for all \(p\). The \(\Lambda\)-chain complex \(C^\Delta_*(\widetilde M)\) is the cellular \(\Lambda\)-chain complex of a lift to the …
reviewed by: Wolfgang Lück
Mathematics Subject Classification ID: 57R45
Mathematics Subject Classification ID: 57R70
zbMATH DE Number: 929694
zbMATH Keywords: Morse function
zbMATH Keywords: Whitney trick
zbMATH Keywords: fundamental group
zbMATH Keywords: Novikov ring
zbMATH Keywords: simple homotopy equivalent
author: A. Pajitnov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 929694