Surgery on the Novikov complex
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 universal covering of some triangulation \(\Delta\) of \(M\). The \(\lambda^-_\xi\)-chain complex is the Novikov-complex associated to the Morse function \(f\) and gradient field \(v\). The paper contains applications to Arnold's conjectures and the theory of fibering a manifold over \(S^1\).
scientific article; zbMATH DE number 929694
scientific article