The Morse-Novikov theory of circle-valued functions and noncommutative localization (Q2736089)

Let \(M\) be a compact smooth manifold, \(f:M\to S^1\) a Morse function with \(c_i(f)\) critical points of index \(i\). Let \(\Sigma^{-1}\mathbb{Z}[\pi_1 (M)]\) be the noncommutative localization of \(\mathbb{Z}[\pi_1(M)]\) with respect to a subset \(\Sigma\). Then a \(\Sigma^{-1}\mathbb{Z}[\pi_1(M)]\)-module \(\widehat C(M,f)\) such that NEWLINE\[NEWLINE\text{rank}_{\Sigma^{-1}\mathbb{Z}[\pi_1(M)]}\widehat{C}_i(M,f) = c_i(f),NEWLINE\]NEWLINE and chain equivalent to \(\Sigma^{-1}C(\widetilde{M})\), is constructed (Main Theorem). Here \(\Sigma^{-1}C(\widetilde{M})\) is the noncommutative localization of \(C(\widetilde M)\), the Morse-Smale complex. \(\widehat C(M,f)\) is a generalization of the Novikov complex \(C^{\text{Nov}}(M,f)\) [\textit{S. P. Novikov}, Russ. Math. Surv. 37, No. 5, 1-56 (1982); translation from Usp. Mat. Nauk 37, No. 5(227) 3-49 (1985; Zbl 0571.58011)]. As a consequence, the generalized Novikov inequalities NEWLINE\[NEWLINEc_i(f)\geq \mu_i(M;R),NEWLINE\]NEWLINE are obtained for any ring morphism \(\rho : \mathbb{Z}[\pi_1(M)]\to R\) sending \(\Sigma\) to invertible matrices over \(R\). Here \(\mu(M,R)\) is the minimum number of generators of degree \(i\) of any finitely generated free \(R\)-module chain complex which is chain homotopy equivalent to \(C(M;R)\). It is noted taking \(R\) to be the ring of rational functions, these inequalities recover the original Novikov inequalities [cf. \textit{M. Farber}, Funct. Anal. Appl. 19, 40-48 (1985; Zbl 0603.58030)].NEWLINENEWLINENEWLINE\(\Sigma\) is defined to be the set of square matrices with entries in \(\mathbb{Z}[\pi_1(M)]\) having the form \(1-ze\) where \(e\) is a square of matrices with entries in \(\mathbb{Z}[\pi]\). Here \(\pi\) is the kernel of \(f_* : \pi_1(M)\to\pi_1(S^1)\). Noncommutative localization is taken in the sense of \textit{P. M. Cohn} [`Free rings and their relations, London, Academic Press (1971; Zbl 0232.16003), 2nd ed. (1985; Zbl 0659.16001)]. To show the Main Theorem, first necessary algebraic machineries on the noncommutative localization of endomorphisms and chain complexes are prepared in Section 2. Then apply these machineries to the handlebody decomposition of \(M\) given by \(f\), the Main Theorem is proved in the last Section 3.NEWLINENEWLINEFor the entire collection see [Zbl 0967.00102].