The Morse-Novikov theory of circle-valued functions and noncommutative localization. (English) Zbl 0983.58007

Farber, M.; Ranicki, A.

Bukhshtaber, V. M. (ed.) et al., Solitons, geometry, and topology: on the crossroads. Collected papers dedicated to the 60th birthday of Academician Sergei Petrovich Novikov. Transl. from the Russian. Moscow: MAIK Nauka/Interperiodica, Proc. Steklov Inst. Math. 225, 363-371 (1999); and Tr. Mat. Inst. Steklova 225, 381-388 (1999).

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 \[ \text{rank}_{\Sigma^{-1}\mathbb{Z}[\pi_1(M)]}\widehat{C}_i(M,f) = c_i(f), \] 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 \(C^{\text{Nov}}(M,f)\) [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 \[ c_i(f)\geq \mu_i(M;R), \] 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. M. Farber, Funct. Anal. Appl. 19, 40-48 (1985; Zbl 0603.58030)].
\(\Sigma\) 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 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.
For the entire collection see [Zbl 0967.00102].

Reviewer: Akira Asada (Takarazuka)

MSC:

58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
57R70	Critical points and critical submanifolds in differential topology
16U20	Ore rings, multiplicative sets, Ore localization

Keywords:

Morse function; noncommutative localization; Novikov complex; Novikov inequalities

Citations:

Zbl 0659.16001; Zbl 0571.58011; Zbl 0603.58030; Zbl 0232.16003