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Morse-Novikov critical point theory, Cohn localization and Dirichlet units. (English) Zbl 0964.57030

Let \(M\) be a smooth manifold, and let \(\omega\) be a closed \(1\)-form on \(M\), \(d\omega = 0.\) If \(\omega\) vanishes at a point \(p \in M\) then \(p\) is called a zero of this form. A zero \(p\in M\) of \(\omega\) is called nondegenerate if there is a neighborhood \(U\) of \(p\) and a smooth function \(f : U \rightarrow \mathbb R\) with a nondegenerate critical point \(p\) such that \(\omega = df\) on \(U.\) A closed \(1\)-form is called Morse if all its zeros are nondegenerate. In the paper under review the author constructs a chain complex counting zeros of a Morse closed \(1\)-form on \(M.\) His construction is closely related with the Novikov complex [S. P. Novikov, Sov. Math., Dokl. 24, 222-226 (1981); translation from Dokl. Akad. Nauk SSSR 260, 31-35 (1981; Zbl 0505.58011)] and is based on the theory of a non-commutative localization developed by P. M. Cohn [Free rings and their relations. Lond. Math. Soc. Monogr. 19 (1985; Zbl 0659.16001)]. The author also describes some inequalities giving topological lower bounds on the minimum number of zeros of Morse closed \(1\)-forms.

MSC:

57R70 Critical points and critical submanifolds in differential topology
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
53C99 Global differential geometry

References:

[1] DOI: 10.1017/S0305004197001734 · Zbl 0894.58012 · doi:10.1017/S0305004197001734
[2] DOI: 10.1007/BF01086024 · Zbl 0603.58030 · doi:10.1007/BF01086024
[3] DOI: 10.1016/S0764-4442(99)80237-7 · Zbl 0942.58021 · doi:10.1016/S0764-4442(99)80237-7
[4] Novikov S. P., Soviet Math. Doklady 24 pp 222– (1981)
[5] DOI: 10.1070/RM1982v037n05ABEH004020 · Zbl 0571.58011 · doi:10.1070/RM1982v037n05ABEH004020
[6] Novikov S. P., Soviet Math. Dokl. 33 pp 551– (1986)
[7] DOI: 10.5802/afst.798 · Zbl 0859.57031 · doi:10.5802/afst.798
[8] DOI: 10.4310/MRL.1996.v3.n4.a12 · Zbl 0883.58036 · doi:10.4310/MRL.1996.v3.n4.a12
[9] Sikorav J.-C., Ann. Scient. Ecole Norm. 19 pp 543– (1986) · Zbl 0621.58018 · doi:10.24033/asens.1517
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