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2002 Controlled connectivity of closed 1–forms
Dirk Schütz
Algebr. Geom. Topol. 2(1): 171-217 (2002). DOI: 10.2140/agt.2002.2.171


We discuss controlled connectivity properties of closed 1–forms and their cohomology classes and relate them to the simple homotopy type of the Novikov complex. The degree of controlled connectivity of a closed 1–form depends only on positive multiples of its cohomology class and is related to the Bieri–Neumann–Strebel–Renz invariant. It is also related to the Morse theory of closed 1–forms. Given a controlled 0–connected cohomology class on a manifold M with n= dimM5 we can realize it by a closed 1–form which is Morse without critical points of index 0, 1, n1 and n. If n= dimM6 and the cohomology class is controlled 1–connected we can approximately realize any chain complex D with the simple homotopy type of the Novikov complex and with Di=0 for i1 and in1 as the Novikov complex of a closed 1–form. This reduces the problem of finding a closed 1–form with a minimal number of critical points to a purely algebraic problem.


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Dirk Schütz. "Controlled connectivity of closed 1–forms." Algebr. Geom. Topol. 2 (1) 171 - 217, 2002.


Received: 3 December 2001; Accepted: 8 March 2002; Published: 2002
First available in Project Euclid: 21 December 2017

zbMATH: 1002.57069
MathSciNet: MR1917049
Digital Object Identifier: 10.2140/agt.2002.2.171

Primary: 57R70
Secondary: 20J05, 57R19

Rights: Copyright © 2002 Mathematical Sciences Publishers


Vol.2 • No. 1 • 2002
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