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Controlled connectivity of closed 1–forms

D Schütz

Algebraic & Geometric Topology 2 (2002) 171–217

arXiv: math.DG/0203283


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 = dimM 5 we can realize it by a closed 1–form which is Morse without critical points of index 0, 1, n 1 and n. If n = dimM 6 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 i 1 and i n 1 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.

controlled connectivity, closed 1–forms, Novikov complex
Mathematical Subject Classification 2000
Primary: 57R70
Secondary: 20J05, 57R19
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Received: 3 December 2001
Accepted: 8 March 2002
Published: 26 March 2002
D Schütz
Department of Mathematics
University College Dublin
Dublin 4