Controlled connectivity of closed 1–forms

Abstract

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 = dim M ≥ 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 = dim M ≥ 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 D i = 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.

Morse Novikov theory and cohomology with forward supports

“…As noted in the introduction, the latter is just a formal complex of Λ Z -modules (see e.g. [P], [Sc2]). Our S-complex realizes the Novikov complex geometrically as a subcomplex of currents, just as the current Morse complex geometrically realizes the Morse complex.…”

Geometric Chain Homotopy Equivalences Between Novikov Complexes

“…That the Novikov complex C * (ω, v) is simple chain homotopy equivalent to ZG ξ ⊗ ZG C ∆ * (M ) is important for finding a minimal number of critical points for a Morse form within a cohomology class, see Latour [8] or [23]. Other applications of torsion are discussed in the next section.…”

“…Let us finish by giving an example of a gradient for which the noncommutative zeta function contains more information than the commutative version. For this we need the following theorem which is proven in [23].…”

On the Whitehead group of Novikov rings associated to irrational homomorphisms

“…In recent years there has also been considerable interest in the simple homotopy type of the Novikov complex; see Latour [12], Pajitnov [18], Damian [4], Schütz [24] or Cornea and Ranicki [3]. Notably, Latour [12] introduced the Whitehead group of the Novikov ring Wh(G; ξ ), a quotient of K 1 ( ZG ξ ) by so called trivial units.…”