Surgery on the Novikov complex

Pazhitnov, A. V.

doi:10.1007/bf00533216

Cited by 21 publications

(40 citation statements)

References 32 publications

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“…A similar theorem has been proven by Pajitnov [24,Th.0.12] in the case of a circle valued Morse function f : M → S 1 . The condition CC 1 is replaced there by the condition that ker(f # : π 1 (M ) → Z) is finitely presented.…”

Section: Theorem 11 Let M N Be a Closed Connected Smooth Manifold Wisupporting

confidence: 73%

Controlled connectivity of closed 1–forms

Schütz

2002

Algebr. Geom. Topol.

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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.

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Section: Theorem 11 Let M N Be a Closed Connected Smooth Manifold Wisupporting

confidence: 73%

Controlled connectivity of closed 1–forms

Schütz

2002

Algebr. Geom. Topol.

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“…Conversely, one can ask if the vanishing of H * (M, u) implies the existence of a nowhere vanishing closed 1-form belonging to the class u ∈ H 1 (M ). For n ≥ 6 this problem was independently solved by F. Latour [19] and A. Pajitnov [24], [25] …”

Section: Morse-novikov Theorymentioning

confidence: 99%

“…2. In the statement of [24] the first two conditions are replaced by: u) is simply equivalent to zero. Actually, one can show (see [20]) that 1 is equivalent to "1 and 2".…”

Section: Moreover If Q Is Odd Condition 2 Is Always Fulfilled (See Rmentioning

confidence: 99%

On the homotopy of finite $ {CW}$-complexes with polycyclic fundamental group

Damian

2008

Trans. Amer. Math. Soc.

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“…The applications of the result of the present will appear in the forthcoming paper [10], which contains the surgery theory for maps M --~ S1 in terms of the Novikov complex.…”

mentioning

confidence: 84%

On the Novikov complex for rational morse forms

Pazhitnov¹

1995

Annales De La Faculté Des Sciences De Toulouse Mathématiques

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Surgery on the Novikov complex

Cited by 21 publications

References 32 publications

Controlled connectivity of closed 1–forms

Controlled connectivity of closed 1–forms

On the homotopy of finite $ {CW}$-complexes with polycyclic fundamental group

On the Novikov complex for rational morse forms

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