On the Whitehead group of Novikov rings associated to irrational homomorphisms

Schütz, Dirk

doi:10.1016/j.jpaa.2006.01.005

Cited by 3 publications

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“…, a k q such that that | detpF 0 q| is minimal possible. I claim that this volume : After this article was completed, I became aware that a similar argument was also used by D. Schütz [19] in his work about K-theory of Novikov rings. equals 1.…”

Section: Theorem 34 For Every Family B Of Vectors In Z K the Cone Cmentioning

confidence: 78%

“…The completion of a twisted polynomial ring can be seen as a ring of special power series (in non-commuting variables). Consider the set of all series of the form (19) λ " 8 ÿ n"0 a n θ In , a n P R, I n P N k , and I n ։ 8.…”

Section: Example 48mentioning

confidence: 99%

“…The next proposition is obvious. Replacing N k by Z k in the formula (19) we obtain the definition of Σ-twisted special Laurent series ring in variables θ 1 , . .…”

Section: Example 48mentioning

confidence: 99%

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On the conical Novikov homology

Pajitnov

2019

European Journal of Mathematics

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Let ω be a Morse form on a closed connected manifold M . Let p :x M Ñ M be a regular covering with structure group G, such that p ˚prωsq " 0. The period homomorphism π1pM q Ñ R corresponding to ω factors through a homomorphism ξ : G Ñ R. The rank of Im ξ is called the irrationality degree of ξ. Denote by Λ the group ring ZG and let p Λ ξ be its Novikov completion. Choose a transverse ω-gradient v. The classical construction of counting the flow lines of v defines the Novikov complex N˚freely generated over p Λ ξ by the set of zeroes of ω. In this paper we introduce a refinement of this construction. We define a subring p Λ Γ of p Λ ξ (depending on an auxiliary parameter Γ which is a certain cone in the vector space H 1 pG, Rq) and show that the Novikov complex N˚is defined actually over p Λ Γ and computes the homology of the chain complex C˚p x M q b Λ p Λ Γ . In the particular case when G « Z 2 , and the irrationality degree of ξ equals 2, the ring p Λ Γ is isomorphic to the ring of series in 2 variables x, y of the form ř rPN arx nr y mr where ar, nr, mr P Z and both nr, mr converge to 8 when r Ñ 8.The algebraic part of the proof is based on a suitable generalization of the classical algorithm of approximating irrational numbers by rationals. The geometric part is a straightforward generalization of the author's proof of the particular case of this theorem concerning the circle-valued Morse maps [15]. As a byproduct we obtain a simple proof of the properties of the Novikov complex for the case of Morse forms of irrationality degree ą 1.The paper contains two appendices. In Appendix 1 we give an overview of the E. Pitcher's work on circle-valued Morse theory (1939). We show that Pitcher's lower bounds for the number of critical points of a circle-valued Morse map coincide with the torsion-free part of the Novikov inequalities (1982). In Appendix 2 we construct a circle-valued Morse map and its gradient such that its unique Novikov incidence coefficient is a power series in one variable with an arbitrarily small convergence radius.

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Section: Theorem 34 For Every Family B Of Vectors In Z K the Cone Cmentioning

confidence: 78%

Section: Example 48mentioning

confidence: 99%

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On the conical Novikov homology

Pajitnov

2019

European Journal of Mathematics

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Foliated eight-manifolds for M-theory compactification

Babalic

Lazaroiu

2015

J. High Energ. Phys.

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We characterize compact eight-manifolds M which arise as internal spaces in N=1 flux compactifications of M-theory down to AdS3 using the theory of foliations, for the case when the internal part of the supersymmetry generator is everywhere non-chiral. We prove that specifying such a supersymmetric background is equivalent with giving a codimension one foliation of M which carries a leafwise G2 structure, such that the O'Neill-Gray tensors, non-adapted part of the normal connection and torsion classes of the G2 structure are given in terms of the supergravity four-form field strength by explicit formulas which we derive. We discuss the topology of such foliations, showing that the C star algebra of the foliation is a noncommutative torus of dimension given by the irrationality rank of a certain cohomology class constructed from the four-form field strength, which must satisfy the Latour obstruction. We also give a criterion in terms of this class for when such foliations are fibrations over the circle. When the criterion is not satisfied, each leaf of the foliation is dense in M.Comment: 63 pages, 2 figure

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Novikov homology of HNN–extensions and right-angled Artin groups

Schütz

2009

Algebr. Geom. Topol.

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We calculate the Novikov homology of right-angled Artin groups and certain HNNextensions of these groups. This is used to obtain information on the homological Sigma invariants of Bieri-Neumann-Strebel-Renz for these groups. These invariants are subsets of all homomorphisms from a group to the reals containing information on the finiteness properties of kernels of such homomorphisms. We also derive information on the homotopical Sigma invariants and show that one cannot expect any symmetry relations between a homomorphism and its negative regarding these invariants. While it was previously known that these invariants are not symmetric in general, we give the first examples of homomorphisms which are symmetric with respect to the homological invariant, but not with respect to the homotopical invariant. 20J05; 20F65, 57R19

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On the Whitehead group of Novikov rings associated to irrational homomorphisms

Cited by 3 publications

References 27 publications

On the conical Novikov homology

On the conical Novikov homology

Foliated eight-manifolds for M-theory compactification

Novikov homology of HNN–extensions and right-angled Artin groups

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