arXiv research

The paper proves two theorems for modified Novikov operators under conformal perturbations.

Study Morse-Novikov cohomology on foliated manifolds and prove Hodge theorem.

Simplified proof of a theorem about 3D shapes.

It is shown that the Novikov inequalities for critical points of closed 1-forms hold with the von Neumann Betti numbers replacing the Novikov numbers. As a corollary we obtain a vanishing theorem for $L^2$ cohomology, generalizing a theorem of W. Lueck. We also prove that von Neumann Betti numbers coincide with the Nov…

This short note shows how the Novikov conjecture for mapping class groups follows from a theorem of Kato and a result theorem of Hamenstadt.

In this paper, we give two Lichnerowicz type formulas for modified Novikov operators. We prove KastlerKalau-Walze type theorems for modified Novikov operators on compact manifolds with (resp.without) boundary. We also compute the spectral action for Witten deformation on 4-dimensional compact manifolds.

We view Dolbeault-Morse-Novikov cohomology H^{p,q}_η(X) as the cohomology of the sheaf Ω_{X,η}^p of η-holomorphic p-forms and give several bimeromorphic invariants. Analogue to Dolbeault cohomology, we establish the Leray-Hirsch theorem and the blow-up formula for Dolbeault-Morse-Novikov cohomology. At last, we conside…

In this article, we first consider the $L^{2}$ \textit{Morse-Novikov cohomology} on a complete Riemannian manifold $M$ equipped with a parallel $1$-form which includes Vaisman manifold. Based on a vanishing theorem of $L^{2}$ \textit{Morse-Novikov cohomology}, we prove that the $L^{2}$-harmonic forms on $M$ are identic…

The article studies cohomology on complex manifolds and proves vanishing theorems.

Generalizes Novikov conjecture results to infinite-dimensional bundles.

Extends Morse-Novikov Homology to include differential graded coefficients and fibration structures.

Novikov theorem extended to rational Pontryagin classes for cyclic group $C_4$.

The weight $θ$-sheaf $\underline{\mathbb{R}}_{X,θ}$ helps us to reinterpret Morse-Novikov cohomologies via sheaf theory. We give several theorems of Künneth and Leray-Hirsch types. As applications, we prove that the $θ$-Lefschetz number is independent of $θ$ and calculate the Morse-Novikov cohomologies of projective bu…

One of the basic objects in the Morse theory of circle-valued maps is Novikov complex - an analog of the Morse complex of Morse functions. Novikov complex is defined over the ring of Laurent power series with finite negative part. The main aim of this paper is to present a detailed and self-contained exposition of the …

A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering, with the monodromy acting on this covering by homotheties. We define three cohomology invariants, the Lee class, the Morse-Novikov class, and the Bott-Chern class, of an LCK-structure. These invariants together play the same …

In this paper we prove the Cheeger-Müller theorem for $L^2$-analytic torsion form under the assumption that there exists a fiberwise Morse function and the Novikov-Shubin invariant is positive.

Abstract: Generalizes stability theories over toric varieties and Novikov type rings.

An excision theorem connects Heegaard Floer homology of 3-manifolds.

We generalize the Novikov inequalities for 1-forms in two different directions: first, we allow non-isolated critical points (assuming that they are non-degenerate in the sense of R.Bott), and, secondly, we strengthen the inequalities by means of twisting by an arbitrary flat bundle. We also obtain an $L^2$ version of …

Novikov inequalities extended to orbifolds.

New knots found with Seifert genus not matching minimal genus Seifert surfaces.

The Morse-Novikov number MN(L) of a smooth link L in the three-dimensional sphere is by definition the minimal possible number of critical points of a regular circle-valued Morse function on the link complement (the term regular means that the Morse function must have nice behaviour in a tubular neighbourhood of L). No…

It is shown that the topological phenomenon "zero in the continuous spectrum", discovered by S.P.Novikov and M.A.Shubin, can be explained in terms of a homology theory on the category of finite polyhedra with values in certain abelian category. This approach implies homotopy invariance of the Novikov-Shubin invariants.…

A well-known property of the signature of closed oriented 4n-dimensional manifolds is Novikov additivity, which states that if a manifold is split into two manifolds with boundary along an oriented smooth hypersurface, then the signature of the original manifold equals the sum of the signatures of the resulting manifol…

We present a new approach to Morse and Novikov theories, based on the deRham Federer theory of currents, using the finite volume flow technique of Harvey and Lawson. In the Morse case, we construct a noncompact analogue of the Morse complex, relating a Morse function to the cohomology with compact forward supports of t…

Survey simplifies embedding theorems for manifolds.

We show that the Cappell-Shaneson version of Pick's theorem for simple lattice polytopes is a consequence of a general relation between characteristic numbers of virtual submanifolds dual to the characteristic classes of a stably almost complex manifold. This relation is analogous to the miraculous cancellation formula…

Let $ω$ be a Morse form on a manifold $M$. Let $p:\hat M\to M$ be a regular covering with structure group $G$, such that $p^*([ω])=0$. Let $ξ:G\to\mathbf{R}$ be the corresponding period homomorphism. Denote by ${\hat Λ}_ξ$ the Novikov completion of the group ring $\mathbf{Z} G$. Choose a transverse $ω$-gradient $v$. Co…

Inspired by the recent works of S. Rao--S. Yang--X.-D. Yang and L. Meng on the blow-up formulae for de Rham and Morse--Novikov cohomology groups, we give a new simple proof of the blow-up formula for Morse--Novikov cohomology by introducing the relative Morse--Novikov cohomology group via sheaf cohomology theory and pr…

In this paper we continue the study of generic properties of the Novikov complex, began in the work "The incidence coefficients in the Novikov complex are generically rational functions" ( dg-ga/9603006). For a Morse map $f:M\to S^1$ there is a refined version of Novikov complex, defined over the Novikov completion of …

The paper explores the geometry of level lines of quasiperiodic functions with many periods.

Given a $1$-cohomology class $u$ on a closed manifold $M$, we define a Novikov fundamental group associated to $u$, generalizing the usual fundamental group in the same spirit as Novikov homology generalizes Morse homology to the case of non exact $1$-forms. As an application, lower bounds for the minimal number of ind…

Let f be a Morse map from a closed manifold to a circle. S.P.Novikov constructed an analog of the Morse complex for f. The Novikov complex is a chain complex defined over the ring of Laurent power series with integral coefficients and finite negative part. This complex depends on the choice of a gradient-like vector fi…

A graph multilink is a link with multiplicities in a homology 3-sphere whose exterior is a graph manifold. In this Note, we compute the Novikov homology of graph multilinks. As a corollary, we give a majoration for the number of Novikov modules on a given graph link.

We give a survey on recent development of the Novikov conjecture and its applications to topological rigidity and non-rigidity. .

An introduction to circle valued Morse theory and Novikov homology, from an algebraic point of view.

Novikov conjecture reduced to Lipschitz cohomology of groups.

We show that the rational Novikov conjecture for a group $Γ$ of finite homological type follows from the mod 2 acyclicity of the Higson compactifcation of an E$Γ$. We then show that for groups of finite asymptotic dimension the Higson compactification is mod p acyclic for all p, and deduce the integral Novikov conjectu…

In this paper we construct a Universal chain complex, counting zeros of closed 1-forms on a manifold. The Universal complex is a refinement of the well known Novikov complex; it relates the homotopy type of the manifold, after a suitable noncommutative localization, with the numbers of zeros of different indices which …

We establish an equivariant generalization of the Novikov inequalities which allow to estimate the topology of the set of critical points of a closed basic invariant 1-form by means of twisted equivariant cohomology of the manifold. We test and apply our inequalities in the case of a finite group. As an application we …

We continue the study of the twisted Novikov homology, introduced in our joint paper with H.Goda (arXiv:math.DG/0312374), and its generalizations. The main applications of the developed algebraic techniques are to the topology of 3-manifolds. We show in particular that the twisted Novikov homology of a 3-manifold M of …

We use noncommutative localization to construct a chain complex which counts the critical points of a circle-valued Morse function on a manifold, generalizing the Novikov complex. As a consequence we obtain new topological lower bounds on the minimum number of critical points of a circle-valued Morse function within a …

We show the Morse-Novikov number of knots in $S^3$ is additive under connected sum and unchanged by cabling.

The present paper contains an interpretation and generalization of Novikov's theory of Morse type inequalities for 1-forms in terms of Conley's theory for dynamical systems.

We consider systems $(M,ω,g)$ with $M$ a closed smooth manifold, $ω$ a real valued closed one form and $g$ a Riemannian metric, so that $(ω,g)$ is a Morse-Smale pair, Definition~2. We introduce a numerical invariant $ρ(ω,g)\in[0,\infty]$ and improve Morse-Novikov theory by showing that the Novikov complex comes from a …

We prove that, if a group is relatively hyperbolic, the parabolic subgroups are virtually nilpotent if and only if there exists a hyperbolic space with bounded geometry on which it acts geometrically finitely. This provides, by use of M. Bonk and O. Schramm embedding theorem, a very short proof of the finiteness of asy…

In the early 2000's Cochran and Harvey introduced non-commutative Alexander polynomials for 3-manifolds. Their degrees give strong lower bounds on the Thurston norm. In this paper we make the case that the vanishing of a certain Novikov-Sikorav homology module is the correct notion of a monic non-commutative Alexander …

We prove that the Morse-Novikov number of a link L in a 3-sphere is less than or equal to twice the tunnel number of L.