arXiv research

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

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…

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

We prove the Novikov conjecture on oriented Cheeger spaces whose fundamental group satisfies the strong Novikov conjecture. A Cheeger space is a stratified pseudomanifold admitting, through a choice of ideal boundary conditions, an L2-de Rham cohomology theory satisfying Poincare duality. We prove that this cohomology …

New metrics with non-negative scalar curvature are always Ricci-flat on certain surgeries.

In this paper, we prove the algebraic K-theory Novikov conjecture for group algebras over the ring of Schatten class operators. The main technical tool in the proof is an explicit construction of the Connes-Chern character.

Generalizes Novikov conjecture results to infinite-dimensional bundles.

We present sufficient conditions for the cohomology of a closed aspherical manifold to be proper Lipschitz in sense of Connes-Gromov-Moscovici [CGM]. The conditions are stated in terms of the Stone-Čech compactification of the universal cover of a manifold. We show that these conditions are formally weaker than the suf…

We use K-area homology to summarize some results about the Novikov conjecture and the Hirzebruch L-class. In fact, we provide necessary and sufficient conditions for closed manifolds to have a homotopy invariant L-class. In order to obtain additional properties we also introduce the cohomology of infinite K--area which…

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

We prove that any countable discrete and torsion free subgroup of a general linear group over an arbitrary field or a similar subgroup of an almost connected Lie group satisfies the integral algebraic K-theoretic (split) Novikov conjecture over \cpt and §, where \cpt denotes the C^*-algebra of compact operators and §de…

We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are presented during the course of the discussion. We also review the status and app…

We construct finite-gap solutions to the modified Novikov-Veselov equations, describe their spectral properties and the reduction to the modified Korteweg--de Vries equation and explain its relation to soliton deformations of tori and the Willmore conjecture.

The paper develops surgery theories for foliations and solves a problem posed by Weinberger.

We develop some basic Lipschitz homotopy technique and apply it to manifolds with finite asymptotic dimension. In particular we show that the Higson compactification of a uniformly contractible manifold is mod $p$ acyclic in the finite dimensional case. Then we give an alternative proof of the Higher Signature Novikov …

Decomposition complexity for metric spaces was recently introduced by Guentner, Tessera, and Yu as a natural generalization of asymptotic dimension. We prove a vanishing result for the continuously controlled algebraic K-theory of bounded geometry metric spaces with finite decomposition complexity. This leads to a proo…

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 …

For a Morse map $f:M\to S^1$ Novikov [11] has introduced an analog of Morse complex, defined over the ring $\ZZZ[[t]][t^{-1}]$ of integer Laurent power series. Novikov conjectured, that generically the matrix entries of the differentials in this complex are of the form $\sum_ia_it^i$, where $a_i$ grow at most exponenti…

We prove the Gromov conjecture on the macroscopic dimension of the universal covering of a closed spin manifold with a positive scalar curvature under the following assumptions on the fundamental group: 1. The Strong Novikov Conjecture holds for $π$. 2. The natural map $per:ko_n(Bπ)\to KO_n(Bπ)$ is injective.

The paper sets new bounds on metrics with positive scalar curvature.

The Arnold conjecture is proven for integers using Floer theory.

Two 4-manifolds are stably diffeomorphic if they become diffeomorphic after connected sum with S^2 x S^2's. This paper shows that two closed, orientable, homotopy equivalent, smooth 4-manifolds are stably diffeomorphic, provided a certain map from the second homology of the fundamental group with coefficients in Z/2 to…

This article will explore the K- and L-theory of group rings and their applications to algebra, geometry and topology. The Farrell-Jones Conjecture characterizes K- and L-theory groups. It has many implications, including the Borel and Novikov Conjectures for topological rigidity. Its current status, and many of its co…

We prove a version of the Arnol'd conjecture for Lagrangian submanifolds of conformal symplectic manifolds: a Lagrangian $L$ which has non-zero Morse-Novikov homology for the restriction of the Lee form $β$ cannot be disjoined from itself by a $C^0$-small Hamiltonian isotopy. Furthermore for generic such isotopies the …

We define and study the signature, A-hat genus and higher signatures of the quotient space of an $S^1$-action on a closed oriented manifold. We give applications to questions of positive scalar curvature and to an Equivariant Novikov Conjecture.

We prove that $Out(F_N)$ is boundary amenable. This also holds more generally for $Out(G)$, where $G$ is either a toral relatively hyperbolic group or a finitely generated right-angled Artin group. As a consequence, all these groups satisfy the Novikov conjecture on higher signatures.

For all systolic groups we construct boundaries which are EZ--structures. This implies the Novikov conjecture for torsion--free systolic groups. The boundary is constructed via a system of distinguished geodesics in a systolic complex, which we prove to have coarsely similar properties to geodesics in CAT(0) spaces.

We show that the Novikov-Shubin invariant of an element of the integral group ring of the lamplighter group Z_2 \wr Z can be irrational. This disproves a conjecture of Lott and Lueck. Furthermore we show that every positive real number is equal to the Novikov-Shubin invariant of some element of the real group ring of Z…

In this short note we apply methods introduced by B. Hanke and T. Shick to prove the vanishing of (low dimensional) higher $A$-genera for spin manifolds admitting a positive scalar curvature metric. Our aim is to provide a short and unified proof for this beautiful result without using the strong Novikov conjecture.

We construct a geometric model for the mapping class group M of a non-exceptional oriented surface of finite type and use it to show that the action of M on the compact Hausdorff space of complete geodesic laminations is topologically amenable. As a consequence, the Novikov higher signature conjecture holds for every s…

A locally conformally Kahler (LCK) manifold is a complex manifold with a Kahler structure on its covering and the deck transform group acting on it by holomorphic homotheties. One could think of an LCK manifold as of a complex manifold with a Kahler form taking values in a local system $L$, called the conformal weight …

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…

Novikov inequalities extended to orbifolds.

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

Knots' Morse-Novikov number behaves additively under connected sum and unchanged by cabling.

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…

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.…

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…

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

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

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…

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…

We prove that the higher harmonic signature of an even dimensional oriented Riemannian foliation of a compact Riemannian manifold with coefficients in a leafwise U(p,q)-flat complex bundle is a leafwise homotopy invariant. We also prove the leafwise homotopy invariance of the twisted higher Betti classes. Consequences …

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…

This paper proves that one-relator groups have a weak Z-structure.