arXiv research

Vanishing theorem for $L^2$-harmonic forms on Riemannian manifolds with parallel 1-form.

Study topological properties of foliations induced by closed 1-forms on orbifolds.

We consider the closed orbit structure of generic gradient flows of Morse closed 1-forms. The torsion of a chain homotopy equivalence between the Novikov complex and the completed simplicial chain complex of the universal cover detects the eta function of the flow. We extend this result to arbitrary Morse closed 1-form…

S.P.Novikov developed an analog of the Morse theory for closed 1-forms. In this paper I suggest an analog of the Lusternik - Schnirelman theory for closed 1-forms.

New method constructs Morse-Novikov complex with infinite series coefficients.

We develop the theory of twisted L^2-cohomology and twisted spectral invariants for flat Hilbertian bundles over compact manifolds. They can be viewed as functions on the first de Rham cohomology of M and they generalize the standard notions. A new feature of the twisted L^2-cohomology theory is that in addition to sat…

The paper studies twisted Morse homology and cohomology on manifolds.

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 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…

In this paper we state and prove Morse type inequalities for Morse functions as well as for closed differential 1-forms. These inequalities involve delocalized Betti numbers. As an immediate consequence, we prove the vanishing of delocalized Betti numbers of manifolds fibering over the circle.

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 $α$ be a Morse closed $1$-form of a smooth $n$-dimensional manifold $M$. The zeroes of $α$ of index $0$ or $n$ are called \emph{centers}. It is known that every non-vanishing de Rham cohomology class $u$ contains a Morse representative without centers. The result of this paper is the one-parameter analogue of the l…

This paper extends Lusternik-Schnirelmann category to non-compact manifolds.

In two previous papers with Yi-Jen Lee, we defined and computed a notion of Reidemeister torsion for the Morse theory of closed 1-forms on a finite dimensional manifold. The present paper gives an a priori proof that this Morse theory invariant is a topological invariant. It is hoped that this will provide a model for …

The paper suggests new topological lower bounds for the number of zeros of closed 1-forms within a given cohomology class. The main new technical tool is the deformation complex, which allows to pass to a singular limit and reduce the original problem with a closed 1-form to a traditional problem with a Morse function.…

We consider the problem of whether it is possible to improve the Novikov inequalities for closed 1-forms, or any other inequalities of a similar nature, if we assume, additionally, that the given 1-form is harmonic with respect to some Riemannian metric. We show that, under suitable assumptions, it is impossible. We us…

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

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

In this paper we study topological lower bounds on the number of zeros of closed 1-forms without Morse type assumptions. We prove that one may always find a representing closed 1-form having at most one zero. We introduce and study a generalization $cat(X,ξ)$ of the notion of Lusternik - Schnirelman category, depending…

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 study the $L^2$ gradient flow of the Yang--Mills functional on the space of connection 1-forms on a principal $G$-bundle over the sphere $S^2$ from the perspective of Morse theory. The resulting Morse homology is compared to the heat flow homology of the space $ΩG$ of based loops in the compact Lie group $G$. An iso…

We extend the Novikov Morse-type inequalities for closed 1-forms in 2 directions. First, we consider manifolds with boundary. Second, we allow a very degenerate structure of the critical set of the form, assuming only that the form is non-degenerated in the sense of Kirwan. In particular, we obtain a generalization of …

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 …

Study homotopy equivalence of spaces of gradient-like flows and Morse functions on surfaces.

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. The proof uses Bismut's modificatio…

We study the cohomology $H^*_{λω}(G/Γ, {\mathbb C})$ of the deRham complex $Λ^*(G/Γ)\otimes{\mathbb C}$ of a compact solvmanifold $G/Γ$ with a deformed differential $d_{λω}=d + λω$, where $ω$ is a closed 1-form. This cohomology naturally arises in the Morse-Novikov theory. We show that for a solvable Lie group $G$ with…

Zeta invariants study Morse forms on Riemannian manifolds, proving smoothness and convergence.

We study the Morse index of self-shrinkers for the mean curvature flow and, more generally, of $f$-minimal hypersurfaces in a weighted Euclidean space endowed with a convex weight. When the hypersurface is compact, we show that the index is bounded from below by an affine function of its first Betti number. When the fi…

We study Lie algebras of type I, that is, a Lie algebra $\mathfrak{g}$ where all the eigenvalues of the operator ad$_X$ are imaginary for all $X\in \mathfrak{g}$. We prove that the Morse-Novikov cohomology of a Lie algebra of type I is trivial for any closed $1$-form. We focus on locally conformal symplectic structures…

Topological complexity for closed 1-forms

Study on Riemannian Poisson warped product spaces and their properties.

This article deals with a continuous closed 1-form defined on a CW-complex. In particular, we show Lusternik-Schnirelmann type theory on continuous closed 1-forms which is related to gradient-like flows. M.Farber defined a continuous closed 1-form and a category with a respect to a cohomology class and constructed a Lu…

The paper explores parallel 1-forms on special Finsler manifolds and their properties.

New insights into cohomology of closed 1-forms.

Indices of vector fields and 1-forms studied for singular varieties and actions.

Lyapunov 1-forms on orbifolds help understand flows on compact spaces.

If a closed 3-manifold M supports a closed, nonsingular, irrational 1-form which linearly deforms into contact forms, then M supports a K-contact form. On the 3-torus, a closed nonsingular 1-form deforms linearly into contact forms if and only if it is a fibration 1-form. on any other 2-torus bundle over the circle, ev…

New metrics produce discrete zero sets for nondegenerate harmonic forms.

The study resolves a conjecture about harmonic forms on compact manifolds.

Calabi surgery modifies Z/2 harmonic 1-forms using 2-valued 1-forms.

The paper proves vanishing and finiteness theorems for p-harmonic 1-forms.

The study finds nondegenerate harmonic 1-forms using symmetry conditions.

Study Euler obstruction of 1-forms on determinantal singularities.

Constructs deformations of Vaisman manifolds preserving foliations.

We use the Frölicher-Nijenhuis formalism to reformulate the inverse problem of the calculus of variations for a system of differential equations of order 2k in terms of a semi-basic 1-form of order k. Within this general context, we use the homogeneity proposed by Crampin and Saunders in [14] to formulate and discuss t…

Study existence of harmonic 1-forms on Calabi-Yau manifolds.

The study examines hyperbolic 3-manifolds with uniform spectral gaps for coclosed 1-forms.

The paper constructs harmonic 1-forms on 3-manifolds with cylindrical necks.