arXiv research

The study tackles realisability of twisted homology classes and introduces new techniques in parametrised homotopy theory.

New constructions show manifold volumes are dense in non-negative reals.

Enhances Pontryagin-Thom theorem for manifold maps.

Proves a Thom isomorphism for foliated differential forms.

Relative Thom polynomials for maps around boundaries established.

Proves a Thom Isotopy Theorem for nonproper semialgebraic maps.

Researchers describe a new Thom form for mapping cones.

In [5] I solved the Thom's conjecture that a proper Thom map is triangulable. In this paper I drop the properness condition in the semialgebraic case and, moreover, in the definable case in an o-minimal structure.

Paper constructs Thom-Smale complex using instantons from Morse functions.

Analytic realization of Thom-Smale complex for G-manifolds.

In this work we analyze the behavior of Massey products of closed manifolds under the blow-up construction. The results obtained in the article are applied to the problem of constructing closed symplectic non-formal manifolds. The proofs use Thom spaces as an important technical tool. This application of Thom spaces is…

New findings contradict the Thom conjecture for high degree hypersurfaces in $CP^3$.

We develop a theory of parametrized geometric cobordism by introducing smooth Thom stacks. This requires identifying and constructing a smooth representative of the Thom functor acting on vector bundles equipped with extra geometric data, leading to a geometric refinement of the the Pontrjagin-Thom construction in stac…

Enhanced GARCH model uses autoencoder for volatility forecasting.

This article is devoted to the study of smooth desingularization, which are customary employed in the definition of De Rham Intersection Cohomology with differential forms. In this paper we work with the category of Thom-Mather simple spaces. We construct a functor which sends each Thom-Mather simple space into a smoot…

These notes are the first chapter of a monograph, dedicated to a detailed proof of the equivariant index theorem for transversally elliptic operators. In this preliminary chapter, we prove a certain number of natural relations in equivariant cohomology. These relations include the Thom isomorphism in equivariant cohomo…

The study calculates the Smith-Thom deficiency of Hilbert squares and provides conditions for maximality.

We define complex cobordism realizations of cohomological Thom polynomials and study their existence, uniqueness and other features. We show that problem is non-trivial on the example of $Σ^1$ singularity.

The first part of this article intends to present the role played by Thom in diffusing Smale's ideas about immersion theory, at a time (1957) where some famous mathematicians were doubtful about them: it is clearly impossible to make the sphere inside out! Around a decade later, M. Gromov transformed Smale's idea in wh…

We demonstrate the usage of explicit form of the Thom class found by Mathai and Quillen for the definition of generating functional of a simple supersymmetric quantum mechanical model.

Proves Arnold-Thom conjecture for surfaces' arrival times.

We study Thom Transversality Theorem using a point of view, suggested by Gromov, which allows to avoid the use of Sard Theorem and gives finer informations on the structure of the set of non-transverse maps.

We give a Pontryagin-Thom-Szucs type construction for non-positive codimensional singular maps, and obtain results about cobordism and bordism groups of -1 codimensional stable maps with prescribed singular fibers.

Develops deformation theory for mapping spaces related to Morse theory and Bott-Thom isomorphism.

Paper confirms Thom's conjecture for nonlinear evolutions on manifolds.

The paper shows geometric realisation over specific groups and knots.

This paper has been withdrawn by the author due a crucial sign error in Theorem B. We present a geometric proof of Thom conjecture, which uses Khovanov homology. Our approach doesn't use any analytic methods and is quite different from proof given by Kronheimer and Mrowka in 1994.

We develop a new purely combinatorial approach to N. Steenrod's problem on realisation of cycles. We prove that every n-dimensional homology class of every topological space can be realised with some multiplicity by an image of a finite-fold covering over the manifold M^n, where M^n is the isospectral manifold of real …

Motivated by the importance of symplectic isotropic realisations in the study of Poisson manifolds, this paper investigates the local and global theory of contact isotropic realisations of Jacobi manifolds, which are those of minimal dimension. These arise naturally when considering multiplicity-free actions in contact…

Let G be a compact Lie group. Let M be a smooth G-manifold and V --> M be an oriented G-equivariant vector bundle. One defines the spaces of equivariant forms with generalized coefficients on V and M. An equivariant Thom form $θ$ on V is a compactly supported closed equivariant form such that its integral along the fib…

We indicate how to combine some classical topology (Thom's work on the Steenrod problem) with some modern topology (simplicial volume) to show that every map between certain manifolds must have degree zero. We furthermore discuss a homotopy theoretic interpretation of parts of our proof, using Thom spaces and Steenrod …

Study shows a modified cobordism category's first derivative is equivalent to a Thom spectrum.

We show that the spectrum constructed by Everitt and Turner as a possible Khovanov homotopy type is a product of Eilenberg-MacLane spaces and is thus determined by Khovanov homology. By using the Dold-Thom functor it can therefore be obtained from the Khovanov homotopy type constructed by Lipshitz and Sarkar.

The paper finds a geometric explanation for coinciding Thom polynomials of cusp and corank-2 singularities.

Proves stable properties of proper maps between manifolds.

The paper connects cut locus, Thom space, and Morse-Bott functions in Riemannian geometry.

The paper proves a theorem about fixed points and relates it to the Nielsen realisation problem.

Study cobordisms of nested manifolds and their invariants.

The study explores which sets of integers can be realized as the degrees of maps between manifolds.

Study Coxeter groups over fusion rings and their geometric realisations.

Introduces quasi-holomorphic maps and their properties.

We prove Nielsen realisation for finite subgroups of the groups of untwisted outer automorphisms of RAAGs in the following sense: given any graph $Γ$, and any finite group $G\leqslant \mathrm{U}^0(A_Γ) \leqslant \mathrm{Out}^0(A_Γ)$, we find a non-positively curved cube complex with fundamental group $A_Γ$ on which $G$…

Given a smooth compact manifold with boundary, we show that the subcomplex of the deformed de Rham complex consisting of eigenspaces of small eigenvalues of the Witten Laplacian is canonically isomorphic to the Thom-Smale complex constructed by Laudenbach. Our proof is based on Bismut-Lebeau's analytic localization tec…

In a previous work we proved the uniqueness and functoriality of primary unfoldings on simple Thom-Mather spaces, which is a functor to the category of smooth manifolds. In this article we extend these results for any stratified Thom-Mather pseudomanifold with arbitary finite length, through a new kind of intermediate …

After a review of several methods designed to produce equivariant cohomology classes, we apply one introduced by Berline, Getzler and Vergne, to get a family of representatives of the universal Thom class of a vector bundle. Surprisingly, this family does not contain the representative given by Mathaï and Quillen. Howe…

The paper constructs infinitely many stably diffeomorphic but non-homotopy equivalent manifolds.

We prove conjectures of Rene Thom and Vladimir Arnold for C^2 solutions to the degenerate elliptic equation that is the level set equation for motion by mean curvature. We believe these results are the first instances of a general principle: Solutions of many degenerate equations behave as if they are analytic, even wh…

New inequality linking geodesic length and volume in complex projective plane.