arXiv research

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.

Relative Thom polynomials for maps around boundaries established.

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

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…

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…

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

Enhances Pontryagin-Thom theorem for manifold maps.

In this paper we extend Y.Eliashberg's $h$-principle to arbitrary generic smooth maps of smooth manifolds. Namely, we prove a necessary and sufficient condition for a continuous map of smooth manifolds of the same dimension to be homotopic to a generic map with a prescribed Thom-Boardman singularity $Σ^I$ at each point…

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…

In this paper, we demonstrate a relation among Seiberg-Witten invariants which arises from embedded surfaces in four-manifolds whose self-intersection number is negative. These relations, together with Taubes' basic theorems on the Seiberg-Witten invariants of symplectic manifolds, are then used to prove the symplectic…

New approach simplifies topological T-duality for torus bundles.

Generalizes Pontryagin's construction for proper maps in stable dimensions.

Introduces quasi-holomorphic maps and their properties.

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.

We introduce an equivariant Pontrjagin-Thom construction which identifies equivariant cohomotopy classes with certain fixed point bordism classes. This provides a concrete geometric model for equivariant cohomotopy which works for any compact Lie group G. In the special case when G is finite or a torus, we show that ou…

We will present proofs for two conjectures stated in arXiv:1808.08073. The first one is that for an arbitrary manifold $W$, the homotopy classes of proper maps $W\times\mathbb{R}^n\to\mathbb{R}^{k+n}$ stabilise as $n\to\infty$, and the second one is that in a stable range there is a Pontryagin--Thom type bijection for …

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…

Let $N$ and $P$ be smooth closed manifolds of dimensions $n$ and $p$ respectively. Given a Thom-Boardman symbol $I$, a smooth map $f:N\to P$ is called an $Ω^{I}$-regular map if and only if the Thom-Boardman symbol of each singular point of $f$ is not greater than $I$ in the lexicographic order. We will represent the gr…

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…

We prove a Thom isomorphism theorem for differential forms in the setting of transverse Lie algebra actions on foliated manifolds and foliated vector bundles.

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…

We use the generalized Pontryagin-Thom construction to analyze the effect of attaching a bypass on the homotopy class of the contact structure. In particular, given a 3-dimensional contact manifold with convex boundary, we show that the bypass triangle attachment changes the homotopy class of the contact structure rela…

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…

Proves Arnold-Thom conjecture for surfaces' arrival times.

We present a cocycle model for elliptic cohomology with complex coefficients in which methods from 2-dimensional quantum field theory can be used to rigorously construct cocycles. For example, quantizing a theory of vector bundle-valued fermions yields a cocycle representative of the elliptic Thom class. This construct…

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.

This paper studies torsion obstructions to complex sections on manifolds.

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.

In this paper we classify the homotopy classes of proper maps $E\rightarrow \mathbb R^k$, where $E$ is a vector bundle over a compact Hausdorff space. As a corollary we compute the homotopy classes of proper maps $\mathbb R^n\rightarrow \mathbb R^k$. We find a stability range of such maps. We conclude with some remarks…

We prove a generalization of Thom's transversality theorem. It gives conditions under which the jet map $f_*|_Y:Y\subseteq J^r(D,M)\ra J^r(D,N)$ is generically (for $f:M\ra N$) transverse to a submanifold $Z\subseteq J^r(D,N)$. We apply this to study transversality properties of a restriction of a fixed map $g:M\ra P$ …

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

Using certain Thom spectra appearing in the study of cobordism categories, we show that the odd half of the Miller-Morita-Mumford classes on the mappping class group of a surface with negative Euler characteristic vanish in integral cohomology when restricted to the handlebody subgroup. This is a special case of a more…

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

Families of smooth closed oriented 4-manifolds with a complex spin structure are studied by means of a family version of the Bauer--Furuta invariants in the context of parametrised stable homotopy theory, leading to a definition of characteristic cohomotopy classes on Thom spectra associated to the classifying spaces o…

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 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 …

The main result of this paper is a sufficient condition in order to have a compact Thom-Mather stratified pseudomanifold endowed with a $\hat{c}$-iterated edge metric on its regular part $q$-parabolic. Moreover, besides stratified pseudomanifolds, the $q$-parabolicity of other classes of singular spaces, such as compac…

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.

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

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

Study cobordisms of nested manifolds and their invariants.

This note announces a general construction of characteristic currents for singular connections on a vector bundle. It develops, in particular, a Chern-Weil-Simons theory for smooth bundle maps $α: E \rightarrow F$ which, for smooth connections on $E$ and $F$, establishes formulas of the type $$ φ\ = \ \text{\rm Res}_φΣ…