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.
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48 results for “Thom transversality”
Smooth maps in o-minimal structures are mostly transverse.
problem Transversality of smooth definable maps in o-minimal structures.
method Definable smooth version of Thom transversality theorem, proving nowhere density of non-transverse maps, and a definable version of Trotman's theorem.
result Non-transverse maps are nowhere dense in the definable smooth topology.
Proves a Thom isomorphism for foliated differential forms.
problem Transverse Lie algebra actions on foliated manifolds and vector bundles.
method Thom isomorphism theorem for differential forms in foliated settings.
result Established a new Thom isomorphism theorem.
In his 1979 paper Trotman proves, using the techniques of the Thom transversality theorem, that under some conditions on the dimensions of the manifolds under consideration, openness of the set of maps transverse to a stratification in the strong (Whitney) topology implies that the stratification is -regular. Here…
Establishes jet transversality for regular maps from flexible manifolds.
problem Transversality for regular maps in algebraic geometry.
method Algebraic version of Forstnerič's theorem for holomorphic maps.
result Genericity theorems for regular maps of maximal ranks.
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 . We apply this to study transversality properties of a restriction of a fixed map $g:M\ra P$ …
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…
Paper constructs Thom-Smale complex using instantons from Morse functions.
problem Constructing Thom-Smale complex for Morse functions.
method Analytic instanton construction using eigenspaces of mapping cone Laplacian.
result Instanton complex is cochain isomorphic to Thom-Smale complex.
Part 2 discusses compatibility between analytic and topological Gysin maps.
problem Compatibility between analytic and topological Gysin maps.
method Detailed discussion using Thom's transversality theorem and Jakob's K-homology theory.
result Equivalence of analytic K-homology theory with the former Gysin maps.
Study of Gaussian random fields on manifolds, focusing on their differential topology.
problem Understanding the differential topology of Gaussian random fields on manifolds.
method Systematic study using Gaussian measures and weak Whitney topology, focusing on convergence in law and transversality.
result The convergence in law of Gaussian random fields is related to the convergence of their covariance structures, with important technical tools like the Thom transversality theorem.
The paper computes a tau-invariant for holomorphic curves in Stein domains and links.
problem Computing tau-invariant for holomorphic curves in Stein domains.
method Using pseudo-holomorphic curves and Stein fillable contact structures.
result New proof of Thom conjecture and topological obstructions for link types.
The paper explores -regularity for real analytic maps and its relation to Milnor fibrations.
problem Understanding the conditions for -regularity in analytic map germs.
method Analyzes Thom regular stratifications and Milnor condition (b) for germs of analytic maps.
result Thom regular stratifications and Milnor condition (b) are crucial for -regularity and open book structures.
This paper shows that an arbitrary generic submanifold in a complex manifold can be deformed into a 1-parameter family of generic submanifolds satisfying strong nondegeneracy conditions. The proofs use a careful analysis of the jet spaces of embeddings satisfying certain nondegeneracy properties, and also make use of t…
Novikov initiated the study of the algebraic properties of quadratic forms over polynomial extensions by a far-reaching analogue of the Pontrjagin-Thom transversality construction of a Seifert surface of a knot and the infinite cyclic cover of the knot exterior. In this paper the analogy is applied to explain the relat…
Geometric regularisation improves statistical models by avoiding degeneracy loci.
problem Non-identifiability, singular information, and moment indeterminacy in statistical models.
method Develops the geometric regularisation of distribution-kernel pairs using Whitney, Thom, and Mather theorems.
result Finite-dimensional weak transversality theorem for generic kernels, avoiding degeneracy strata of high codimension.
The paper establishes criteria for symplectic surfaces in 4-manifolds.
problem Finding symplectic surfaces in 4-manifolds with bounded genus.
method Criterion for minimization of genus among surfaces with given homology and boundary.
result Symplectic surfaces minimize slice genus for their boundary in contact 3-manifolds.
Study on Morse homology for reflection actions on manifolds.
problem Finding metrics making equivariant Morse-Smale pairs stable.
method Counting broken trajectories to define equivariant Thom-Smale-Witten complexes.
result Stable Morse-Smale condition is generic for metrics.
The paper explores density of stable mappings and their properties.
problem Density of stable mappings in different dimensions.
method Infinitesimal and algebraic methods to prove density of proper stable and topologically stable mappings.
result Density of topologically stable mappings holds for any pair (n,p), and for proper stable mappings if (n,p) is in nice dimensions.
For some geometries including symplectic and contact structures on an n-dimensional manifold, we introduce a two-step approach to Gromov's h-principle. From formal geometric data, the first step builds a transversely geometric Haefliger structure of codimension n. This step works on all manifolds, even closed. The seco…
C^0-topology in Morse theorymath.DG
Let be a Morse function on a closed manifold , and be a Riemannian gradient of satisfying the transversality condition. The classical construction (due to Morse, Smale, Thom, Witten), based on the counting of flow lines joining critical points of the function associates to these data the Morse comple…
Enhances Pontryagin-Thom theorem for manifold maps.
problem Identifying map spaces with moduli spaces of submanifolds.
method Space-level enhancement of Pontryagin-Thom theorem.
result Maps from manifolds to Thom spaces identified with moduli spaces of submanifolds.
Relative Thom polynomials for maps around boundaries established.
problem Understanding singularities in maps around boundaries.
method Introducing and analyzing Thom polynomials relative to prescribed maps around boundaries, establishing structure theorems and correction terms.
result Unified framework for invariants of immersions and singularities of their extensions.
Develops a new geometric cobordism theory using smooth Thom stacks.
problem Creating a versatile geometric cobordism theory for various geometric data.
method Introducing smooth Thom stacks and identifying a smooth representative of the Thom functor.
result The new theory generalizes parametrized cobordism and includes families of geometric data.
Proves a Thom Isotopy Theorem for nonproper semialgebraic maps.
problem Nonproper semialgebraic maps without properness.
method Proves a Thom Isotopy Theorem for nonproper semialgebraic maps.
result Validates the Thom Isotopy Theorem for nonproper semialgebraic maps.
Mapping cone Thom formsmath.DG
Researchers describe a new Thom form for mapping cones.
problem Developing a new Thom form for mapping cones.
method Using the mapping cone covariant derivative and Berezin integral, they explicitly write down the Thom form.
result The Thom form is closed with respect to the mapping cone differentiation, integrates to 1 along the fiber, and satisfies the transgression formula.
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.
Overview of algebraic geometry for almost complex manifolds.
problem Developing algebraic geometry for almost complex manifolds without genericity.
method Reviewing results based on pseudoholomorphic maps and intersection theory.
result Introduction of birational morphism between almost complex manifolds.
Constructs a Dirac-Schrödinger operator on Thom-Mather stratified spaces.
problem Computing an index for orbit spaces of semi-free -actions.
method Pushing down an -invariant first order transversally elliptic operator to the quotient space.
result The constructed operator's index coincides with Lott's signature.
Proves Arnold-Thom conjectures for motion by mean curvature.
problem Degenerate elliptic equations and motion by mean curvature.
method Analytic behavior of solutions for C^2 solutions.
result First instances of a general principle in degenerate equations.
We show some generic (robust) properties of smooth surfaces immersed in the real 3-space (Euclidean, affine or projective), in the neighbourhood of a {\em godron} (term due to R.Thom): an isolated parabolic point at which the (unique) asymptotic direction is tangent to the parabolic curve. With the help of these proper…
Analytic realization of Thom-Smale complex for G-manifolds.
problem Realizing Thom-Smale complex for G-manifolds with Lie group action.
method Using G-invariant Witten instanton complex associated with a Morse-Bott function.
result Generalized Thom-Smale complex for G-manifolds including horizontal direction influence.
Paper proves stability of solutions for specific hyperbolic systems.
problem Stability of solutions to hyperbolic conservation laws.
method Proof of generic structural stability using differential topology.
result Riemann solutions are preserved under perturbations of flux, left, and right states.
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…
On the Thom conjecture in math.GT
New findings contradict the Thom conjecture for high degree hypersurfaces in .
problem Finding the simplest smooth simply connected 4-manifold in homologous to a degree hypersurface .
method Comparing values of manifolds in the same homology class as .
result For all , there exists a manifold with .
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…
Extends Y.Eliashberg's h-principle to generic maps with prescribed Thom-Boardman singularities.
problem Homotoping maps with specific singularities.
method Proves a condition for continuous maps to be homotopic to generic maps with prescribed Thom-Boardman singularities.
result Necessary and sufficient condition for homotopy in dimension 3.
A self-transverse immersion of a smooth manifold M^{k+2} in R^{2k+2} has a double point self-intersection set which is the image of an immersion of a smooth surface, the double point self-intersection surface. We prove that this surface may have odd Euler characteristic if and only if k is congruent to 1 modulo 4 or k+…
The study calculates the Smith-Thom deficiency of Hilbert squares and provides conditions for maximality.
problem Calculating the Smith-Thom deficiency of Hilbert squares and conditions for maximality.
method Using Mayer-Vietoris mapping and rank calculations.
result Established necessary and sufficient conditions for maximality of Hilbert squares in projective complete intersections.
Combining classical and modern topology, shows maps between certain manifolds must have degree zero.
problem Degree zero maps between certain manifolds
method Combining Thom's work on the Steenrod problem with simplicial volume, using Thom spaces and Steenrod powers
result Every map between certain manifolds must have degree zero
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 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 singularity.
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.
problem Existence of limit tangents for gradient flow lines of surfaces.
method Gradient flow lines of mean curvature flows with neck or cylindrical singularities.
result Proves Arnold's conjecture for all mean convex mean curvature flows of surfaces.
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.
problem Studying weak homotopy equivalences and decompositions of vector bundles.
method Morse theory on path spaces, deformation theory, Clifford representations, Bott-Thom isomorphism.
result Stable decompositions of vector bundles over sphere bundles derived from Clifford representations.
The study tackles realisability of twisted homology classes and introduces new techniques in parametrised homotopy theory.
problem When a twisted homology class is realised by a submanifold.
method Introducing cobordism classes twisted by a coefficient system, defining a twisted Thom space, and constructing a parametrised Postnikov tower.
result A twisted homology class is realisable if and only if its Poincaré dual is the image of the twisted Thom class under a parametrised map.
Paper confirms Thom's conjecture for nonlinear evolutions on manifolds.
problem Thom's gradient conjecture for nonlinear evolution equations.
method Extending and settling the conjecture in infinite dimensional problems using Łojasiewicz, L. Simon, and Kurdyka-Mostowski-Parusinski's foundational works.
result Uniqueness of the limiting direction and characterization of convergence rates for both classical and infinite dimensional settings.
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.