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András Szűcs

Publications and source records attributed to András Szűcs.

10 recordsLinked to original sources
arXiv2025

Quasi-holomorphic maps

We introduce a new notion, called quasi-holomorphic maps. These are real smooth maps equipped with a structure that imitates the singularities and singularity stratifications of holomorphic maps on the source and target manifolds, although the manifolds themselves carry no global complex structures. Some important examples of quasi-holomorphic maps are branched coverings and links of finitely determined holomorphic map germs. We show a Pontryagin--Thom type construction for a ``universal'' quasi-holomorphic map with prescribed multisingularities, from which all such maps can be induced, and a similar result for maps with prescribed singularities. Applying this, we prove that the Thom polynomials of holomorphic singularities determine the cohomology classes represented by the singular loci of not only holomorphic but quasi-holomorphic maps as well. As another application we define the cobordism groups of quasi-holomorphic maps with restricted multisingularities, whose classifying space was given by the above construction. We completely compute the free parts of these cobordism groups and in some special cases also obtain results on their torsion parts.

math.GT↗
arXiv2024

On coincidences of Morin and first order Thom--Boardman singular loci

It is well-known that the Thom polynomial in Stiefel--Whitney classes expressing the cohomology class dual to the locus of the cusp singularity for codimension-$k$ maps and that of the corank-$2$ singularity for codimension-$(k-1)$ maps coincide. The aim of the present paper is to find out whether there is any geometric explanation to this seemingly mysterious coincidence. We thank László Fehér for posing us this interesting question that we answer here in the positive, and motivated by it we search for further similar coincidences of the loci of the corank-$r$ and Morin singular points, but found such only for special classes of maps. Finally we compute the cohomology classes dual to the singularity strata for any Morin map. (This result--for holomorphic maps--was presented by Kazarian but the proof was left unpublished.)

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arXiv2019

On the classifying spaces of cobordisms of singular maps

The classifying spaces of cobordisms of singular maps have two fairly different constructions. We expose a homotopy theoretical connection between them. As a corollary we show that the classifying spaces in some cases have a simple product structure.

math.GT↗
arXiv2017

Singularities and stable homotopy groups of spheres I

We establish an interesting connection between Morin singularities and stable homotopy groups of spheres. We apply this connection to computations of cobordism groups of certain singular maps. The differentials of the spectral sequence computing these cobordism groups are given by the composition multiplication in the stable homotopy groups of spheres.

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arXiv2016

Realizing homology classes up to cobordism

It is known that neither immersions nor maps with a fixed finite set of multisingularities are enough to realize all mod 2 homology classes in manifolds. In this paper we define the notion of realizing a homology class up to cobordism; it is shown that for realization in this weaker sense immersions are sufficient, but maps with a fixed finite set of multisingularities are still insufficient.

math.AT↗
arXiv2013

Links of singularities up to regular homotopy

The abstract link L_d of the complex isolated singularity x^2 + y^2 + z^2 + v^{2d} = 0 is diffeomorphic to S^3 \times S^2. We classify the embedded links of these singularities up to regular homotopies precomposed with diffeomorphisms of S^3 \times S^2. Let us denote by i_d the inclusion of L_d in S^7. We show that for arbitrary diffeomorphisms φ_d of S^3 \times S^2 with L_d the compositions i_d \circ φ_d are image regularly homotopic for two values d_1 and d_2 if and only if d_1-d_2 is even.

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arXiv2011

On the cobordism groups of cooriented, codimension one Morin maps

Cobordism groups of cooriented fold maps of codimension 1 are computed completely. Namely their odd torsion part coincides with that of the stable homotopy group of spheres in the same dimension, while the 2-primary part is the kernel of the Kahn-Priddy map. (The Kahn-Priddy map is an epimorhism of the stable homotopy group of the infinite dimensional real projective space onto the 2-primary part of the stable homotopy group of spheres). Analogous results - modulo small primes - are obtained for cusp maps and more complicated Morin maps as well.

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arXiv2008

Cobordism of singular maps

We prove a conjecture due to M. Kazarian, connecting two classifying spaces in singularity theory. These spaces are: - Kazarian's space (generalizing Vassiliev's algebraic complex and) showing which cohomology classes are represented by singularity strata. - Author's space $X_τ$ giving homotopy representation of cobordisms of singular maps with a given list of allowed singularities \cite{R--Sz}. As a consequence we obtain the ranks of cobordism groups of singular maps with a given list of allowed singularities, and also their $p$-torsion parts for big primes $p.$ Further we give complete answer to the problem of elimination of singularities by cobordisms. Obtain very clear homotopical description of the classifying space $X_τ.$ We reveal some connection of the torsion parts of these cobordism groups to the stable homotopy groups of spheres and values of Thom polynomials.

math.GT↗
arXiv2001

Regular homotopy classes of immersions of 3-manifolds into 5-space

We give geometric formulae which enable us to detect (completely in some cases) the regular homotopy class of an immersion with trivial normal bundle of a closed oriented 3-manifold into 5-space. These are analogues of the geometric formulae for the Smale invariants due to Ekholm and the second author. As a corollary, we show that two embeddings into 5-space of a closed oriented 3-manifold with no 2-torsion in the second cohomology are regularly homotopic if and only if they have Seifert surfaces with the same signature. We also show that there exist two embeddings $F_0$ and $F_8 : T^3 \hookrightarrow {\bf R}^5$ of the 3-torus $T^3$ with the following properties: (1) $F_0 \sharp h$ is regularly homotopic to $F_8$ for some immersion $h : S^3 \looparrowright {\bf R}^5$, and (2) the immersion $h$ as above cannot be chosen from a regular homotopy class containing an embedding.

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