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András Csépai

Publications and source records attributed to András Csépai.

9 recordsLinked to original sources
arXiv2026

On quasi-holomorphic homotopies of immersions of 3-manifolds into 5-manifolds

The notion of a quasi-holomorphic homotopy of two immersions of a $3$-manifold into a $5$-manifold extends the notion of their regular homotopy by also allowing the homotopy to pass through instances of maps with an isolated singularity around which the path of the homotopy forms a cross-cap (complex Whitney umbrella). We describe the local form of such homotopies and explain connections with the theory of holomorphic map germs from $\mathbb{C}^2$ to $\mathbb{C}^3$. Our main result is a complete a description of how the fundamental group of the complement of the image of an immersion of a $3$-manifold into a $5$-manifold (i.e. its knot group) changes under a quasi-holomorphic homotopy. As a corollary we will see that certain quasi-holomorphic homotopies of the standard embedding of the $3$-sphere into the $5$-sphere do not change its knot group.

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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.

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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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arXiv2023

Computations on cobordism groups of projected immersions

We use a classifying space construction by Szűcs and Terpai to extend Salomonsen's exact sequence from cobordism groups of immersions to cobordism groups of hyperplane projections of immersions with prescribed singularities. We apply this to obtain results on these cobordism groups in small and large codimensional cases.

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arXiv2023

Analogues of the Atiyah--Wall exact sequences for cobordism groups of singular maps

Classical results of Rohlin, Dold, Wall and Atiyah yield two exact sequences that connect the oriented and unoriented (abstract) cobordism groups $Ω_n$ and $\mathfrak{N}_n$. In this paper we present analogous exact sequences connecting the oriented and unoriented cobordism groups of maps with prescribed singularities. This gives positive answer to a fifteen-year-old question posed by Szűcs and has interesting consequences even in the case of cobordisms of immersions.

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arXiv2020

Stable Pontryagin-Thom construction for proper maps II

In arXiv:1905.07734 we presented a construction that is an analogue of Pontryagin's for proper maps in stable dimensions. This gives a bijection between the cobordism set of framed embedded compact submanifolds in $W\times\mathbb{R}^n$ for a given manifold $W$ and a large enough number $n$, and the homotopy classes of proper maps from $W\times\mathbb{R}^n$ to $\mathbb{R}^{k+n}$. In the present paper we generalise this result in a similar way as Thom's construction generalises Pontryagin's. In other words, we present a bijection between the cobordism set of submanifolds embedded in $W\times\mathbb{R}^n$ with normal bundles induced from a given bundle $ξ\oplus\varepsilon^n$, and the homotopy classes of proper maps from $W\times\mathbb{R}^n$ to a space $U(ξ\oplus\varepsilon^n)$ that depends on the given bundle. An important difference between Thom's construction and ours is that we also consider cobordisms of non-compact manifolds after indroducing a suitable notion of cobordism relation for these.

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arXiv2020

Cobordisms of singular maps

The notion of cobordism of singular maps was introduced around 1980 by A. Szűcs and U. Koschorke independently. As an application, Szűcs used it to compute cobordism groups of immersions and embeddings in dimensions where the classical theory did not succeed. His method of studying cobordism groups of singular maps involves the investigation of classifying spaces that were constructed by him, first in a few special cases, then, with the help of the work of Rimányi, in complete generality. Since then quite a few results and (partial) computations were performed in this theory. The present thesis collects and organises these results of Szűcs, Terpai and other coauthors towards and on the computation of cobordism groups of singular maps.

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arXiv2019

Stable Pontryagin-Thom construction for proper maps

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 proper maps $W\times\mathbb{R}^n\to\mathbb{R}^{k+n}$. The second one actually implies the first one and we shall prove the second one by giving an explicit construction.

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