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Boldizsar Kalmar

Publications and source records attributed to Boldizsar Kalmar.

16 recordsLinked to original sources
arXiv2025

Morse functions constructed by random walks

We construct random Morse functions on surfaces by random walk and compute related distributions. We study the space of Morse functions through these random variables. We consider subspaces characterized by the surfaces with boundary obtained by cutting the closed domain surface of the Morse function at the levels of regular values. We consider Morse functions having a bounded number of critical points and one single local minimum. We find a small set of Morse functions which are close enough to any other Morse function in the sense that they share the same characterizing surfaces with boundary.

math.PR↗
arXiv2022

Concordance of decompositions given by defining sequences

We study the concordance and bordism of decompositions associated with defining sequences and we relate them to some invariants of toroidal decompositions and to the cobordism of homology manifolds. These decompositions are often wild Cantor sets and they arise as nested intersections of knotted solid tori. We show that there are at least uncountably many concordance classes of such decompositions in the 3-sphere.

math.GT↗
arXiv2021

Decomposition space theory

In these notes we give a brief introduction to decomposition theory and we summarize some classical and well-known results. The main question is that if a partitioning of a topological space (in other words a decomposition) is given, then what is the topology of the quotient space. The main result is that an upper semi-continuous decomposition yields a homeomorphic decomposition space if the decomposition is shrinkable (i.e. there exist self-homeomorphisms of the space which shrink the partitions into arbitrarily small sets in a controllable way). This is called Bing shrinkability criterion. It is applied in major 4-dimensional results: in the disk embedding theorem and in the proof of the 4-dimensional topological Poincare conjecture. It is extensively applied in constructing approximations of manifold embeddings in dimension at least 5, see for example Edwards's cell-like approximation theorem. We prove for example that every cell-like subset in a 2-dimensional manifold is cellular, that Antoine's necklace is a wild Cantor set, that in a complete metric space a usc decomposition is shrinkable if and only if the decomposition map is a near-homeomorphism and that every manifold has collared boundary.

math.GT↗
arXiv2020

Fold cobordisms and a Poincare-Hopf type theorem for the signature

We give complete geometric invariants of cobordisms of framed fold maps. These invariants consist of two types. We take the immersion of the fold singular set into the target manifold together with information about non-triviality of the normal bundle of the singular set in the source manifold. These invariants were introduced in the author's earlier works. Secondly we take the induced stable partial framing on the source manifold whose cobordisms were studied in general by Koschorke. We show that these invariants describe completely the cobordism groups of framed fold maps into R^n. Then we are looking for dependencies between these invariants and we study fold maps of 4k-dimensional manifolds into R^2. We construct special fold maps which are representatives of the fold cobordism classes and we also compute cobordism groups. We obtain a Poincare-Hopf type formula, which connects local data of the singularities of a fold map of an oriented 4k-dimensional manifold M to the signature of M. We also study the unoriented case analogously and prove a similar formula about the twisting of the normal bundle of the fold singular set.

math.GT↗
arXiv2017

Interlacement of double curves of immersed spheres

We characterize those unions of embedded disjoint circles in the 2-sphere which can be the multiple point set of a generic immersion of the 2-sphere into 3-dimensional space in terms of the interlacement of the given circles. Our result is the one higher dimensional analogue of Rosenstiehl's characterization of words being Gauss codes of self-crossing plane curves. Our proof uses a result of Lippner and we further generalize the ideas of Fraysseix and Ossona de Mendez, which leads us to directed interlacement graphs of paired trees and their local complementation.

math.GT↗
arXiv2013

Singular maps on exotic 4-manifold pairs

We show examples of pairs of smooth, compact, homeomorphic 4-manifolds, whose diffeomorphism types are distinguished by the topology of the singular sets of smooth stable maps defined on them. In this distinction we rely on results from Seiberg-Witten theory.

math.GT↗
arXiv2012

Maps on 3-manifolds given by surgery

Suppose that the 3-manifold M is given by integral surgery along a link L in S^3. In the following we construct a stable map from M to the plane, whose singular set is canonically oriented. We obtain upper bounds for the minimal numbers of crossings and non-simple singularities and of connected components of fibers of stable maps from M to the plane in terms of properties of L.

math.GT↗
arXiv2010

Relations among characteristic classes and existence of singular maps

We obtain relations among the characteristic classes of a manifold M admitting corank one maps. Our relations yield strong restrictions on the cobordism class of M and also nonexistence results for singular maps of the projective spaces. We obtain our results through blowing up a manifold along the singular set of a smooth map and perturbing the arising non-generic corank one map.

math.GT↗
arXiv2008

Fold maps and immersions from the viewpoint of cobordism

We obtain complete geometric invariants of cobordism classes of oriented simple fold maps of (n+1)-dimensional manifolds into an n-dimensional manifold N in terms of immersions with prescribed normal bundles. We compute that this cobordism group of simple fold maps is isomorphic to the direct sum of the (n-1)th stable homotopy group of spheres and the (n-1)th stable homotopy group of the infinite dimensional projective space. By using geometric invariants defined in the author's earlier works, we also describe the natural map of the simple fold cobordism group to the fold cobordism group by natural homomorphisms between cobordism groups of immersions. We also compute the ranks of the oriented (right-left) bordism groups of simple fold maps.

math.GT↗
arXiv2008

Cobordism of fold maps, stably framed manifolds and immersions

We give complete geometric invariants of cobordisms of fold maps with oriented singular set and cobordisms of even codimensional fold maps. These invariants are given in terms of cobordisms of stably framed manifolds and cobordisms of immersions with prescribed normal bundles defined by the author in his earlier works.

math.GT↗
arXiv2008

Cobordisms of fold maps of 4-manifolds into the space

We compute the oriented cobordism group of fold maps of 4-manifolds into the space with all the possible restrictions (and also with no restriction) to the singular fibers. We also give geometric invariants which describe completely the cobordism group of fold maps.

math.GT↗
arXiv2007

Cobordism invariants of fold maps

This is a survey paper of author's results on cobordism groups and semigroups of fold maps and simple fold maps. The results include: establishing a relation between fold maps and immersions through geometrical invariants of cobordism classes of fold maps and simple fold maps in terms of immersions with prescribed normal bundles, detecting stable homotopy groups of spheres as direct summands of the cobordism semigroups of fold maps, Pontryagin-Thom type construction for -1 codimensional fold maps and estimations about the cobordism classes of manifolds which have fold maps into stably parallelizable manifolds. In the last section some of these results are extended and we show that our invariants also detect stable homotopy groups of the classifying spaces $BO(k)$ as direct summands of the cobordism semigroups of fold maps.

math.GT↗
arXiv2007

Fold cobordisms and stable homotopy groups

We show that the cobordism groups of negative codimensional folds maps contain direct sums of stable homotopy groups of Thom spaces of vector bundles like the circle and the infinite dimensional projective space. We give geometrical invariants which detect these direct summands.

math.GT↗