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Tamás Terpai

Publications and source records attributed to Tamás Terpai.

12 recordsLinked to original sources
arXiv2026

The boundary of the Milnor fibre and a linking invariant of finitely determined germs

The image of a finitely determined holomorphic germ $Φ$ from $\mathbb{C}^2$ to $\mathbb{C}^3$ defines a hypersurface singularity $(X,0)$, which is in general non-isolated. We show that the diffeomorphism type of the boundary of the Milnor fibre $\partial F$ of $X$ is a topological invariant of the germ $Φ$. We establish a correspondence between the gluing coefficients (so-called vertical indices) used in the construction of $\partial F$ and a linking invariant $L$ of the associated sphere immersion introduced by T. Ekholm and A. Szűcs. For this we provide a direct proof of the equivalence of the different definitions of $L$. Since $L$ can be expressed in terms of the cross cap number $C(Φ)$ and the triple point number $T(Φ)$ of a stable deformation of $Φ$, we obtain a relation between these invariants and the vertical indices. This is illustrated on several examples.

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 (This result--for holomorphic maps--was presented by Kazarian but the proof was left unpublished.)

math.GT↗
arXiv2023

Decompositions of the positive real numbers into disjoint sets closed under addition and multiplication

The main purpose of this paper is to prove that the positive real numbers can be decomposed into finitely many disjoint pieces which are also closed under addition and multiplication. As a byproduct of the argument we determine all the possible decompositions of the transcendental extension of the rational field of rank one into two pieces. Further, we prove that the positive elements of a real algebraic extensions of the rational numbers are indecomposable into two pieces.

math.NT↗
arXiv2020

Frozen percolation on the binary tree is nonendogenous

In frozen percolation, i.i.d. uniformly distributed activation times are assigned to the edges of a graph. At its assigned time, an edge opens provided neither of its endvertices is part of an infinite open cluster; in the opposite case, it freezes. Aldous (2000) showed that such a process can be constructed on the infinite 3-regular tree and asked whether the event that a given edge freezes is a measurable function of the activation times assigned to all edges. We give a negative answer to this question, or, using an equivalent formulation and terminology introduced by Aldous and Bandyopadhyay (2005), we show that the recursive tree process associated with frozen percolation on the oriented binary tree is nonendogenous. An essential role in our proofs is played by a frozen percolation process on a continuous-time binary Galton Watson tree that has nice scale invariant properties.

math.PR↗
arXiv2019

Theorems of Carathéodory, Helly, and Tverberg without dimension

We prove a no-dimensional version of Carathédory's theorem: given an $n$-element set $P\subset \Re^d$, a point $a \in \conv P$, and an integer $r\le d$, $r \le n$, there is a subset $Q\subset P$ of $r$ elements such that the distance between $a$ and $\conv Q$ is less than $\diam P/\sqrt {2r}$. A general no-dimension Helly type result is also proved with colourful and fractional consequences. Similar versions of Tverberg's theorem and some of their extensions are also established.

math.MG↗
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.

math.GT↗
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↗