arXiv subjects

Masato Tanabe

Publications and source records attributed to Masato Tanabe.

5 recordsLinked to original sources

arXiv2026

Thom polynomials relative to prescribed maps around the boundary

Thom polynomials are universal cohomological obstructions to the appearance of singularities of given types in differentiable maps. As an application, various invariants of immersions have been expressed in terms of singularities of their extension maps, known as singular Seifert surfaces. To place these results in a unified framework, we aim in this paper to establish a relative version of Thom polynomial theory. Our results consist of four parts. (1) We introduce Thom polynomials relative to prescribed maps around the boundary (or a closed codimension-zero submanifold) that avoid singularities of given types. (2) We show a structure theorem for Thom polynomials relative to framable immersions. It expresses them as the sum of the term obtained by substituting Kervaire's relative characteristic classes into the absolute Thom polynomial and a universal correction term. (3) We determine correction terms in several cases, not only reinterpreting earlier works as instances of relative Thom polynomials but also recovering some of them. Most earlier formulas are summarized as the vanishing of correction terms. (4) We give suggestive evidence for the relative Thom polynomials of multi-singularity types $A_0^k$, with an application.

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arXiv2024

Unstability problem of real analytic maps

As well-known, the $C^\infty$ stability of proper $C^\infty$ maps is characterized by the infinitesimal $C^\infty$ stability. In the present paper we study the counterpart in real analytic context. In particular, we show that the infinitesimal $C^\omega$ stability does not imply $C^\omega$ stability; for instance, a Whitney umbrella $\mathbb{R}^2 \to \mathbb{R}^3$ is not $C^\omega$ stable. A main tool for the proof is a relative version of Whitney's Analytic Approximation Theorem which is shown by using H. Cartan's Theorems A and B.

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arXiv2024

Regular homotopy classes of links of simple singularities and immersions associated with their Dynkin diagrams

Our aim is to determine the regular homotopy classes of immersions related to Arnol'd's simple singularities. For every type of simple singularities, we determine the regular homotopy class of the inclusion map of the link into the 5-sphere. We further show that the inclusion map is regularly homotopic to the immersion associated with the corresponding Dynkin diagram, which was constructed by Kinjo. We prove these by computing the complete invariants of the immersions given by Wu and Saeki--Sz\H{u}cs--Takase. As an application, we also determine the Smale invariants of Kinjo's immersions.

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arXiv2023

Real analytic extension of functions on normal crossings

We consider a compact $C^\omega$ manifold $X$ and finitely many regular $C^\omega$ submanifolds $Y_1, \dots, Y_q$ of $X$, which are closed subsets in $X$, such that the union of $Y_j$'s has only normal crossings. We show that every continuous function on the union which is of class $C^\omega$ on each $Y_j$ can be extended to a $C^\omega$ function on $X$. A crucial feature of our proof is to employ basic tools of real analytic geometry -- Cartan Theorems A and B.

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arXiv2022

Canonical stratification of definable Lie groupoids

Our aim is to precisely present a tame topology counterpart to canonical stratification of a Lie groupoid. We consider a definable Lie groupoid in semialgebraic, subanalytic, o-minimal over $\mathbb{R}$, or more generally, Shiota's $\mathfrak{X}$-category. We show that there exists a canonical Whitney stratification of the Lie groupoid into definable strata which are invariant under the groupoid action. This is a generalization and refinement of results on real algebraic group action which J. N. Mather and V. A. Vassiliev independently stated with sketchy proofs. A crucial change to their proofs is to use Shiota's isotopy lemma and approximation theorem in the context of tame topology.

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