Topological $${\mathcal{K}}$$ -classification of finitely determined map germs

Costa, João Carlos Ferreira; Nuño–Ballesteros, J. J.

doi:10.1007/s10711-012-9789-y

Cited by 5 publications

(5 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the last part we give a complete description of those map germs with Boardman symbol Σ 2,1 and provide a complete topological classification of this type of map germ up to multiplicity 6 (Theorem 4.13). This partly answers problems (3) and (4).…”

Section: Introductionmentioning

confidence: 53%

“…However, in this case, f is not topologically equivalent to the cone of the link in the classical sense. Instead of this, we have to consider a generalized version of the cone that turns out to be topologically equivalent to f (see [3] for details). The topological classification of this class of map germs will be considered in a forthcoming paper [2].…”

Section: Finitely Determined If and Only If There Is A Representativementioning

confidence: 99%

“…In that case the link is a stable map γ : M → S 1 , where M is now a compact surface with boundary and genus zero. However, we need a generalized version of the cone to describe the topology of f (see [3]). The topological classification of map germs with non-isolated zeros will be considered in a forthcoming paper [2].…”

Section: Introductionmentioning

confidence: 99%

“…The topological classification of map germs with non-isolated zeros will be considered in a forthcoming paper [2]. Some recent papers treat the topological classification of finitely determined map germs f : (R n , 0) → (R p , 0) by looking at the topological type of the link (see, for instance, [3,11,[14][15][16]). However, as far as we know, this is the first time that it is considered for the n > p case.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Reeb Graph of a Map Germ from ℝ³ to ℝ² with Isolated Zeros

Batista¹,

Costa²,

Nuño–Ballesteros³

2016

Proceedings of the Edinburgh Mathematical Society

Self Cite

View full text Add to dashboard Cite

We consider finitely determined map germs f : (ℝ3, 0) → (ℝ2, 0) with f–1(0) = {0} and we look at the classification of this kind of germ with respect to topological equivalence. By Fukuda's cone structure theorem, the topological type of f can be determined by the topological type of its associated link, which is a stable map from S2 to S1. We define a generalized version of the Reeb graph for stable maps γ : S2→ S1, which turns out to be a complete topological invariant. If f has corank 1, then f can be seen as a stabilization of a function h0: (ℝ2, 0) → (ℝ, 0), and we show that the Reeb graph is the sum of the partial trees of the positive and negative stabilizations of h0. Finally, we apply this to give a complete topological description of all map germs with Boardman symbol Σ2, 1.

show abstract

Section: Introductionmentioning

confidence: 53%

Section: Finitely Determined If and Only If There Is A Representativementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Reeb Graph of a Map Germ from ℝ³ to ℝ² with Isolated Zeros

Batista¹,

Costa²,

Nuño–Ballesteros³

2016

Proceedings of the Edinburgh Mathematical Society

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is important to cite that recently many papers treat that the problem of topological classification of finitely determined map germs [ f ] : (R n , 0) → (R p , 0) by looking at the topological type of the link (cf. Costa and Nuño-Ballesteros 2013;Moya-Pérez and Nuño-Ballesteros 2010, 2014, 2015. However, as far as we know, this paper is the first one which considers the case of non isolated zeros.…”

Section: Introductionmentioning

confidence: 99%

The Reeb Graph of a Map Germ from $$\mathbb {R}^3$$ R 3 to $$\mathbb {R}^2$$ R

Batista

Costa

Nuño–Ballesteros

2017

Bull Braz Math Soc, New Series

Self Cite

View full text Add to dashboard Cite

We consider the topological classification of finitely determined map germs [ f ] : (R 3 , 0) → (R 2 , 0) with f −1 (0) = {0}. The case f −1 (0) = {0} was treated in another recent paper by the authors. The main tool used to describe the topological type is the link of [ f ], which is obtained by taking the intersection of its image with a small sphere S 1 δ centered at the origin. The link is a stable map γ f : N → S 1 , where N is diffeomorphic to a sphere S 2 minus 2L disks. We define a complete topological invariant called the generalized Reeb graph. Finally, we apply our results to give a topological description of some map germs with Boardman symbol 2,1 .

show abstract