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

The Reeb Graph of a Map Germ from $$\mathbb {R}^3$$ R 3 to $$\mathbb {R}^2$$ R 2 with Non... We consider the topological classification of finitely determined map germs $$[f]:(\mathbb {R}^3,0)\rightarrow (\mathbb {R}^2,0)$$ [ f ] : ( R 3 , 0 ) → ( R 2 , 0 ) with $$f^{-1}(0)\ne \{0\}$$ f - 1 ( 0 ) ≠ { 0 } . The case $$f^{-1}(0) = \{0\}$$ 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 $$S^1_\delta $$ S δ 1 centered at the origin. The link is a stable map $$\gamma _f:N\rightarrow S^1$$ γ f : N → S 1 , where N is diffeomorphic to a sphere $$S^2$$ 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 $$\Sigma ^{2,1}$$ Σ 2 , 1 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Brazilian Mathematical Society, New Series Springer Journals

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

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References (23)

Publisher
Springer Journals
Copyright
Copyright © 2017 by Sociedade Brasileira de Matemática
Subject
Mathematics; Mathematics, general; Theoretical, Mathematical and Computational Physics
ISSN
1678-7544
eISSN
1678-7714
DOI
10.1007/s00574-017-0058-4
Publisher site
See Article on Publisher Site

Abstract

We consider the topological classification of finitely determined map germs $$[f]:(\mathbb {R}^3,0)\rightarrow (\mathbb {R}^2,0)$$ [ f ] : ( R 3 , 0 ) → ( R 2 , 0 ) with $$f^{-1}(0)\ne \{0\}$$ f - 1 ( 0 ) ≠ { 0 } . The case $$f^{-1}(0) = \{0\}$$ 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 $$S^1_\delta $$ S δ 1 centered at the origin. The link is a stable map $$\gamma _f:N\rightarrow S^1$$ γ f : N → S 1 , where N is diffeomorphic to a sphere $$S^2$$ 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 $$\Sigma ^{2,1}$$ Σ 2 , 1 .

Journal

Bulletin of the Brazilian Mathematical Society, New SeriesSpringer Journals

Published: Nov 16, 2017

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