6533b7d3fe1ef96bd1261374
RESEARCH PRODUCT
Combinatorial Models in the Topological Classification of Singularities of Mappings
J. J. Nuño-ballesterossubject
PhysicsImage (category theory)010102 general mathematicsDimension (graph theory)Boundary (topology)Stable map01 natural sciencesManifold010101 applied mathematicsCombinatoricsCone (topology)0101 mathematicsTopological conjugacyWord (group theory)description
The topological classification of finitely determined map germs \(f:(\mathbb R^n,0)\rightarrow (\mathbb R^p,0)\) is discrete (by a theorem due to R. Thom), hence we want to obtain combinatorial models which codify all the topological information of the map germ f. According to Fukuda’s work, the topology of such germs is determined by the link, which is obtained by taking the intersection of the image of f with a small enough sphere centered at the origin. If \(f^{-1}(0)=\{0\}\), then the link is a topologically stable map \(\gamma :S^{n-1}\rightarrow S^{p-1}\) (or stable if (n, p) are nice dimensions) and f is topologically equivalent to the cone of \(\gamma \). When \(f^{-1}(0)\ne \{0\}\), the situation is more complicated. The link is a topologically stable map \(\gamma :N\rightarrow S^{p-1}\), where N is a manifold with boundary of dimension \(n-1\). However, in this case, we have to consider a generalized version of the cone, so that f is again topologically equivalent to the cone of the link diagram. We analyze some particular cases in low dimensions, where the combinatorial models are provided by objects which are well known in Computational Geometry, for instance, the Gauss word or the Reeb graph.
year | journal | country | edition | language |
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2018-01-01 |