Invariants of Stable Maps from the 3-Sphere to the Euclidean 3-Space

Abstract

In the present work, we study the decompositions of codimensionone transitions that alter the singular set the of stable maps of S 3 into R 3 , the topological behaviour of the singular set and the singularities in the branch set that involves cuspidal curves and swallowtails that alter the singular set. We also analyse the effects of these decompositions on the global invariants with prescribed branch sets.2010 Mathematics Subject Classification. 57R45, 58K15, 58K65.

A new sum of graphs and caterpillar trees

“…They found this formula in two ways: first, derived as a special case of an application of Pólya's enumeration Theorem which counts graphs with integer-weighted points; secondly, by an appropriate labelling of the lines of the caterpillar. In this paper, we gave a new proof of this result using a new sum of graphs, introduced in [3,2] and studied with great attention in [4].…”

Grafos pesados y aplicaciones estables de 3-variedades en R3

“…Los grafos con pesos son una herramienta para el estudio de aplicaciones entre variedades usada por varios matemáticos [3], [4], [9], [11], [6]. En [9] introdujeron grafos con pesos en vértices y aristas, asociándolos a las aplicaciones estables de 3-variedades cerradas y orientadas en R 3 , como invariantes topológicos globales.…”

“…Cabe indicar que en [11], [5] y [6] los autores estudiaron y presentaron otras técnicas para construir aplicaciones estables, usando las transiciones de codimensión 1, presentadas en [7] y [2].…”