The geometrical information encoded by the Euler obstruction of a map

TL;DR

This paper explores how the Euler obstruction of a complex map encodes topological information about singular spaces and relates it to invariants like the Brasselet number and the number of cusps in generic perturbations.

Contribution

It establishes new relationships between the Euler obstruction of a map, local Euler obstructions, and Chern numbers for higher-dimensional singular spaces.

Findings

Relation between Euler obstruction and Brasselet number.

Connection of Chern numbers with cusps in perturbations.

Extension of topological invariants to higher-dimensional singular spaces.

Abstract

In this work we investigate the topological information captured by the Euler obstruction of a map, $f:(X,0)\to (\mathbb{C}^{2},0)$, with $(X,0)$ a germ of a complex $d$-equidimensional singular space, with $d > 2$, and its relation with the local Euler obstruction of the coordinate functions and, consequently, with the Brasselet number. Nevertheless, under some technical conditions on the departure variety we relate the Chern number of a special collection related to the map-germ $f$ at the origin with the number of cusps of a generic perturbation of $f$ on a stabilization of $(X,f)$.