Radial index and Poincar\'e-Hopf index of 1-forms on semi-analytic sets

TL;DR

This paper explores the relationship between the radial index of 1-forms on semi-analytic sets and the classical Poincaré-Hopf index, extending index theory to singular sets in real space.

Contribution

It establishes a connection between the radial index on semi-analytic sets and Poincaré-Hopf indices of vector fields in R^n, broadening index theory for singular sets.

Findings

Relates radial index to Poincaré-Hopf indices for semi-analytic sets

Provides formulas connecting indices on singular and regular sets

Extends classical index theory to new classes of semi-analytic sets

Abstract

The radial index of a 1-form on a singular set is a generalization of the classical Poincar\'e-Hopf index. We consider different classes of closed semi-analytic sets in R^n that contain 0 in their singular locus and we relate the radial index of a 1-form at 0 on these sets to Poincar\'e-Hopf indices at 0 of vector fiels defined on R^n.