The Hopf type theorem for equivariant gradient local maps

TL;DR

This paper introduces a new degree-type invariant for equivariant gradient local maps under compact Lie group actions, establishing a bijection with an infinite direct sum of integers, advancing the understanding of equivariant degree theory.

Contribution

It constructs a novel invariant for equivariant gradient local maps and proves a bijection with an infinite sum of integers, extending equivariant degree theory.

Findings

The invariant provides a complete classification of equivariant gradient otopy classes.

The invariant establishes a bijection with a countably infinite direct sum of integers.

The construction applies to real finite dimensional orthogonal representations of compact Lie groups.

Abstract

We construct a degree-type otopy invariant for equivariant gradient local maps in the case of a real finite dimensional orthogonal representation of a compact Lie group. We prove that the invariant establishes a bijection between the set of equivariant gradient otopy classes and the direct sum of countably many copies of $\mathbb{Z}$.