A characteristic-index inequality for closed embeddings of locally compact groups

TL;DR

This paper introduces a characteristic-index inequality for closed embeddings of locally compact groups, showing that the characteristic index of a subgroup is bounded by that of the larger group, enhancing understanding of their structural relationships.

Contribution

It establishes a new inequality relating the characteristic indices of closed subgroups and their ambient groups in locally compact connected groups.

Findings

Characteristic index of subgroups is dominated by that of the ambient group.

Provides a structural inequality for closed embeddings.

Enhances understanding of subgroup and group relationships in topology.

Abstract

The characteristic index of a locally compact connected group $G$ is the non-negative integer $d$ for which we have a homeomorphism $G\cong K\times \mathbb{R}^d$ with $K\le G$ maximal compact. We prove that the characteristic indices of closed connected subgroups are dominated by those of the ambient groups.