On subgroups of first homology

TL;DR

This paper explores new theorems about the first homology of various topological spaces, including dichotomies, a strong abelianization concept, and properties of the Hawaiian earring's fundamental group.

Contribution

It introduces new results on first homology, including dichotomies, a strong abelianization notion, and analysis of Borel subgroups in the Hawaiian earring's fundamental group.

Findings

Dichotomies for first homology of Peano continua

A new notion of strong abelianization for path connected metric spaces

Hawaiian earring's fundamental group contains Borel subgroups of various types

Abstract

We prove several new theorems regarding first homology. Some dichotomies for first homology of Peano continua are presented, as well as a notion of strong abelianization for arbitrary path connected metric spaces. We also show that the fundamental group of the Hawaiian earring has Borel subgroups of almost all multiplicative and additive types.