On Hawaiian homology groups

TL;DR

This paper introduces Hawaiian homology, a new homology theory for classifying pointed topological spaces, which simplifies calculations and provides insights into Hawaiian groups of complex spaces.

Contribution

It defines Hawaiian homology, demonstrates its advantages over Hawaiian groups, and explores its applications in analyzing wild topological spaces.

Findings

Hawaiian homology is isomorphic to the abelianization of Hawaiian groups for certain spaces.

Hawaiian homology simplifies calculations compared to Hawaiian groups.

Applications to wild spaces reveal new structural information.

Abstract

In this paper, we introduce a kind of homology which we call Hawaiian homology to study and classify pointed topological spaces. The Hawaiian homology group has advantages of Hawaiian groups. Moreover, the first Hawaiian homology group is isomorphic to the abelianization of the first Hawaiian group for path-connected and locally path-connected topological spaces. Since Hawaiian homology has concrete elements and abelian structure, its calculations are more routine. Thus we use Hawaiian homology groups to compare Hawaiian groups, and then we obtain some information about Hawaiian groups of some wild topological spaces.