Thick Spanier groups and the first shape group

TL;DR

This paper establishes that for certain topological spaces, the kernel of the shape group homomorphism is exactly the Spanier group, providing a new perspective on the fundamental group structure of these spaces.

Contribution

It introduces a novel approach to analyze the kernel of the shape group homomorphism, proving its equivalence to the Spanier group for a broad class of spaces.

Findings

The kernel of the shape group homomorphism equals the Spanier group for locally path connected, paracompact Hausdorff spaces.

Provides a new method to explore the fundamental group via shape theory.

Clarifies the relationship between shape groups and Spanier groups in topological spaces.

Abstract

We develop a new route through which to explore $\ker\Psi_X$, the kernel of the $\pi_1$-shape group homomorphism determined by a general space $X$, and establish, for each locally path connected, paracompact Hausdorff space $X$, $\ker\Psi_X$ is precisely the Spanier group of $X$.