Homotopy groups of shrinking wedges of non-simply connected CW-complexes

TL;DR

This paper investigates the homotopy groups of shrinking wedges of non-simply connected CW-complexes, using advanced topological theories to characterize and understand their algebraic invariants.

Contribution

It introduces a canonical homomorphism linking the homotopy groups of the wedge to product and sum of individual groups, with conditions for injectivity and applications to specific spaces.

Findings

Characterization of the homotopy groups of shrinking wedges.

Conditions under which the homomorphism is injective.

Shrinking wedge of aspherical complexes is aspherical.

Abstract

In this paper, we study the homotopy groups of a shrinking wedge $X$ of a sequence $\{X_j\}$ of non-simply connected CW-complexes. Using a combination of generalized covering space theory and shape theory, we construct a canonical homomorphism $$\Theta:\pi_n(X)\to\prod_{j\in\mathbb{N}}\bigoplus_{\pi_1(X)/\pi_1(X_j)}\pi_n(X_j),$$ characterize its image, and prove that $\Theta$ is injective whenever each universal cover $\widetilde{X}_j$ is $(n-1)$-connected. These results (1) provide a characterization of the $n$-th homotopy group of the shrinking wedge of copies of $\mathbb{RP}^n$, (2) provide a characterization of $\pi_2$ of an arbitrary shrinking wedge, and (3) imply that a shrinking wedge of aspherical CW-complexes is aspherical.