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

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.