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 $$\Theta :{\pi }_n(X)\to \prod _{j\in \mathbb{N}}\underset{{\pi }_1(X)/{\pi }_1(X_j)}{⨁}{\pi }_n(X_j),$$ characterize its image, and prove that $\Theta$ is injective whenever each universal cover ${\tilde{X}}_j$ 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.

62 pages, 8 figures