Sequential $n$-connectedness and infinite factorization in higher homotopy groups

TL;DR

This paper introduces a new concept called sequential n-connectedness to analyze higher homotopy groups of complex spaces, leading to novel computations and characterizations, including for Hawaiian earrings.

Contribution

It develops new methods based on sequential n-connectedness and Whitney Covering Lemma to characterize higher homotopy groups of Peano continua and infinite CW-complex attachments.

Findings

Established an injection from th homotopy groups to an infinite product of direct sums.

Characterized the image of the homomorphism using generalized covering space theory.

Provided new computations for th homotopy groups of specific complex spaces.

Abstract

A space $X$ is "sequentially $n$-connected" at $x\in X$ if for every $0\leq k\leq n$ and sequence of maps $f_1,f_2,f_3,\dots:S^k\to X$ that converges toward a point $x\in X$, the maps $f_m$ contract by a sequence of null-homotopies that converge toward $x$. We use this property, in conjunction with the Whitney Covering Lemma, as a foundation for developing new methods for characterizing higher homotopy groups of finite dimensional Peano continua. Among many new computations, a culminating result of this paper is: if $Y$ is a space obtained by attaching an infinite shrinking sequence $A_1,A_2,A_3,\dots$ of $(n-1)$-connected CW-complexes to a one-dimensional Peano continuum $X$ along a sequence of points in $X$, then there is an injection $\Phi:\pi_n(Y)\to \prod_{j=1}^{\infty}\bigoplus_{\pi_1(X)}\pi_n(A_j)$ that is canonical after a certain choice of paths in $X$ is made. Moreover, we characterize the image of $\Phi$ using generalized covering space theory. As a case of particular interest, this provides a characterization of $\pi_n(\mathbb{H}_1\vee \mathbb{H}_n)$ where $\mathbb{H}_n$ denotes the $n$-dimensional Hawaiian earring.