Sequential n-connectedness and infinite deformations of n-loops

Abstract

A space X is “sequentially n-connected” at \(x\in X\) if for every \(0\leqslant k\leqslant n\) and sequence of k-loops \(f_1,f_2,f_3,\ldots :S^k\rightarrow X\) that converges toward the point x, the maps \(f_m\) contract by a sequence of null-homotopies that converge toward x. Unlike standard local contractibility conditions, the sequential n-connectedness property is closed under forming infinite products and infinite shrinking wedges. We use this property, in conjunction with the Whitney Covering Lemma, to construct homotopies that simultaneously perform infinite deformations of n-loops and, ultimately, allow us to continuously deform arbitrary n-loops into maps with simpler forms. As a direct application, we extend the computation of the n-th homotopy group of a shrinking wedge of certain \((n-1)\)-connected spaces due to K. Eda and K. Kawamura.