Higher homotopy invariants for spaces and maps

Abstract

For a pointed topological space $\mathit{X}$, we use an inductive construction of a simplicial resolution of $\mathit{X}$ by wedges of spheres to construct a “higher homotopy structure” for $\mathit{X}$ (in terms of chain complexes of spaces). This structure is then used to define a collection of higher homotopy invariants which suffice to recover $\mathit{X}$ up to weak equivalence. It can also be used to distinguish between different maps $f:\mathit{X}\to \mathit{Y}$ which induce the same morphism $f_{\ast }:{\pi }_{\ast }\mathit{X}\to {\pi }_{\ast }\mathit{Y}$.