Higher homotopy invariants for spaces and maps

TL;DR

This paper introduces a new higher homotopy structure for pointed spaces using simplicial resolutions, enabling the definition of invariants that distinguish spaces and maps beyond traditional homotopy groups.

Contribution

It develops an inductive simplicial resolution method to construct higher homotopy invariants that fully classify spaces up to weak equivalence and differentiate maps with identical induced homotopy group morphisms.

Findings

Higher homotopy invariants can recover spaces up to weak equivalence.

The invariants distinguish between maps with the same induced homotopy group morphism.

A new inductive construction of simplicial resolutions of spaces.

Abstract

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