#### Volume 21, issue 5 (2021)

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Higher homotopy invariants for spaces and maps

### David Blanc, Mark W Johnson and James M Turner

Algebraic & Geometric Topology 21 (2021) 2425–2488
##### Abstract

For a pointed topological space $\mathbf{X}\phantom{\rule{-0.17em}{0ex}}$, we use an inductive construction of a simplicial resolution of $\mathbf{X}$ by wedges of spheres to construct a “higher homotopy structure” for $\mathbf{X}$ (in terms of chain complexes of spaces). This structure is then used to define a collection of higher homotopy invariants which suffice to recover $\mathbf{X}$ up to weak equivalence. It can also be used to distinguish between different maps $f:\mathbf{X}\to \mathbf{Y}$ which induce the same morphism ${f}_{\ast }:{\pi }_{\ast }\mathbf{X}\to {\pi }_{\ast }\mathbf{Y}\phantom{\rule{-0.17em}{0ex}}$.

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##### Keywords
higher homotopy operation, homotopy invariants, $\Pi$–algebra, simplicial resolution
##### Mathematical Subject Classification 2010
Primary: 55Q35
Secondary: 18G30, 55P15, 55U35