1998/09/26 by Justin R Smith, Justin R. Smith, Smith, Justin R. · 1 citation
Mathematics · #55R91 (Primary) 18G30 (Secondary) #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.AT #msc:18G30 #msc:55R91
paper · pdf · doi:10.48550/arxiv.math/9809151
26 pages. LaTeX2e with XYPic, version 3.3. Uses topmacs.sty file
arxiv created 1998/09/26 · openalex publication_date 1998/09/26 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper proves that the functor C(*) that sends pointed, simply-connected CW-complexes to their chain-complexes equipped with diagonals and iterated higher diagonals, determines their integral homotopy type --- even inducing an equivalence of categories between the category of CW-complexes up to homotopy equivalence and a certain category of chain-complexes equipped with higher diagonals. Consequently, C(*) is an algebraic model for integral homotopy types similar to Quillen's model of rational homotopy types. For finite CW complexes, our model is finitely generated. Our result implies that the geometrically induced diagonal map with all ``higher diagonal'' maps (like those used to define Steenrod operations) collectively determine integral homotopy type.