m-structures determine integral homotopy type

Details m-structures determine integral homotopy type m-structures determine integral homotopy type

1998-09-26

arxiv.org/abs/math/9809151v1

Mathematics

Algebraic Topology

26 pages. LaTeX2e with XYPic, version 3.3. Uses topmacs.sty file

Scientific paper

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.

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