Title: m-structures determine integral homotopy type
Author: Justin R. Smith (jsmith@mcs.drexel.edu)
Comments: 26 pages. LaTeX2e with XYPic, version 3.3. Uses XYPic fonts.
MSC-class: 55R91 (Primary) 18G30 (Secondary)
This paper proves that the functor $\mathscr{C}(*)$ 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, $\mathscr{C}(*)$ 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.