Quasi-commutative cochains in algebraic topology

Max Karoubi

Published 2005-09-13Version 1

We introduce a new algebraic concept of an algebra which is "almost" commutative (more precisely "quasi-commutative differential graded algebra" or ADGQ, in French). We associate to any simplicial set X an ADGQ - called D(X) - and show how we can recover the homotopy type of the topological realization of X from this algebraic structure (assuming some finiteness conditions). The theory is sufficiently general to include also ringed spaces X. The construction is by itself interesting since it uses the difference calculus (instead of the differential calculus of Sullivan's theory) and a new type of tensor product, called "reduced tensor product". Although we dont have a minimal model yet, it is relatively easy to define the cup i-products and the Steenrod operations on the category of ADGQ's. Homotopy groups can be deduced from the "iterated Hochschild homology" of D(X). The determination of the homotopy type from this algebraic structure uses in an essential way recent results of M. Mandell.