ABSTRACT
METHODES QUANTIQUES EN TOPOLOGIE ALGEBRIQUE
English summary : Quantum methods in Algebraic Topology
by Max Karoubi
http://www.math.jussieu.fr/~karoubi/
In this paper, we present a new version of cochains in Algebraic
Topology, starting with "quantum differential forms". This version provides
many examples of modules over the braid group, together with a control of
the non commutativity of cup-products on the cochain level. If the quantum
parameter q is equal to 1, we essentially recover the commutative
differential graded algebra of de Rham-Sullivan forms on a simplicial set.
For topological applications, we may take either q = 1 if we are dealing
with rational coefficients or q = 0 in the general case. This construction
can
be generalized easily in a sheaf context.
From this viewpoint, we extract a new structure of "neo-algebra".
This structure is detailed in section III of the English summary (section 7
in the detailed French version). This is a special case of the notion of
partial algebra introduced by I. Kriz and P. May in their book (Asterisque
Nr. 233, 1995). To a simplicial set X we can associate in a functorial way
a neo-algebra, which cohomology is canonically isomorphic to the usual one
with coefficients in k (k might be an arbitrary commutative ring). As a
differential graded algebra, this neo-algebra is related to the usual
algebra of cochains C*(X) by a (zigzag) sequence of natural
quasi-isomorphisms.
Using in an essential way some recent results of M.-A. Mandell ,
http://www.lehigh.edu/~dmd1/algtop.html , one may then show that this
neo-algebra (up to quasi-isomorphisms and under some mild finiteness and
nilpotence conditions) determines the homotopy type of X. The proof relies
on the basic fact that it may be also provided with an infty -algebra
structure which is related to the classical one on C*(X) by a sequence of
natural quasi-isomorphisms.
On a more practical level, we give a method how to compute Steenrod
operations in mod. p cohomology, as well as homotopy groups of X from the
neo-algebraic data which we are describing. More details will be published
soon.