Quantum Methods in Algebraic Topology

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 [1][12]. For topological applications, we may take either q = 1 if we are dealing with rational coefficients or q = 0 in the general case. In both cases, the quantum formulas are simpler (if q = 0 for instance, the quantum exponential e q (x) is just the function 1/1-x).