Infinity structures and higher products in rational homotopy theory

Abstract

Rational homotopy theory classically studies the torsion free phenomena in the homotopy category of topological spaces and continuous maps. Its success is mainly due to the existence of relatively simple algebraic models that faithfully capture this non-torsion homotopical information. Infinity structures are algebraic gadgets in which some axioms hold up to a hierarchy of coherent homotopies. We will work particularly with two of the most important of these, namely, A-infinity and L-infinity algebras. The former can be thought of as a differential graded algebra (DGA, henceforth, or CDGA if it is commutative) where the associativity law holds up to a homotopy which is determined by a 3-product whose associativity up to homotopy is again given by a 4-product, and so on. The latter can be seen as a differential graded Lie algebra (DGL, henceforth) where similarly, the Jacobi identity holds up to a homotopy which is determined by a 3-product, and so on. In this work, we use infinity structures to shed light on classical matters of rational homotopy theory, and beyond. When attempting to classify (or, more modestly, just to distinguish) rational homotopy types, one is naturally led to consider secondary operations in homotopy or cohomology. Among these, we focus on the higher Whitehead and Massey products, complementing the usual Whitehead product in homotopy andcup product in cohomology, respectively. These fundamental homotopical invariantsare at the very heart of the theory, but their manipulation can be difficult at times. Infinity structures also classify rational homotopy types and are sometimes more amenable to computations or better adapted to the problem at hand than are the secondary operations. The main achievement of this thesis is the description of the precise relationship between the higher arity operations of an infinity structure governing a given rational homotopy type and the higher products in homotopy and cohomology of it.