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Description
Quid of iterated loop spaces ?
In algebraic topology, the first invariant of homotopy type one defines is the fundamental group of a space X. The group structure comes from a product that endows the loop space of X with a so called A∞ structure.
In the late 50’s, J.D. Stasheff showed that any (nice) space with such A∞ structure has the homotopy type of some loop space. This was refined and extended in 1972 by J.P. May. His recognition principle characterizes the weak homotopy type of any iterated loop space as algebras over some little cubes operad.
In a first part, after some recollections of homotopy theory and motivations, we shall do preliminary observations leading to Stasheff’s result. The second part is dedicated to the notion of an operad which is key in both the statement and the proof of May’s theorem.
