Summary

Sometimes a topological space X is endowed with a continuous product (think for example of a Lie group): in this case one can say a lot about the topological invariants of X. For example, its homology becomes a graded ring and its fundamental group is abelian. However, many interesting spaces X are endowed with several natural operations with multiple inputs and one output, and there is no canonical way to choose one operation. An operad helps us keeping track of all possible operations that we want to consider: this notion naturally arose in the 1970s from the study of iterated loop spaces by Boardman, Vogt and May.

In the context of linear algebra, operads classify various types of (multilinear) operations that one can put on a module, and the relations that we want to impose among these operations (think of an algebra structure on a module, with associativity and commutativity as properties).

This seminar aims to provide an introduction to operad theory, with a focus on its applications to algebraic topology. We will in particular see how operads help us understanding the homology of spaces and sequences of spaces, and detect properties of their homotopy type.

One of our main goals will be the recognition principle of May, stating that if X has an action of the operad E_d of little d-cubes (and assuming that X is grouplike), then X is homotopy equivalent to a d-fold loop space of another space. We will also consider the following situation: we have a space X which naturally decomposes as a disjoint union of spaces X_n indexed by natural numbers; an operad O acts on X, and for all n there are natural maps X_n → X_n+1 coming from this action. In this case, the action of O can help us understand how the different spaces X_n are interrelated, and obtain some information on the stable homology of X, i.e. the colimit of H_∗(X_n) for n going to infinity. We will in particular discuss the group completion theorem for the homology of certain algebras over the operad E₁.

Finally, we will consider the surface operad M introduced by Tillmann and we will see that if X has an action of an operad with homological stability (for example M) and X is grouplike, then X is homotopy equivalent to an infinite loop space.

Plan of the talks

All talks, also the ones at exceptional dates, will take place 14:15–15:45.