PhD Course "Operads in algebraic topology"

Time and location: Thursdays 13.15-15.00, place alternating between:
room 37, house 5, Kraeftriket, SU (even weeks)
and room 3418, Linstedtsvaegen 25, 4th floor, KTH (odd weeks).
The first meeting took place the 28th of January 2016 at SU.
Course given by Alexander Berglund and Stephanie Ziegenhagen.

In this course we will treat two important results in algebraic topology: the recognition principle for iterated loop spaces, due to Boardman-Vogt and May, and Mandell's theorem, which roughly states that the homotopy type of a topological space is faithfully encoded by the E-infinity algebra structure on its singular cochain complex.

What these results have in common is that they use operads in an essential way. Operads were introduced in the 1970's to understand iterated loop spaces. The recognition principle states that a connected space has the weak homotopy type of an n-fold loop space if and only if it is an algebra over the little n-cubes operad. After giving an introduction to operads, the first part of the course will treat the recognition principle, following [May].

Since iterated and infinite loop spaces are ubiquitous in algebraic topology, it is not surprising that the little n-cubes operad and its algebraic variants, called E_n-operads, have many other applications. In the second part of the course we will treat Mandell's result on the equivalence between the homotopy category of nilpotent finite type p-complete spaces and the homotopy category of a certain class of E-infinity algebras. This equivalence is analogous to Sullivan's rational homotopy theory: the rational homotopy type of a space is faithfully encoded by a commutative differential graded algebra (essentially the de Rham complex with the wedge product of differential forms). In positive characteristic, however, we have to take higher homotopies on the singular cochain complex into account; these higher homotopies are closely related to the homotopies admitted by an infinite loop space.

If time admits, we will in the last part of the course have a look at current developments, with the precise topics depending on the interests of the students. Possible topics include Kontsevich graph complexes or Koszul duality for E_n-operads.

References include:

[BV] Michael Boardman, Rainer Vogt, "Homotopy invariant algebraic structures on topological spaces", Lecture Notes in Mathematics 347, Springer-Verlag (1973).
[Ma1] Michael Mandell, "E-infinity Algebras and p-Adic Homotopy Theory", Topology 40 (2001), no. 1, 43-94.
[Ma2] Michael Mandell, "Cochains and Homotopy Type", Publ. Math. IHES, 103 (2006), 213-246.
[MSS] Martin Markl, Steven Shnider, James Stasheff, "Operads in Algebra, Topology and Physics", Mathematical Surveys and Monographs 96, AMS (2002).
[May] Jon Peter May, "The geometry of iterated loop spaces", Lecture Notes in Mathematics 271, Springer-Verlag (1972).
2015-12-06, Stephanie Ziegenhagen