In Spring 2023 Niall Taggart and I will be running a seminar for MSc students on iterated loop spaces. Here is the syllabus. The seminar will be held on Mondays, 15.15-17h, starting February 6.
The aim of the seminar is to understand the algebraic structure of (iterated) loop spaces using the language of operads. Specifically, we will see that loop spaces are essentially equivalent to spaces equipped with a group structure 'up to homotopy'. We will prove May's recognition theorem, stating that an n-fold loop space is essentially the same thing as a space with such a group structure up to homotopy satisfying further commutativity constraints, described precisely by the operad of little n-cubes. This seminar is an opportunity to learn about operads and algebra up to homotopy, which have found applications in many different parts of mathematics.
Schedule
Feb 13, MIN 2.05 (Gijs): Introduction and overview.
Feb 20, BBG083 (Julie): Operads, algebras, and basic examples. Julie's exercises.
Feb 27, BBG061 (Esther): Monads and operads. Esther's exercises.
Mar 6, BBG001 (Megan): The little n-cubes operad and n-fold loop spaces. Megan's exercises.
Mar 13, BBG083 (Marien): Fibrations, the long exact sequence for homotopy groups, quasifibrations. Setting up the proof of the approximation theorem. Marien's exercises.
Mar 20, BBG083 (René): The proof of the approximation theorem. René's exercises.
Mar 27, BBG071 (Rodin): Simplicial spaces and geometric realization. Rodin's exercises.
Apr 3, BBG071 (Melle): Bar constructions. Melle's exercises.
Apr 17, MIN 2.05 (Niall): Proof of the recognition theorem.
Apr 24, BBG065 (Gijs): E-infinity spaces, infinite loop spaces, and spectra.
May 1, BBG079 (Megan): Examples of infinite loop spaces. Megan's exercises.
May 8, BBG079 (Julie): Segal's approach to infinite loop spaces. Postponed.
May 15, BBG079 (Melle): Cohomology operations.
May 22, MIN 2.05 (Esther): Homology operations for E-infinity spaces. Esther's exercises.
June 5, BBG079 (Marien): The Snaith splitting. Marien's exercises.
June 12, MIN 2.05 (Julie): Segal's approach to infinite loop spaces. Julie's exercises.
June 19, MIN 2.05 (René & Rodin): Homology of the little n-cubes operad. Exercises.