FSF3609 Operads in Algebraic Topology 7.5 credits

• Operads and algebras over operads

• The little n-cubes operad, A-infinity and E-infinity operads

• Iterated loop spaces

• Approximation of free iterated loop spaces via the little n-cubes operad

• The simplicial bar construction

• The recognition principle

• The E-infinity algebra structure on singular cochains

• Mandell's theorem

The course should provide the students with

• thorough knowledge of operads and standard constructions related to them, especially in the categories of topological spaces and chain complexes

• familiarity with  the little n-cubes operad and algebraic and topological variants thereof

• a precise understanding of the relationship between iterated loop spaces and algebras over the little n-cube operad

• Understand how the singular cochain complex of a topological space is an algebra over an E-infinity operad

• Understand the main steps in the proof of Mandell's theorem

The course requires knowledge of algebraic topology and homological algebra. Most of the background needed has been for example covered by SF2735, except for a basic knowledge of simplicial structures and model categories that we require.

• Michael Boardman, Rainer Vogt, "Homotopy invariant algebraic structures on topological spaces", Lecture Notes in Mathematics 347, Springer-Verlag (1973).

• Michael Mandell, "E-infinity Algebras and p-Adic Homotopy Theory", Topology 40 (2001), no. 1, 43-94.

• Michael Mandell, "Cochains and Homotopy Type", Publ. Math. IHES, 103 (2006), 213-246.

• Martin Markl, Steven Shnider, James Stasheff, "Operads in Algebra, Topology and Physics", Mathematical Surveys and Monographs 96, AMS (2002).

• Peter May, "The geometry of iterated loop spaces", Lecture Notes in Mathematics 271, Springer-Verlag (1972).

If the course is discontinued, students may request to be examined during the following two academic years.

• INL1 - Assignment, 7.5 credits, grading scale: P, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

Examination will consist of the students giving presentations about related topics at the end of the term.

Presentation of the chosen topic.

Wojciech Chachólski

• All members of a group are responsible for the group's work.
• In any assessment, every student shall honestly disclose any help received and sources used.
• In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.