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Robin Stoll: Operads and the recognition principle for loop spaces

Time: Fri 2019-11-29 13.00 - 14.00

Location: Kräftriket, house 5, room 32

Participating: Robin Stoll

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The goal of this talk is to precisely formulate the so called recognition principle for loop spaces (a classical result originally due to Boardman and Vogt in the 60s, though we will follow the approach of May from the 70s), and, if time permits, to sketch parts of its proof. It states that a topological space \(X\) "is" the space of maps from the \(n\)-sphere to some other space if and only if \(X\) admits the structure of a group "up to homotopy" that is "commutative up to homotopy to the \(n\)-th degree".

To make this precise we will introduce the notion of an operad, focusing specifically on the so called little-disks operads as they play a central role in the theorem we focus on. These notions play an important role in modern algebraic topology in general (mainly for encoding complicated compositional structure in a nice way), so that hopefully the talk will on the way provide an outlook over some of the methods used throughout the field.