SMC Colloquium

Finite loop spaces

by Jesper Grodal (University of Copenhagen)

Room 3721

Room 3721

Department of Mathematics, KTH, Lindstedtsvägen 25, 7th floor

Hilbert's 5th problem, in its most basic form, asks if every compact topological group, which admits the structure of a smooth manifold, is a Lie group. In this form, it was answered affirmatively by von Neumann in 1929. If one takes a homotopical interpretation of the word "admits", the question is more subtle, and one is led to the notion of a finite loop space. These turn out not quite to be Lie groups, but nevertheless posses a rich enough structure to admit a classification. My talk will outline this story, which starts with a 1941 paper of Hopf: "Uber die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen" and ends close to the present.