UBC Mathematics Colloquium

Jeff Smith


Group actions on spheres

Fri., Nov. 20, 2009, 3:00pm, MATX 1100


A discussion of symmetry is good for the fluff section of any grant proposal. In this talk I will discuss finite symmetry groups of spheres.  These questions have been part of topology from the beginning.  Poincare showed that the binary icosahedral group acts freely on the 3-sphere and that the quotient is a homology 3-sphere. This led to his famous question about simply connected three manifolds.

Traditionally, three kinds of group actions on spheres have been studied, linear actions, smooth actions and continuous actions. The linear actions of a group are its orthogonal representations. A smooth action of a group is a homomorphism to Diff S^n and a topological actions is a homomorphism to Homeo S^n (the topological group of homeomorphisms from the sphere to itself). These correspond to three of the geometries of the sphere. Homotopy theory studies the most fundamental geometry, the one where only the toughest invariants are left.

There are several ways to define the homotopy actions of a group. In the easiest definition, a homotopy action of G on S^n is an action of G on a space X that is homotopy equivalent to S^n. But X can be complicated; it need not be a manifold or even finite dimensional.

The linear actions of G are the orthogonal representations of G.  The smooth and topological actions are only completely understood when the group is acting freely, the so called spherical space form problem, and only a few finite groups can act freely on a sphere. Much to our surprise, Grodal-Smith have completely classified the homotopy actions of a finite group on a sphere. The surprise arises because the classification of homotopy actions is equivalent to computing

[BG,B Aut S^n]

where Aut S^n is the topological monoid of self equivalences of S^n; B Aut S^n and BG are the classifying spaces. Both are infinite dimensional and classical techniques are useless. The entry into computation is the modern technology for studying homotopy fixed points.

Back to the colloquium page.