UBC Mathematics Colloquium
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
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.