William Browder, Lecture 4

A homological approach to problems of compact group actions:
Actions on products of spheres; extending Heller's theorem.

Speaker: William Browder. Lecture 4.

Abstract: We prove a number of special cases of the so called:

Rank Conjecture: If G = (Z/p)^r acts freely on
the product (S^n_1) x .... x (S^n_k), then r < k, ie, the rank of G must be
less than or equal to the number of spheres.
For example we have:
Theorem The conjecture is true if the dimension of each of the spheres is either 1 or n.
Heller's 1957 generalization for the product of two spheresof the Smith theorem
says that (Z/p)^3 cannot act freely on (S^n) x (S^m). (An easy proof follows
from results of Lecture 2.) So one would like to at least prove the Rank Conjecture
in case the dimensions of the spheres are concentrated in two dimensions.
Theorem The conjecture is true in case p = 2 and the dimensions are concentrated in
3 and n.
In this case we see the Steenrod Algebra playing a central role.
We may prove more general staements of this type with two dimensions of spheres,
but the restrictons on dimension become more complicated.
In case only one of the dimension is off, we can prove a numbe of results of this type.