Countably compact free Abelian groups:size and the square
by
Artur Hideyuki Tomita
Universidade de Sao Paulo

In this talk we discuss about the following questions:

A) (Dikranjan and Shakmatov) For which \kappa is there a group topology on the free Abelian group of size \kappa that makes it countably compact?

Their motivation was Tkachenko's CH example mentioned below and their result that there are no countably compact free groups.

It was shown that c is such a cardinal under CH (Tkachenko [Izvestia, 1990]), Martin's Axiom (Tomita [CMUC, 1998]) and MA_countable (Tomita and Watson, manuscript).

As the known countably compact free Abelian groups are constructed using a technique related to the construction of countably compact groups without non-trivial convergent sequences, the following also is a natural question:

B) For which \kappa is there a group topology on some group of size \kappa that makes it countably compact and without non-trivial convergent sequences?

C) (D. Grant) Is there a Wallace semigroup (a countably compact topological semigroup with two-sided cancellation that is not a topological group)whose square is countably compact?

Due to the fact that the known examples of Wallace semigroups are related to the construction of countably compacat free Abelian groups (Robbie and Svetlichnyi [PAMS, 1996], Tomita [Canadian Math. Bull., 1996] and Tomita [Topology Proc., 1997], it is natural also to ask

D) Is there a topological free Abelian group whose square is countably compact?

In a joint work with P. Koszmider, we show that 2^c is a cardinal as in A) in a forcing model.

Using MA_countable we show that there is an ultrafilter p such that for each \kappa = \kappa^\omega, there exists a group of size \kappa and p-compact (in particular, countably compact) group topology on it that does not have non-trivial convergent sequences.

Date received: June 30, 2000