A. J. Sieradski, Omega-groups, 15 February, 1998
Abstract
The algebraic topology of locally contractible spaces, such as, CW complexes
and manifolds, is fully expressible using discrete group theory. But for
wild spaces, by which we mean metric spaces with arbitrarily small essential
features, new group theory terminology is required. For this purpose, this
paper offers the theory of omega-groups. In essence, these are groups
whose elements are assigned non-negative real weights and form countable
products of all possible order-type whenever their weights tend to zero as
their depth of appearance in the order-type increases without bound.
Omega-groups are to discrete groups as wild spaces are to tame spaces, e.g.,
as totally disconnected spaces are to discrete spaces. Section 1 has the
definitions. Section 2 introduces a weighted combinatorial group theory
to present omega-groups. Section 3 illustrates the theory with motivating
examples of the fundamental group of wild metric cell complexes.