Speaker:

Andreas Zastrow

Thursday, July 25, 2013 - from 17:00
to 17:25

Abstract:

Computing the classical algebaric invariants for wild (e.g.\
non-triangulable) spaces has yield one or the other psychologically
surprising result (e.g.\ Milnor's \& Barratt's example of a wild two-dimensional
complex with a non-trivial three-dimensinal singular homologiy group, the
fact that a one-point union of simply connected spaces may be non-simply
connected, or the fact that the Hawaiian Earring has a non-free fundamental
group and an astonishingly complicated 3-summands formula for its first
singular homology group, the latter shown by Eda in 2000). In the the very
recent years by Diestel, Spr\"ussel and Georgakopoulos papers have been
published which may be interpreted as a kind of search for a homology theory
which behaves more sensible on wild spaces than does classical singular
homology theory.
Now the analysis of some of these phenomena has left the impression that
the origin for all the strange behaviour of the classical algebraic
invariants is that they are based on classical finite arithmetics, while
wild spaces often have a topological structure which can only suitably
be mirrored by infinite algebraic operations. Since Milnor-Thurston
homology theory is based on considering measures and thus allows to some
extent infinite summation, although it was created by a complete different
motivation, the idea arose to examine how these homolopy groups
behave on wild topological spaces. Milnor-Thurston homology theory
fulfils the Eilenberg-Steenrod-Axioms and thus coincides
with singular homology theory on triangulable spaces, but it is also
known that it does not automatically coincide for arbitrary spaces.
During the years 2011--12 Przewocki found a way to compute the
Milnor-Thurston homology groups for the Warsaw Circle, with the result that
the first and all higher homology group are trivial and thus coincide
with singular homology, but the zeroth homology group is uncountably
dimensional.
This result motivated the current research on the the general behaviour
of zeroth Milnor-Thurston homology groups. While the Warsaw Circle
already shows that the dimension of the zeroth Milnor-Thurston group
is not automatically the number of
path-components and in particular can be much bigger, it was meanwhile
shown by Przewocki that in case of Peano-Continua
the zerorth Milnor-Thurston group is always one-dimensional
and thus coincides with singular homology theory. It remains
the question whether it can be smaller than the singular homology group,
which would make it necesary to construct measures on one-simplices that
connect points in different path components. While by a result of
Przewocki this can be ruled out for spaces with measurable
path-components, that talk is about to present an example showing
that this can well happen for spaces with non-measurable path components.
The talk shall briefly explain the basic concept and original motivation
of the construction of Milnor-Thurston homology theory,
then sum up Przewocki's results on zeroth Milnor-Thurston homology groups,
and then focus on the last above-mentioned result.