Abstract. Around 1900 Poincaré (the father of Algebraic Topology)
wanted to invent a tool for showing that certain nice spaces (so
called manifolds, i.e locally Euclidean spaces were topologically
different, i.e. there was no bijection between them, continuous in
both directions. His idea was to “count the submanifolds in the
space”, in the sense, that two submanifolds should be considered
equivalent if they together bound another submanifold in the space.
Soon he realized that this was a dead end, and turned to an algebraic
way of constructing the tool (the so called homology groups) using
free Abelian groups generated by the simplices of the space.
A few decades later Steenrod raised the question: “How far is this
algebraic realization from the original geometric idea?”. More precisely: Can we
obtain any homology class as a continuos image of a manifold? Rhene
Thom (Fields medal 1954) answered this question to the positive in
case of coefficients and partially positively for integer
But this was only the first step towards the original geometric idea
of Poincare. The second step would be to show that we can choose the
continuos map as a nice map: an embedding, or at least locally
embedding, or as a map having only simple singularities.
Last year with a young English topologist Mark Grant we showed that
immersions (i.e. locally embedding maps) are not sufficient for
realizing all Z2-homology classes. Moreover for any finite set of
multisingularities the maps having only multisingularities from this
list are insufficient to realize any homology class. In this sense
homologies are infinitely complex and the algebraic realization is far
away from the original geometric intuition, so the devil won again.
It is still open whether any finite set of local singularities is
sufficient for realizing any homology class.
The proof for immersions uses a formula describing the homology class
of the singularity of a smooth map. The proof for the
multisingularities uses the classifying spaces of the singular maps
with a given set of allowed multisingularities.
Reference: Grant, Mark; András, Szücs: On realizing homology classes
by maps of restricted complexity. Bull. London. Math. Soc. 45 (2013),
You can watch the lecture here: