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

coefficients.

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),

no.2, 329-34

You can watch the lecture here:

https://www.youtube.com/watch?v=mioOARGePmQ&index=4&list=PL_0phSnA7tyQJD...