Bordism, stratifolds and the representability of homology classes.

Abstract:
Bordism theory is one of the central tools of algebraic topology. In its simplest form it studies the question: When is a closed manifold the boundary of another manifold. This question was solved by Rene Thom in the 50’s and it has many applications.
A natural problem that arises is the so called Steenrod’s representability problem: Is every homology class of a space represented by the continuous image of the fundamental class of a manifold?. This question was solved also by Rene Thom. Every homology class with Z/2Z-coefficients is represented, but there are homology classes with integer coefficients that are not represented.
In this talk I will give a brief introduction to bordism theory and discuss stratifolds, a theory of smooth stratified spaces defined by Matthias Kreck to represent homology with integer coefficients as a bordism theory. I will introduce Z/kZ-stratifolds to represent homology with Z/kZ-coefficients.