A zoo of geometric homology theories

TL;DR

This paper explores a variety of bordism-based homology theories using stratifolds, highlighting their construction, simple examples, and potential future applications despite current computational complexity.

Contribution

Introduces a unified framework of geometric homology theories via stratifolds, expanding the understanding of bordism groups with singularities and their special cases.

Findings

Simple examples of these homology groups are provided.

Computations are complex, and applications are yet to be developed.

Potential connections to algebraic cycles and the Griffith group are suggested.

Abstract

The theories in our zoo are all bordism groups, which generalize the case of smooth manifolds by allowing singularities. There are many concepts of manifolds with singularities one could use here. For our pupose the objects the author introduced some years ago, which are called stratifolds, work particularly well. The zoo comes from forcing certain strata indexed by the subset $A$ to be empty. Special cases are ordinary singular homology and singular bordism. Despite their simple construction computations of these groups seem to be very complicated. We give a few simple examples. Thus there are no interesting applications so far and the zoo looks a bit like a curiosity. But one never knows for what these theories might be good in the future. We mention a concrete question which might be useful in connection with the Griffith group consisting of algebraic cycles in a smooth algebraic variety over the complex numbers which vanish in singular homology. I dedicate these notes to my friend Egbert Brieskorn. Egbert is (in a very different way like our common teacher Hirzebruch) a person which had a great influence on me. When I had to make a complicated decision I often had him in front of my eyes and asked myself: What would Egbert suggest? Conversations with him were always intense and fruitful. I miss him very much.