Homologies are infinitely complex

TL;DR

The paper demonstrates that stratified sets of finite complexity cannot realize all homology classes in smooth manifolds for any codimension greater than one, highlighting the infinite complexity of homologies.

Contribution

It establishes that finite complexity stratified sets are insufficient for realizing all homology classes, revealing the necessity of infinite complexity.

Findings

Finite complexity stratified sets cannot realize all homology classes for k>1

Results apply to smooth manifolds and generic maps with co-oriented double points

Highlights the infinite complexity inherent in homology classes

Abstract

We show that for any k>1, stratified sets of finite complexity are insufficient to realize all homology classes of codimension k in all smooth manifolds. We also prove a similar result concerning smooth generic maps whose double-point sets are co-oriented.