Cobordism of nested manifolds

TL;DR

This paper develops a nested cobordism theory for manifolds with embedded submanifolds, extending classical constructions and linking it to link cobordism invariants, with applications to stable cobordism groups.

Contribution

It introduces a nested Pontryagin-Thom construction and connects nested cobordism groups to link cobordism invariants, expanding the understanding of manifold embeddings.

Findings

Identified a nested analog of the Pontryagin-Thom construction.

Established homotopy equivalences relating nested manifolds and links.

Provided an alternative proof of Wall's splitting result for stable nested cobordism groups.

Abstract

We study cobordisms of nested manifolds, which are manifolds together with embedded submanifolds, which can themselves have embedded submanifolds, etc. We identify a nested analog of the Pontryagin-Thom construction. Moreover, when the highest-dimensional manifold has a normal bundle with a framed direction, we find spaces homotopy equivalent to the nested Pontryagin-Thom spaces that relate nested manifolds up to cobordism with links up to cobordism. This gives rise to nested cobordism invariants coming from previously studied cobordism invariants of links. In addition, we provide an alternative proof of a result by Wall about the splitting of the stable nested cobordism groups.