Codimension 2 embeddings and finite localization of spaces

TL;DR

This paper develops a localization framework for classifying codimension 2 embeddings in manifolds by analyzing their complements through homology equivalences and homotopy groups.

Contribution

It introduces a localization functor tailored to codimension 2 embeddings and characterizes local objects via homotopy groups, advancing classification methods.

Findings

Constructed a localization functor for homology equivalences.

Characterized local objects using homotopy groups.

Provides tools for classifying concordance classes of embeddings.

Abstract

In order to classify concordance classes of codimension 2 embeddings in a manifold M, we need to determine the complement of such an embedding. These complements are spaces over M well defined up to some homology equivalence. We construct a localization functor corresponding to this class of homology equivalences and we give a characterization of local objects in terms of homotopy groups.