Some results on extension of maps and applications

TL;DR

This paper explores the extension of maps via obstruction theory, classifies homotopy classes, and applies these results to vector bundle equivalence and homotopy of embeddings up to surgery.

Contribution

It introduces a non classical approach to map extension, provides a new proof of Adachi's theorem, and relates embeddings' homotopy to normal bundle equivalence.

Findings

Classification of homotopy classes of maps.

A simple proof of Adachi's theorem on vector bundles.

Conditions under which embeddings are homotopic up to surgery.

Abstract

This paper concerns extension of maps using obstruction theory under a non classical viewpoint. It is given a classification of homotopy classes of maps and as an application it is presented a simple proof of a theorem by Adachi about equivalence of vector bundles. Also it is proved that, under certain conditions, two embeddings are homotopic up to surgery if and only if the respective normal bundles are $SO$-equivalent.