A generalization of the Hopf degree theorem

TL;DR

This paper extends the classical Hopf degree theorem to classify sections of nontrivial oriented sphere bundles over manifolds, broadening the understanding of homotopy classes beyond trivial bundles.

Contribution

It provides a generalized framework for the Hopf theorem applicable to nontrivial sphere bundles, expanding the classification of sections in algebraic topology.

Findings

Generalization of the Hopf theorem to nontrivial sphere bundles

Classification of sections of sphere bundles over manifolds

Extension of homotopy class characterization

Abstract

The Hopf theorem states that homotopy classes of continuous maps from a closed connected oriented smooth $n$-manifold $M$ to the $n$-sphere are classified by their degree. Such a map is equivalent to a section of the trivial $n$-sphere bundle over $M$. A generalization of the Hopf theorem is obtained for sections of nontrivial oriented $n$-sphere bundles over $M$.