A generalization of the Hopf degree theorem

Matthew D. Kvalheim

Published 2022-03-19Version 1

The Hopf degree theorem states that homotopy classes of continuous maps from a smooth connected closed $n$-manifold $M$ to the $n$-sphere are classified by their degree when $M$ is oriented and by their mod $2$ degree when $M$ is nonorientable. Such a map is equivalent to a section of the trivial $n$-sphere bundle over $M$. A generalization of the Hopf degree theorem is obtained for the case that the sphere bundle over $M$ is nontrivial.