Construction of sphere maps with given degrees and a new proof of Morse index formula

TL;DR

This paper introduces a method to construct higher-dimensional sphere maps from lower-dimensional ones, provides explicit formulas for sphere maps with specified degrees, and offers a new proof of the Morse index formula, a generalized Poincaré-Hopf theorem.

Contribution

It presents a novel construction procedure for sphere maps of higher dimensions and a new proof of the Morse index formula, enhancing understanding of topological degree and index theory.

Findings

Explicit formula for smooth sphere maps with given degree

Construction method for higher-dimensional sphere maps

New proof of the Morse index formula

Abstract

This note presents a procedure of constructing a higher dimensional sphere map from a lower dimensional one and gives an explicit formula for smooth sphere map with a given degree. As an application a new proof of a generalized Poincare-Hopf theorem called Morse index formula is also presented.