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

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.

💡 Research Summary

The paper tackles two intertwined problems in differential topology: (i) how to construct smooth maps on higher‑dimensional spheres with a prescribed topological degree, starting from a lower‑dimensional sphere map, and (ii) how to use this construction to give a streamlined proof of the Morse index formula, a generalization of the Poincaré‑Hopf theorem for manifolds with boundary.

The first part introduces a “dimension‑raising” operation. Let f : Sⁿ⁻¹ → Sⁿ⁻¹ be a smooth map of degree d. The authors consider the product Sⁿ⁻¹ ×