Geometric formulas for Smale invariants of codimension two immersions

TL;DR

This paper introduces three geometric formulas for the Smale invariant of certain high-dimensional sphere immersions, linking them to characteristics of maps from lower-dimensional manifolds and establishing new homotopy invariance results.

Contribution

It provides novel formulas for the Smale invariant in high codimension and explores their implications for regular homotopy classes and cusp counts.

Findings

Formulas express Smale invariant via geometric characteristics.

Non-regular homotopic immersions in different spaces are distinguished.

Minimum cusp counts are established for homotopies between immersions.

Abstract

We give three formulas expressing the Smale invariant of an immersion f of a (4k-1)-sphere into (4k+1)-space. The terms of the formulas are geometric characteristics of any generic smooth map g of any oriented 4k-dimensional manifold, where g restricted to the boundary is an immersion regularly homotopic to f in (6k-1)-space. The formulas imply that if f and g are two non-regularly homotopic immersions of a (4k-1)-sphere into (4k+1)-space then they are also non-regularly homotopic as immersions into (6k-1)-space. Moreover, any generic homotopy in (6k-1)-space connecting f to g must have at least a_k(2k-1)! cusps, where a_k=2 if k is odd and a_k=1 if k is even.