Multiple points of immersions

TL;DR

This paper introduces an obstruction theory for regular homotopies of immersions between smooth manifolds, focusing on eliminating k-fold points, with a complete obstruction in certain dimension ranges.

Contribution

It constructs a new obstruction in framed bordism groups that determines when an immersion can be regularly homotoped to remove k-fold points.

Findings

Obstruction takes values in framed bordism group

Obstruction is complete when (k+1)(n+1) ≤ km

Provides criteria for eliminating k-fold points in immersions

Abstract

Given smooth manifolds $V^n$ and $M^m$, an integer $k$, and an immersion $f:V\looparrowright M$, we have constructed an obstruction for existence of regular homotopy of $f$ to an immersion $f':V\looparrowright M$ without $k$-fold points. This obstruction takes values in certain framed bordism group, and for $(k+1)(n+1)\leq km$ turns out to be complete.