Bounding immersion classes of codimension one (Q1200047)

Let \(M\) be a closed \(n\)-manifold and \(X\) an arbitrary \((n+1)\)-manifold. A framed immersion \((f:M\to X,\nu=\) its nonzero normal vector field) is said to be bounding if there exist a compact \((n+1)\)-manifold \(W\) with boundary \(M\) and an extension of \(f\) to an immersion \(W\to X\) with \(\nu\) pointing outward along \(M\). This paper deals with the question: When is a framed immersion regularly homotopic to a bounding one? The main result is: Theorem. If \(n\geq 2\) then a framed immersion class \(\alpha=[(f,\nu)]:M\to X\) is bounding iff the framed bordism class of \((f,\nu)\) vanishes, as an element of the framing bordism group of framed immersions of \(n\)- manifolds into \(X\). Analogous results follow when \(\alpha\) is an immersion class with trivial normal bundle and both \(M\) and \(X\) are oriented or either \(M\) or \(X\) is unoriented.