Obstructions to embedding $n$-manifolds in $(2n-1)$-manifolds

Abstract:

Suppose $f:({M^n},\partial {M^n}) \to ({Q^{2n - 1}},\partial {Q^{2n - 1}})$ is a proper PL map between PL manifolds ${M^n}$ and ${Q^{2n - 1}}$ of dimension $n$ and $2n - 1$ respectively, $M$ compact. J. F. P. Hudson has shown that associated with each such map $f$ that is an embedding on $\partial M$ is an element $\bar \alpha (f)$ in ${H_1}(M;{Z_2})$ when $n$ is odd and an element $\bar \beta (f)$ in ${H_1}(M;Z)$ when $n$ is even. These elements are invariant under a homotopy relative to $\partial M$. We show that, under slight additional assumptions on $M,Q$ and $f,f$ is homotopic to an embedding if and only if $\bar \alpha (f) = 0$ for $n$ odd and $\bar \beta (f) = 0$ for $n$ even. This result is used to give a sufficient condition for extending an embedding $f:\partial {M^n} \to \partial {B^{2n - 1}}$ (${B^{2n - 1}}$ denotes $(2n - 1)$-dimensional ball) to an embedding $F:({M^n},\partial {M^n}) \to ({B^{2n - 1}},\partial {B^{2n - 1}})$.