Locally flat imbeddings of topological manifolds in codimension three.

Abstract:

This paper presents an imbedding theorem for one topological manifold ${M^n}$ in another topological manifold ${Q^q}$, provided that the codimension $(q - n)$ is at least three. The result holds even if the manifolds are of the recently discovered non-piecewise-linear type. Denote the boundaries of $M$ and $Q$ by $\dot M$ and $\dot Q$ respectively. Suppose that $M$ is $2n - q$ connected and $Q$ is $2n - q + 1$ connected. It is then proved that any map $f:(M,\dot M) \to (Q,\dot Q)$ such that $f|\dot M$ is a locally flat imbedding is homotopic relative to $\dot M$ to a proper locally flat imbedding $g:M \to Q$. It is also shown that if $M$ is closed and $2n - q + 1$ connected and $Q$ is $2n - q + 2$ connected, then any two homotopic locally flat imbeddings are locally flatly concordant.