Abstract
Suppose $e\colon M\to X$ is a map from a (finite dimensional) manifold to a locally compact space. Conditions on $e$ are given under which it can be extended to a proper map $e’\colon M’\to X$ by adding boundary to $M$. This theorem is applied to construct a mapping cylinder neighborhood for a $1$-LC ANR in a manifold, resolutions of manifold factors, block bundle approximations to approximation fibrations, local flatness of embeddings, and locally flat approximations of wild embeddings.
