Topology and its Applications
A core of a (noncompact) manifold is a submanifold with the property that the inclusion of the submanifold into the manifold is a homotopy equivalence. It is shown by example that a manifold may fail to contain a compact core even though the manifold has the homotopy type of a finite complex.
Venema, Gerard A., "A manifold that does not contain a compact core" (1998). University Faculty Publications. 548.