It is proved that, from the
viewpoint of “geometric” homology theory, an arbitrary embedding of a closed surface
S in any 3-manifold with trivial first homology group looks exactly like the standard
embedding of S in the euclidean 3-space. A consequence: every compact subset of a
3-manifold with trivial first homology group can be embedded in a homology
3-sphere. Necessary and sufficient (homological) conditions are given for a compact
3-manifold to be embeddable in some acyclic 3-manifold (or in some homology
3-sphere).
