[For the entire collection see Zbl 0564.00014.]
The main result is the following theorem: Let A be an oriented, connected 3-manifold such that \(H_ 2(A,\partial A)=0,\) let T be a compact connected, oriented surface, \(f: T\to A\) a proper map such that \(f|_{\partial T}\) is an imbedding. If the oriented f(\(\partial T)\) bounds a (possibly non-compact) sub-surface of \(\partial A\) then f(\(\partial T)\) bounds a compact sub-surface S of \(\partial A\) with genus \(S\leq genus T.\)
The non-obvious case is A non-compact. For imbeddings \(f: T\to A\) the theorem stated was proven by J. Stallings [Math. Z. 184, 1-17 (1983; Zbl 0509.57004)] in his search for a topological proof of the famous conjecture on the solvability of non-singular sets of equations in groups.