3-manifolds with \(H_ 2(A,\partial A)=0\) and a conjecture of Stallings. (English) Zbl 0585.57011

Knot theory and manifolds, Proc. Conf., Vancouver/Can. 1983, Lect. Notes Math. 1144, 138-145 (1985).
[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.
Reviewer: V.Turaev


57N10 Topology of general \(3\)-manifolds (MSC2010)
20F05 Generators, relations, and presentations of groups
57Q35 Embeddings and immersions in PL-topology