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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

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