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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.

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 |