On Scott’s core theorem. (English) Zbl 0709.57012

Rubinstein, J. H.; Swarup, G. A.

Bull. Lond. Math. Soc. 22, No. 5, 495-498 (1990).

G. P. Scott proved in “Compact submanifolds of 3-manifolds” [J. Lond. Math. Soc., II. Ser. 7, 246-250 (1973; Zbl 0266.57001)] that if M is a 3-manifold with finitely-generated fundamental group, then there is a compact submanifold N in M such that the inclusion of N in M induces an isomorphism of fundamental groups. Such an N is called a core of M. The proof uses an earlier theorem of G. P. Scott [ibid. 6, 437-440 (1973; Zbl 0254.57003)] that finitely-generated 3-manifold groups are finitely presented, as well as some new delicate constructions and arguments. In this note we give a short direct proof of the core theorem of Scott’s first mentioned paper assuming the main result of the other one. Our method is suitable for generalizations [see R. S. Kulkarni and P. B. Shalen, Adv. Math. 76, No.2, 155-169 (1989; Zbl 0684.57019); D. McCullough, Q. J. Math., Oxf. II. Ser. 37, 299-307 (1986; Zbl 0628.57008)] and we prove also an extension which implies McCullough’s [loc. cit.].

MSC:

57N10	Topology of general \(3\)-manifolds (MSC2010)
57M05	Fundamental group, presentations, free differential calculus

Keywords:

3-manifold with finitely-generated fundamental group; compact submanifold; core; finitely presented

Citations:

Zbl 0266.57001; Zbl 0254.57003; Zbl 0684.57019; Zbl 0628.57008