Scott core theorem

In mathematics, the Scott core theorem is a theorem about the finite presentability of fundamental groups of 3-manifolds due to G. Peter Scott, (Scott 1973). The precise statement is as follows:

Given a 3-manifold (not necessarily compact) with finitely generated fundamental group, there is a compact three-dimensional submanifold, called the compact core or Scott core, such that its inclusion map induces an isomorphism on fundamental groups. In particular, this means a finitely generated 3-manifold group is finitely presentable.

A simplified proof is given in (Rubinstein & Swarup 1990), and a stronger uniqueness statement is proven in (Harris & Scott 1996).

References

• Harris, Luke; Scott, G. Peter (1996), "The uniqueness of compact cores for 3-manifolds", Pacific Journal of Mathematics 172 (1): 139–150, doi:10.2140/pjm.1996.172.139, ISSN 0030-8730, MR 1379290
• Rubinstein, J. H.; Swarup, G. A. (1990), "On Scott's core theorem", The Bulletin of the London Mathematical Society 22 (5): 495–498, doi:10.1112/blms/22.5.495, MR 1082023
• Scott, G. Peter (1973), "Compact submanifolds of 3-manifolds", Journal of the London Mathematical Society. Second Series 7 (2): 246–250, doi:10.1112/jlms/s2-7.2.246, MR 0326737