Scott core theorem
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Short description: On the finite presentability of fundamental groups of 3-manifolds
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), and a stronger uniqueness statement is proven in (Harris Scott).
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, http://projecteuclid.org/getRecord?id=euclid.pjm/1102366188
- 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
- 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
![]() | Original source: https://en.wikipedia.org/wiki/Scott core theorem.
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