mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Scott core theorem is a theorem about the finite presentability of

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

s of

3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...

s due to

G. Peter Scott Godfrey Peter Scott, known as Peter Scott, (born 1944) is a British mathematician, known for the Scott core theorem. Scott received his PhD in 1969 from the University of Warwick under Brian Joseph Sanderson. Scott was a professor at the Univers ...

, . The precise statement is as follows: Given a 3-manifold (not necessarily

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

) with finitely generated fundamental group, there is a compact three-dimensional

submanifold In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which p ...

, called the compact core or Scott core, such that its

inclusion map In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iot ...

induces an

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...

on fundamental groups. In particular, this means a finitely generated 3-manifold group is finitely presentable. A simplified proof is given in , and a stronger uniqueness statement is proven in .

References

* * * 3-manifolds Theorems in group theory Theorems in topology {{topology-stub