Nielsen Isomorphism Theorem
The Nielsen isomorphism theorem basically states that the algebraic topology of a closed Riemann surface is the same as its geometry.
More precisely, let R be a closed hyperbolic surface. Let G be the Fuchsian group of R and let be a faithful representation of G, and let be discrete. Then define the set
and add to this set a topology of pointwise convergence, so that A(G) is an algebraic topology.
The Nielsen isomorphism theorem: For any there exists a homeomorphism h of the upper half-plane H such that for all .
Most of the material here is copied, not very accurately, out of the book below (see page 12).
Read more about this topic: Fuchsian Model
Famous quotes containing the word theorem:
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)