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Structure Theorems for Riemann and Topological Surfaces
Author(s) -
Álvarez Venancio,
Rodríguez José M.
Publication year - 2004
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610703004836
Subject(s) - uniformization theorem , riemann surface , geometric function theory , mathematics , riemann hypothesis , topology (electrical circuits) , pure mathematics , riemann–hurwitz formula , combinatorics
The classification theorem of compact surfaces states that every topological orientable compact surface is homeomorphic to a sphere or to a ‘torus’ of genus g , with g =1,2,…. It is proved in the paper that every hyperbolic Riemann surface except for D\{0} can be decomposed into basic pieces of only a few different types: Y‐pieces, funnels and half‐disks. As a corollary of this result, the generalization of the classification theorem to non‐compact surfaces is obtained.