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Journal of the London Mathematical Society 2004 69(1):153-168; doi:10.1112/S0024610703004836
© 2004 by London Mathematical Society
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© The London Mathematical Society

Structure Theorems for Riemann and Topological Surfaces

Venancio Álvarez and José M. Rodríguez

Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga Campus de Teatinos, 29071 Málaga, Spain, nancho{at}anamat.cie.uma.es
Departamento de Matemáticas, Universidad Carlos III de Madrid Avenida de la Universidad, 30, 28911 Leganés, Spain, jomaro{at}math.uc3m.es

Received 26 November 2000. Revision received 16 April 2003.

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.


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