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Triangulations and homology of Riemann surfaces

2002; American Mathematical Society; Volume: 131; Issue: 2 Linguagem: Inglês

10.1090/s0002-9939-02-06470-5

1088-6826

Peter Buser, Mika Seppälä,

Homotopy and Cohomology in Algebraic Topology

We derive an algorithmic way to pass from a triangulation to a homology basis of a (Riemann) surface. The procedure will work for any surfaces with finite triangulations. We will apply this construction to Riemann surfaces to show that every compact hyperbolic Riemann surface $X$ has a homology basis consisting of curves whose lengths are bounded linearly by the genus $g$ of $X$ and by the homological systole. This work got started by comments presented by Y. Imayoshi in his lecture at the 37th Taniguchi Symposium which took place in Katinkulta near Kajaani, Finland, in 1995.