Dotted torus

Eversion of a dotted torus.

In mathematics , a dotted torus is an area that can be obtained from a torus by removing a point (or the equivalent of a circular disk ).

Correspondingly, a double, triple or n-point dotted torus is a surface that is obtained from a torus by removing 2, 3 or n points.

topology

Wedge product of two circles

The dotted torus is homotopy equivalent to the wedge product of two circles. The n-fold dotted torus is homotopy-equivalent to the wedge product of n + 1 circles.

Accordingly, the fundamental group of the dotted torus is a free group with 2 generators and the fundamental group of the n-fold dotted torus is a free group with n + 1 generators.

Hyperbolic geometry

Hyperbolic metrics on the dotted torus can be constructed by gluing two ideal triangles . The Teichmüller space of hyperbolic metrics on the dotted torus is 2-dimensional, and 2n-dimensional on the n-fold dotted torus.

literature

• Sario, L .; Nakai, M .: Classification theory of Riemann surfaces. The basic teachings of the mathematical sciences, Volume 164 Springer-Verlag, New York-Berlin 1970

Web links

• Minsky: Punctured torus groups