A double torus.

In mathematics, a genus-2 surface (less frequently known as the double torus) is a topological object formed by the connected sum of two torii. That is to say, from each of two torii the interior of a disk is removed, and the boundaries of the two disks are identified (glued together), forming a double torus.

This is the simplest case of the connected sum of n-tori. A connected sum of tori is an example of a two dimensional manifold. According to the classification theorem for 2-manifolds, every compact connected 2-manifold is either a sphere, a connected sum of tori, or a connected sum of projective planes.

Double torus knots are studied in knot theory.

Example

The Bolza surface is the most symmetric hyperbolic surface of genus 2.

References

• James R. Munkres, Topology, Second Edition, Prentice-Hall, 2000, ISBN 0-13-181629-2.
• William S. Massey, Algebraic Topology: An Introduction, Harbrace, 1967.

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