Relationship Between the Genus and Euler Characteristic of a Surface

The genus of a surface is the greatest number of distinct continuous non intersecting closed curves which can be drawn on a surface without separating it into distinct regions. For each division of a surface to become a map, the division has to include edges such that all regions are simply connected, so the eulercharacteristic of a surface is related to its genus.

The genus and euler characteristic for some surfaces are given in the table.

Surface

Genus

Euler Characteristic

Sphere

0

2

Torus

1

0

Two Hole Torus

2

-2

A Torus With n Holes

In fact, for any surface the genus g is relation to the Euler characteristicvia the equation

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