Euler Characteristic of Surfaces Joined Together

Generally, ifopen surfacesare joined together along their boundaries to form a surfacethen the Euler characteristic of the resulting surface is

Example: Find the Euler characteristic ofcylinders joined end to end, and two discs.

Obviously the cylinders joined end to end, closed at each end by two discs is a closed surface homeomorphic to a sphere.

A cylinder can be thought of as a surface - having one face - with two boundary edges and one edge from the top to the bottom and vertices, one on each boundary edge.

Hence

A disc can be thought of as a surface wit one face, once boundary edge and one vertex on the boundary.

Hence

Example: Find the Euler characteristic ofa sphere with n holes and n discs.

The discs are joined to the sphere along the boundary edges of the holes.

The Euler characteristic of a sphere with n holes is

As argued above, the Euler characteristic of a disc is 1.

Hence

The union of the sphere withholes and n discs is obviously a sphere. The Euler characteristic of a sphere is 2.