Topology of Surfaces

The genus of a surface is the number of holes it has. Gluing two tori together along a common boundary component creates a surface with two holes, so the genus is 2.

A torus is a surface that can be obtained by rotating a circle around another circle. It has one hole, so its genus is 1.

The Euler characteristic of a surface is a topological invariant that is equal to the number of vertices minus the number of edges plus the number of faces. For a surface with genus $g$, the Euler characteristic is $2 - 2g$.

A surface is orientable if it is possible to consistently choose a normal vector at each point on the surface. The Klein bottle is not orientable because it is impossible to choose a normal vector that is continuous everywhere on the surface.

The fundamental group of a surface is the group of homotopy classes of closed loops on the surface. The fundamental group of a torus is $Z^2$, which means that it is generated by two independent loops.

A sphere is a surface that can be obtained by bending a flat disk without tearing or gluing. It is homeomorphic to any other surface that can be obtained by bending a flat disk without tearing or gluing.

Gluing two Möbius strips together along a common boundary component creates a surface with one hole, so the genus is 1.

A Klein bottle is a surface that can be obtained by gluing two Möbius strips together along a common boundary component. It has two holes, so its genus is 2.

The Euler characteristic of a surface with genus $g$ and $n$ boundary components is $2 - 2g - n$.

A surface is compact if it is closed and bounded. The plane is not compact because it is not bounded.

The fundamental group of a Klein bottle is $Z_2$, which means that it is generated by a single loop that can be continuously deformed to its own inverse.

A torus is a surface that can be obtained by rotating a circle around another circle. It is homeomorphic to any other surface that can be obtained by rotating a circle around another circle.

Gluing three Möbius strips together along a common boundary component creates a surface with two holes, so the genus is 2.

A projective plane is a surface that can be obtained by gluing two Möbius strips together along a common boundary component. It has three holes, so its genus is 3.

The Euler characteristic of a surface with genus $g$ and $n$ boundary components is $2 - 2g - n$.