Projective Space as Simplicial Complex

The simplest representation of a projective space would be like this:

There is only one vertex, although it is drawn twice.

There are two directed edges: 'a' and 'b' which go from 'x' to 'x'.

Here we insert a second vertex 'y' part of the way along the previous arrows.

This may look similar to triangulations of other surfaces such as the torus, but note that now the direction of the edges is now clockwise around the outside.

We still have each vertex appearing on the diagram twice. In order to represent this in ordinary space we need to merge the duplicates.

To do this we will embed the rectangle in 3D space. We can then lift up the edges called 'x' and bring them together. At the same time we can lower the points 'y' and bring them together.

Projective Plane - No boundaries, every edge is connected to two faces:

(3) -> projectivePlane()

        +1  2  3+
        |1  3  4|
        |1  2  6|
        |1  5  6|
   (3)  |1  4  5|
        |2  3  5|
        |2  4  5|
        |2  4  6|
        |3  4  6|
        +3  5  6+
   Type: FiniteSimplicialComplex(VertexSetAbstract)

Above is my attempt to draw, with each vertex appearing once, this makes the diagram very cluttered and hard to understand. Its easier to draw if nodes 4,5 and 6 appear twice as below:

Note: edges 4->5, 5->6 and 6->4 connect two faces but they wind in the same direction.

Graphs, Surfaces and Homology - Peter Giblin.
Builds up to homology groups via graphs and simplicial complexes.