Projective Space as Simplicial Complex

A one sided surface which cannot be embedded in 3D space without intersecting itself. There are no external boundaries, every edge connects two faces. (A punctured projective plane is equivalent to a Möbius band)

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'.

projective space

If we apply 'a' we go from 'x' to 'x' but we are not in the state we started in, we can then apply 'b' to get back to where we started. This diagram does not seem to fully capture this behavior so we will elaborate it. First we will represent it as a triangulation.

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.

projective space

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.

Off topic - for information about the tools used to produce this diagram - see page here.

 

projective

This attempts to produce a space with negative curvature everywhere (like a saddle - going up in the x dimension and down in the y dimension). Its not possible for a finite shape to have that property everywhere so I conclude that its not possible to do this in 3D space.

A Minimal Triangulation of the Surface of a Projective Plane

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)

projective plane

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:

projective plane

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

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

 


metadata block
see also:
  • I have put the code here.
Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Mathematics++: Selected Topics Beyond the Basic Courses (Student Mathematical Library) Kantor, Ida.

Chapters:

  1. Measure
  2. High Dimensional Geometry
  3. Fourier Analysis
  4. Representations of Finite Groups
  5. Polynomials
  6. Topology

Chapter 6 - Topology. Contains a relatively gentle introduction to homology.

 

This site may have errors. Don't use for critical systems.

Copyright (c) 1998-2017 Martin John Baker - All rights reserved - privacy policy.