Delta Complexes

Delta Complex

When we looked at cell complexes we got a chain of 'face maps' between each dimension and the next lower one. homology face maps

In homology we treat this as a chain of abelian groups (more detail on the page here).

Sometimes, rather than defining faces, of any dimension, by their vertices It is useful to define each dimention in terms of the next lower dimenstion, for example, when generating homotopy groups such as the fundamental group.

To do this we represent the complex as a sequence of 'face maps' each one indexed into the next.

Geometric Realisation and Delta Complexes

There is a close connection between geometric realisation and delta complexes.

When we embed a simplicial set or cell complex into a topological space we have a dimension for each vertex and the vertex is represented by a vector from 0 to 1 in that dimension. We can do the same for lines, triangles ... There in now a linear mapping between vertices, lines, triangles... (that is a matrix). So a topological shape is now represented by a sequence of matrices.


More information:


The ordering of the indexes is important. The indexes of any face are always ordered acording to their order in the next order face.

As an example consider the representation for a single tetrahedron:

The usual representation would be: (1,2,3,4)
which is very efficient but it does not allow us to refer to its edges and triangles.  

Here we have indexed:

  • The 4 vertices (in red)
  • The 6 edges (in green)
  • The 4 triangles (in blue)
indexed representation
This representation holds all these indexes so that we don't have to keep creating them and they can be used consistently.

Face Maps

So the tetrahedron indexes its 4 triangles, each triangle indexes its 3 edges and each edge indexes its 2 vertices.

Each face table therefore indexes into the next. To show this I have drawn some of the arrows (I could not draw all the arrows as that would have made the diagram too messy).

indexed table

Since edges, triangles, tetrahedrons, etc. are oriented we need to put the indexed in a certain order.

Edges are directed in the order of the vertex indices. That is low numbered index to high numbered index: edge numbering

For triangles we go in the order of the edges, except the middle edge is reversed.

On the diagram the edges are coloured as follows:

  • green arrows - not reversed
  • purple arrows - reversed

This gives the triangles a consistent winding

indexed triangle
However the whole tetrahedron is not yet consistent because some adjacent edges go in the same direction and some go in opposite directions. indexed tetrahedron
This gives the orientation of the faces as given by the right hand rule. That is: if the thumb of the right hand is outside the tetrahedion, pointing toward the face, then the fingers tend to curl round in the direction of the winding. right hand rule

The higher order faces may continue on in this manor, however it is difficult to visualize.

So to represent this structure we use the following:

Rep := Record(VERTS:VS,SIMP:List(List(Integer)))



Here is an example of a tetrahedron (2-sphere). indexed table
(1) -> DELTA := DeltaComplex(VertexSetAbstract)

   (1)  DeltaComplex(VertexSetAbstract)
                                                                   Type: Type
(2) -> b4 := sphereSolid(3)$SimplicialComplexFactory

   (2)  [1  2  3  4]
                             Type: FiniteSimplicialComplex(VertexSetAbstract)
(3) -> c4 := deltaComplex(b4)

   edge:[[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]] triangle:
                                        Type: DeltaComplex(VertexSetAbstract)








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


  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.


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