Delta vs simplicial complexes

[ForMyThunder]

What is the difference between a delta-complex and a simplicial complex? Hatcher's book says that simplicial complexes are uniquely determined by their vertices. Could someone clarify this? Thanks.

 

[quasar987]

Well, a simplicial complex can be specified as a set of vertices, together with the specification of which vertices are to make up a simplex. From this raw data, one can then construct a topological space by gluing simplices accordingly. If there is a homeomorphism btw this space and a space X, this is called a triangulation of X.

The main difference btw simplicial and delta complexes is that in simplicial complexes, there is the restriction that two simplices must intersect in a common face (or not at all), whereas delta complexes do not have this restriction. So a delta-triangulation on a space X will typically have less triangles than a triangulation, and is so it is easier to find one, and computations (such as the Euler characteristic or the homology) are easier to perform.

For instance, the minimal delta-triangulation on the 2-torus has only 2 2-triangles, 3 1-triangles, and 1 0-triangle. The minimal "simplicial triangulation" of the torus has... well, I don't know, but the most obvious (to me) triangulation of the 2-torus has 18 2-triangles alone. (http://rip94550.files.wordpress.com/2008/07/triangulation-18.png)

 

[micromass]

Hi ForMyThunder!

The standard n-simplex has vertices (1,0,0,...,0),(0,1,0,...,0),...,(0,0,0,...,1). Now, given a map $\sigma:\Delta^n\rightarrow X$ of our Delta-complex, then we can call

$$\sigma (1,0,0,...,0),~\sigma (0,1,0,...,0),...,\sigma (0,0,0,...,1)$$

are the vertices of these maps. A simplicial complex is such that no two maps $\sigma_\alpha$ and $\beta$ have the same set of vertices!

For example, consider the square [0,1]x[0,1]. Then the points (0,0),(1,0) and (0,1) form a triangle which is homeomorphic to $\Delta^2$, so take that as a first map. The points (1,0),(0,1) and (1,1) also determine a map. Continuing further gives us a simplicial complex, because every collection of points belongs to at most 1 map.

However, if we would take another map from $\Delta^2$ to the triangle (0,0), (1,0), (0,1) and adjoing it to our complex, then there would be two maps with vertices (0,0), (1,0) and (0,1). This would not form a simplicial complex.

Hope that helped!

 

[ForMyThunder]

Thanks! I understand now.

 

[blue_raver22]

A delta-complex and a simplicial complex are both types of geometric structures used in mathematics and topology to study spaces and their properties. While they are similar in some ways, there are also key differences between them.

A simplicial complex is a collection of geometric objects called simplices that are glued together along their faces. These simplices can be thought of as higher dimensional versions of triangles, with a simplex of dimension n having n+1 vertices. A simplicial complex is uniquely determined by its vertices, meaning that if you know the set of vertices and how they are connected, you can construct the entire complex.

On the other hand, a delta-complex is a more general type of structure that allows for more flexibility in the shapes and connections between its elements. It is made up of simplices, but these simplices can also be distorted or stretched in various ways, allowing for more complex and diverse shapes to be formed. Unlike simplicial complexes, delta-complexes are not uniquely determined by their vertices, as the same set of vertices can be used to construct multiple different delta-complexes.

In summary, the main difference between a delta-complex and a simplicial complex lies in their level of flexibility and the uniqueness of their construction. While simplicial complexes are rigidly determined by their vertices, delta-complexes allow for more variation and are not uniquely determined by their vertices alone.