# Geometric realization of topology

## Main Question or Discussion Point

Hello,

Suppose that I have a cell complex and I want to define it's geometric realization, I can do it via mapping such that assign coordinates to 0-cells. however how can i do that for edges, faces and volumes. is there is ageneral formulas for lines, faces and volumes.

Regards

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Hello,

Suppose that I have a cell complex and I want to define it's geometric realization, I can do it via mapping such that assign coordinates to 0-cells. however how can i do that for edges, faces and volumes. is there is ageneral formulas for lines, faces and volumes.

Regards
For a simplicial complex there is a simplicial map of the complex into the standard simplex in n-dimensions - where n is the number of vertices in the complex.

Dear wofsy,

I did not understand your reply " I am not an expert in this field". Actually my problem lies in representing geometrical objects. Suppose that I have a cell complex (cw-complex) such that each cell in this complex can be a assigned geometrical realization.

fro example

vertix ----> point

edge ----> line (curved )

face ----> surface

...

if I am talking about linear complex, it is enough to assign points to vertexes. however suppose that the mesh that I want to represent is curvilinear mesh. this is not possible.

Regards

Dear wofsy,

I did not understand your reply " I am not an expert in this field". Actually my problem lies in representing geometrical objects. Suppose that I have a cell complex (cw-complex) such that each cell in this complex can be a assigned geometrical realization.

fro example

vertix ----> point

edge ----> line (curved )

face ----> surface

...

if I am talking about linear complex, it is enough to assign points to vertexes. however suppose that the mesh that I want to represent is curvilinear mesh. this is not possible.

Regards
Not sure what you are talking about. A linear complex can be embedded in the standard simplex in n dimensions. The algorithm is trivial. But maybe you are trying to do something else.

Dear wofsy,

Can you please direct me to this algorithm and some explanation. It seems that we are talking about different things.

Thanks

The standard n simplex has n+1 vertices and is a subset of R^n. Any subset of vertexes spans a lower dimensional simplex so any linear simplicial complex with n or less vertices is a subset of this one.

Derar Wofsy,

The standard n simplex has n+1 vertices and is a subset of R^n. Any subset of vertexes spans a lower dimensional simplex so any linear simplicial complex with n or less vertices is a subset of this one.
I think we are talking about diffrent things, I am interested in cw-complex. the defenesion of cw-complex is purly based on topology not geometry. so my question is how to add geometric realization to cw-complexes.

Regards

Derar Wofsy,

I think we are talking about diffrent things, I am interested in cw-complex. the defenesion of cw-complex is purly based on topology not geometry. so my question is how to add geometric realization to cw-complexes.

Regards
I didn't mean to give you a complete answer. But a simplicial complex is a cw-complex and it may be that any cw-complex has a simplicial decomposition. I gave you a geometric realization of an arbitrary simplicial complex whether defined topologically or combinatorally.

in general I am not sure - at best you can have an existence theorem