Geometric realization of topology

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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
 

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  • #2
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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.
 
  • #3
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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
 
  • #4
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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.
 
  • #5
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Dear wofsy,

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

Thanks
 
  • #6
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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.
 
  • #7
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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
 
  • #8
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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
 

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