Question about a complex regarded as a topological space

[kakarotyjn]

Definition. A complex K,when regarded as a topological space,is called a polyhedron and written |K|.

I think it is easy to understand the definition,but there are some theorem and problems involving it confused me.

1.Let K be a simplicial complex in $$E^n$$,if we take the simplexes of K separately and give their union the identification topology,then we obtain exactly |K|.

And also I don't understand explicitly about identification topology,I only know a set A in Y is open if and only if $$\pi ^{ - 1} (A)$$ is open in X for which $$\pi$$
is the map from X to the identification space Y.

What exactly does it means by saying 'if we take simplexes of K separately and give their union the identification topology'?What occurs to the simplexes of K by this?

2.Check that |CK| and C|K| are homeomorphic spaces.

Does it need proof? I think |CK|=C|K|,isn't it? If not,what's the difference between them?

Thank you very much!

 

[lavinia]

Definition. A complex K,when regarded as a topological space,is called a polyhedron and written |K|.

I think it is easy to understand the definition,but there are some theorem and problems involving it confused me.

1.Let K be a simplicial complex in $$E^n$$,if we take the simplexes of K separately and give their union the identification topology,then we obtain exactly |K|.

And also I don't understand explicitly about identification topology,I only know a set A in Y is open if and only if $$\pi ^{ - 1} (A)$$ is open in X for which $$\pi$$
is the map from X to the identification space Y.

What exactly does it means by saying 'if we take simplexes of K separately and give their union the identification topology'?What occurs to the simplexes of K by this?

2.Check that v| are homeomorphic spaces.

Does it need proof? I think |CK|=C|K|,isn't it? If not,what's the difference between them?

Thank you very much!

1. The simplices separately each have the topology they inherit as polyhedra in Euclidean space. These polyhedra are then glued together along facets of the simplices to produce the topological realization of the simplicial complex.

2.what are |CK| and C|K?

 

[kakarotyjn]

Thank you,lavinia! It seems a little clearer to me

CK is the cone on K.

 

[lavinia]

2.Check that |CK| and C|K| are homeomorphic spaces.

Does it need proof? I think |CK|=C|K|,isn't it? If not,what's the difference between them?

I think C|K| is the cone on a topological space. |K| denotes a topological space not a simplicial complex.

CK is a simplicial complex not a topological space. |CK| is its topological realization.

 

[kakarotyjn]

I can't figure out why we distinguish the two concepts so preciously,why not regard a simplex not only a simplex but also a topological space?

 

[lavinia]

I can't figure out why we distinguish the two concepts so preciously,why not regard a simplex not only a simplex but also a topological space?

A simplicial complex is a scaffholding overlayed upon a topological space. It is not a topological structure in and of itself. A given space will have infinite many different simplicial decompositions.

Simplicial complexes have there own theory, independent of their topology, e.g. they have homology theories.

An interesting question for manifolds is whether two simplicial decompositions are equivalent i.e. whether each can be subdivided into the same complex. If only the topology were interesting then this would be true. But there are examples where this fails this is, there are simplicial decompositions of the same topological space that are not equivalent.