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I am a bit confused about delta complex structure and CW complex structrue.
I was wondering is n-dimensional disks homeomorphic two n dimensional triangles?
I mean if that is the case then I can make every Delta complex structure to a CW complex structure and vice versa. If this is the case one could ask why do we have the delta complex structure? Could the hole homology theory been built from CW complexes? or lets say with some other complex structure i.e from n dimensional squares?

TheGeometer
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    Indeed, you can build homology theory for CW-complexes directly. Delta complexes are useful because they are like simplicial complexes but, because the gluing conditiens are weaker, allow us to construct spaces using fewer cells. On the other hand, CW complexes allow much greater liberty at how you glue things and that can complicate some other things. You pick the construction thatdoes what you need in the situation you want to handle, and that is why it is good to have options. – Mariano Suárez-Álvarez Mar 12 '15 at 22:25
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    There is something in between delta complexes and CW-complexes: regular CW-complexes. They are often quite nice. – Mariano Suárez-Álvarez Mar 12 '15 at 22:28
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    Every Delta complex can be expressed as a CW complex, indeed. One major difference is that Delta complexes are much more combinatorial than CW complexes. Actually, a Delta complex is just a bunch of discrete data, and the space we equip with that structure is the realization of the Delta complex. That way, simplicial homology can be defined purely combinatorial and does not involve any topology actually. – Stefan Hamcke Mar 12 '15 at 22:29
  • I see, thanks a lot good to clarify that. – TheGeometer Mar 12 '15 at 22:32

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