Non planar simplicial homology?

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In summary, the conversation discusses a complex with nodes defined on a plane but actually exists in 3 dimensions, and the question of whether computing the 1st or 2nd homology group would work for finding holes in the complex. It is clarified that the embedding dimension does not affect the computation of the first homology group, as long as the chains are of the correct dimension. The definition of homology does not consider the embedding of the complex in an ambient space.
  • #1
Coolphreak
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Let's say I have a complex like this:
http://img267.imageshack.us/img267/5606/nonplanarpfum9.th.png

The original nodes are defined on the same plane, but the actual complex exists in 3 dimensions. If I want to find holes in the complex, would computing the 1st homology group work in this non planar case? There are 3 "holes" in this complex. Would computing the 2nd homology group work? However, I do not have any 3D components, just 2D components embedded in 3 space.
 
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  • #2
A simplicial complex is a set of simplices. The embedding of it into some space is immaterial (or it wouldn't be a very good homology theory). Just work out H_1.
 
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  • #3
so even with self intersecting edges (non planar graphs), the first homology group can be correctly computed no matter the embedding dimension? As long as the chains are of the correct dimension?
 
  • #4
Does the definition of homology have anything to do with its embedding in an ambient space? No. There is absolutely nothing in the definition that even takes this into account. It is the quotient space of the kernel of one map by the image of another.
 

1. What is non planar simplicial homology?

Non planar simplicial homology is a mathematical concept used in topology to study the properties of non-planar, or curved, surfaces. It involves breaking down a curved surface into smaller, simpler pieces called simplices, and then analyzing the relationships between these pieces to understand the overall structure of the surface.

2. How does non planar simplicial homology differ from planar simplicial homology?

The main difference between non planar and planar simplicial homology is the shape of the surface being studied. Planar simplicial homology is used for flat, two-dimensional surfaces, while non planar simplicial homology is used for non-planar, three-dimensional surfaces.

3. What is the significance of studying non planar simplicial homology?

Studying non planar simplicial homology allows for a deeper understanding of the topology, or shape, of curved surfaces. It also has practical applications in fields such as computer graphics and physics, where curved surfaces are often encountered.

4. Can non planar simplicial homology be applied to real-world objects?

Yes, non planar simplicial homology can be applied to real-world objects that have curved surfaces, such as spheres, tori, or other three-dimensional shapes. By breaking down these surfaces into smaller simplices, we can gain a better understanding of their underlying structure.

5. Are there any limitations to using non planar simplicial homology?

One limitation of non planar simplicial homology is that it only applies to surfaces that can be decomposed into simplices, and therefore cannot be used for more complex shapes such as fractals. Additionally, the accuracy of the analysis may be limited by the resolution of the simplices used.

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