Non planar simplicial homology?

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In the discussion on non-planar simplicial homology, participants explore the implications of computing homology groups for a complex defined in a 2D plane but existing in 3D space. The first homology group, H_1, can be computed accurately despite the complex's non-planar nature, as the definition of homology does not depend on the embedding in an ambient space. The presence of self-intersecting edges does not affect the computation of H_1, provided the chains are of the correct dimension. The second homology group, H_2, is also considered, but the focus remains on the validity of H_1 calculations in this context. Overall, the discussion emphasizes that homology theory is robust against the complexities of spatial embedding.
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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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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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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?
 
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.
 

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