Non planar simplicial homology?

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Discussion Overview

The discussion centers on the computation of homology groups for a simplicial complex that is defined in a planar manner but exists in three-dimensional space. Participants explore whether the non-planarity of the complex affects the calculation of the first and second homology groups, particularly in the context of identifying "holes" in the complex.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions if computing the first homology group would be valid in a non-planar case, noting the presence of three "holes" in the complex.
  • Another participant asserts that the embedding of a simplicial complex into a space does not affect homology theory, suggesting to compute H_1 regardless of the embedding.
  • A further inquiry is made about whether self-intersecting edges in non-planar graphs impact the computation of the first homology group, emphasizing the importance of maintaining correct dimensional chains.
  • One participant clarifies that the definition of homology does not consider the embedding in an ambient space, stating it is based on the quotient space of the kernel of one map by the image of another.

Areas of Agreement / Disagreement

Participants express differing views on the implications of non-planarity for homology computations. While some argue that embedding does not affect the calculations, others raise questions about specific cases, indicating that the discussion remains unresolved.

Contextual Notes

There are unresolved assumptions regarding the nature of the simplicial complex and the specific definitions of homology being applied. The discussion also touches on the implications of self-intersecting edges and their role in homology calculations.

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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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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