Non planar simplicial homology?

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SUMMARY

This 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. The participants confirm that computing the first homology group (H_1) is valid regardless of the non-planar embedding, as the definition of homology does not depend on the ambient space. The second homology group (H_2) can also be computed, but the focus remains on the first group due to the presence of three identified holes in the complex. The definition of homology strictly involves the quotient space of the kernel of one map by the image of another, making the embedding dimension irrelevant.

PREREQUISITES
  • Understanding of simplicial complexes and simplices
  • Familiarity with homology theory and its definitions
  • Knowledge of algebraic topology concepts
  • Basic grasp of quotient spaces and kernel-image relationships
NEXT STEPS
  • Study the computation of simplicial homology groups in various dimensions
  • Explore the implications of non-planar embeddings on homology
  • Learn about the relationship between homology and topological spaces
  • Investigate advanced topics in algebraic topology, such as persistent homology
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Mathematicians, particularly those specializing in algebraic topology, students studying topology concepts, and researchers interested in the properties of simplicial complexes and their homological characteristics.

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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