Non planar simplicial homology?

1. Oct 27, 2007

Coolphreak

Let's say I have a complex like this:

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.

2. Oct 27, 2007

matt grime

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.

Last edited: Oct 27, 2007
3. Oct 27, 2007

Coolphreak

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. Oct 28, 2007

matt grime

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.