1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Non planar simplicial homology?

  1. Oct 27, 2007 #1
    Let's say I have a complex like this:
    http://img267.imageshack.us/img267/5606/nonplanarpfum9.th.png [Broken]

    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.
    Last edited by a moderator: May 3, 2017
  2. jcsd
  3. Oct 27, 2007 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    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
  4. Oct 27, 2007 #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?
  5. Oct 28, 2007 #4

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Non planar simplicial homology?
  1. Planar graph (Replies: 7)

  2. Planar knots in 4-D (Replies: 3)

  3. Homology maps (Replies: 7)

  4. Homology of RP(2) (Replies: 2)