Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Why is homology isomorphic to reduced homology plus Z?

  1. Jul 5, 2011 #1
    Working through Hatcher...
    For any space X, we have an augmented chain complex

    [itex]...\rightarrow C_1(X) \rightarrow C_0(X)\rightarrow \mathbb{Z}\stackrel{\epsilon}{\rightarrow}0[/itex]
    Hathcer says that since [itex]\epsilon[/itex] induces a map [itex]H_0(X)\rightarrow \mathbb{Z}[/itex] with kernel [itex]\tilde{H}_0(X)[/itex], we get an isomorphism [itex]H_0(X)\simeq \tilde{H}_0(X)\oplus \mathbb{Z}[/itex]

    Where is this isomorphism coming from? I understand where the induced map on [itex]H_0(X)[/itex] comes from...

    Thanks
     
  2. jcsd
  3. Jul 5, 2011 #2

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    There is a short exact sequence 0-->H(reduced)_0-->H_0-->Z-->0, and Z being free, it splits. That is, H_0=H(reduced)_0 x Z.
     
  4. Jul 7, 2011 #3
    thanks!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Why is homology isomorphic to reduced homology plus Z?
  1. Homology question (Replies: 2)

  2. Simplicial homology (Replies: 2)

Loading...