Working through Hatcher...(adsbygoogle = window.adsbygoogle || []).push({});

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

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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?

Loading...

Similar Threads - homology isomorphic reduced | Date |
---|---|

A Is there a natural paring between homology and cohomology? | Feb 12, 2017 |

Meaning of isomorphism/diffeomorphism ## f: R^n\to M^m## | Oct 1, 2015 |

What About Cech Homology? | Sep 16, 2014 |

Why Are Homology Groups Not MUCH Larger? | May 16, 2014 |

Null-homologous framing? | Jan 21, 2014 |

**Physics Forums - The Fusion of Science and Community**