So I'm having a little trouble with the part of Van Kampen's theorem my professor presented to us. He called this the easy 1/2 of Van Kampen's theorem.(adsbygoogle = window.adsbygoogle || []).push({});

Theorem (1/2 of Van Kampen's) - Let X_{,x}=A_{,x}U B_{,x}(sets with basepoint x) where A and B are open in X and A[itex]\bigcap[/itex]B is path-connected. Then [itex]\pi[/itex]_{1}(X) is generated by [itex]\pi[/itex]_{1}(A) and [itex]\pi[/itex]_{1}(B).

[itex]\pi[/itex]_{1}(A) and [itex]\pi[/itex]_{1}(B) are not necessarily subsets of [itex]\pi[/itex]_{1}(X), at least in general. So if anyone can enlighten me on exactly what the Professor meant. I would think he just means the embedding of [itex]\pi[/itex]_{1}(A) and [itex]\pi[/itex]_{1}(B) in [itex]\pi[/itex]_{1}(X) but I don't think, at least in general, the homomorphism induced by the inclusion is injective.

Thanks very much.

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

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

# Van Kampen's Theorem

Loading...

Similar Threads - Kampen's Theorem | Date |
---|---|

A Stokes' theorem on a torus? | Apr 27, 2017 |

I Proof of Stokes' theorem | Apr 8, 2017 |

I Generalisation of Pythagoras theorem | Jul 12, 2016 |

A Non-Abelian Stokes theorem and variation of the EL action | May 31, 2016 |

Another implememntation of van Kampen thoerem. | Sep 27, 2008 |

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