1.0 KiB
1.0 KiB
Math4202 Topology II (Lecture 30)
Algebraic Topology
We skipped a few chapters about Jordan curve theorem, which will be your final project soon. LOL, I will embedded the link once I'm done.
Seifert-Van Kampen Theorem
The Seifert-Van Kampen Theorem
Let X=U\cup V be a union of two open subspaces. Suppose that U\cap V, U,V are path connected. Fix x_0\in U\cap V.
Let H be a group (arbitrary). And now we assume \phi_1,\phi_2 be a group homomorphism, and \phi_1:\pi_1(U,x_0)\to H, and \phi_2:\pi_1(V,x_0)\to H.
Let i_1,i_2,j_1,j_2,i_{12} be group homomorphism induced by the inclusion maps.
Assume this diagram commutes.
\phi_1\circ i_1=\phi_2\circ i_2
There is a group homomorphism \Phi:\pi_1(X,x_0)\to H making the diagram commute. \Phi\circ j_1=\phi_1 and \Phi\circ j_2=\phi_2.
We may change the base point using conjugations.