07081300/Class notes for Thursday, January 24
INCOMPLETE AND UNEDITED: Completion coming soon.

Typed Notes
The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.
Proof of Van Kampen
Let G_i = \pi_1(U_i) H = \pi_1(Y_1\cap U_2) G=\pi_1(U_1\cup U_2)
We aim to show that G=G_1*_H G_2
Hence, we want to define two maps:
\Phi:G_1*_H G_2\rightarrow G and
\Psi:G\rightarrow G_1*_H G_2
such that they are inverses of each other.
Now, recall the commuting diagram:
Further, let b_i alternate between 1 and 2 for successive i's.
Hence we define \Phi via for \alpha_i\in G_{b_i},
[\alpha_1][\alpha_2]\ldots[\alpha_n]\rightarrow [j_{b_1 *}\alpha_1\cdot\ldots\cdot j_{b_n *}\alpha_n]
Clearly this is well defined. We need to check the relations in G_1*_H G_2 indeed hold. Well, the identity element corresponds to the identity path so the relation that removes identities holds.