Fundamental group of complement of image of diagonal embedding from $S_g$ to $S_g \times S_g$?

Question

Let $S_g$ be a surface of genus $g \ge 0$. Let $\Delta \subset S_g \times S_g$ be the image of the diagonal embedding $x \mapsto (x, x)$. Let $X$ be the complement of $\Delta$ in $S_g \times S_g$. My question is, what is $\pi_1(X)$?

Answer 1

This group is known as the "2 strand pure braid group of $S_g$", and is described in Birman's book "Braids, Links, and Mapping Class Groups".

Here's a brief account: using some basic homotopy theory, one can obtain a short exact sequence for $\pi_1(X)$.

There is a fibration $p : X \to S_g$ defined by $p(x,y)=x$ for all $(x,y)$. The fiber over a base point $x_0$ is all pairs $(x_0,y)$ such that $y \ne x_0$, and the fiber is identified with the once punctured surface $S_{g,1}=S_g-\{x_0\}$.

Applying the theorem on the long exact sequence of a fibration we obtain a short exact sequence $$1 \to \pi_1(S_{g,1}) \to \pi_1(X) \to \pi_1(S_g) \to 1 $$ where the kernel is isomorphic to the free group of rank $2g$. This is still not a complete description, but it should get you started.