Equivalence relation on a 3-genus surface.

Question

Construct a genus-3 surface as follows

1. First let $T_1$ be a torus with a hole in it, i.e. that the boundary of $T_1$ is $S_1$ which we will call $\Gamma_1$
2. Next make a copy $T^{\prime}_1$ with boundary circle $\Gamma_2$
3. Now let $T_2$ be a torus with two holes in it,
4. Now "stitch together" $T_2$ by identifying its boundary to the circles $\Gamma_1$ and $\Gamma_2$. element-wise. Call this space $\Sigma$

I have the following equivalence relation $\sim$: Let $p\in T_2\setminus \Gamma_1 \cup \Gamma_2, x\sim y$ if there is a path on $\Sigma \setminus \Gamma_1 \cup \Gamma_2$ from $x$ to $y$ through $p$. Put all points path connected to $p$ and all points from $ \Gamma_1 , \Gamma_2$ into the class $[p]$ and all other points are left as singletons.

First I must draw a picture of the space $\Sigma /\sim$, mine is an image of two torus' (tori?) joined at a single point.

I then must describe 4 distinct loops on the quotient space from 1 point who's path-homotopy classes are distinct (I do not need to prove this), I chose to go around the outside and inside of each torus, starting at point [p].

Finally I must describe the fundamental group $\pi_1( \Sigma /\sim)$. This is where I'm struggling as I don't know what to do for this, I'm assuming Van Kampen will help but I don't know what partition to make.

Is my working so far ok and what partitions will (nicely) work for Van Kampen?

Answer 1

For Van Kampen, you want to take $T_1 \cup$ a small open neighbourhood of [p] in $T'_1$, and $T'_1 \cup$ a small open neighbourhood of [p] in $T_1$. Then you perform the pushout from their intersection. This is a very applicable technique for finding the fundamental groups of spaces (that you want to use Van Kampen on.)

The intersection is an interesting one. I think it is homotopy retact to a single point, and so it trivial, so the push out is a free product, so nice enough to calculate (I'll leave that to you.)

This has been a very interesting space to consider, thank you for posting about it. Your working, by my reckoning is sound (though only you know if it's rigourous enough!)