Computing fundamental group of boundary-identified polygons

Question

Compute the fundamental group of each of the following boundary-identified polygons.

I imagine I need to use Van-Kampen's theorem, but I'm not sure what sort of open sets I should be choosing. Could someone please walk me through one of these?

Answer 1

Hatcher explains how to compute the fundamental group of a CW complex. Each of the three spaces in your problem are the result of taking a 1d CW complex (that is, a topological graph) and gluing the boundary of a 2-cell (a disk) to the 1d complex. A disk $D$ has a cover given by $D-\{x\}$ and $D^{\circ}$, where $x\in D^{\circ}$, and this cover has the nice property that $D-\{x\}$ deformation retracts onto $\partial D$, that $D^{\circ}$ deformation retracts onto $x$, and that $(D-\{x\})\cap D^{\circ}$ deformation retracts onto a space homeomorphic to $S^1$.

Anyway, puncture the interior of a polygon to get one set in the cover, and let the interior of the polygon be the other set. (When there are more polygons, you have to be more careful than just using the interiors of each polygon, for basepoint considerations.)

Let $X$ be the third example, let $x\in X$ be in the interior of the triangle, and let $U\subset X$ be an open disk neighborhood of $x$ in the interior of the triangle (the interior of the triangle itself is sufficient). The space $X-x$ deformation retracts onto the loop labeled $a$, which I say is a loop because the identification diagram implies that all three vertices are the same vertex, hence $\pi_1(X-x)\cong \mathbb{Z}$, which we will say is generated by $a$ itself. The space $U$ is contractible, so $\pi_1(U)$ is trivial. The intersection $(X-x)\cap U$ deformation retracts to a circle, so $\pi_1((X-x)\cap U)\cong\mathbb{Z}$. Let $t$ represent the generator of this fundamental group. Using some basepoint convention, we see that $t$ included into $X-x$ is $aa^{-1}a=a$, and that $t$ included into $U$ is $1$. Hence, by the van Kampen theorem, $\pi_1(X)\cong\langle a\mid a=1\rangle\cong 1$.

For the other cases, since there is a nonempty boundary, you could get away with deformation retracting the polygon onto a graph.