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Saturday, June 16, 2018

The Fundamental Group of a Sphere with Two Points Identified.

This is exercise 1.2.7. from Hatcher's book.
Lemma (Proposition 1.26.(a), Page 50 of Hatcher's Algebraic Topology).
If $Y$ is a cell complex obtained from a cell complex $X$ by attaching $2-$cells $e_{\alpha}^2$, $\alpha\in \Lambda$ by attaching maps $\phi_\alpha: S^1\to X$, then the inclusion $X\hookrightarrow Y$ induces a surjection $\pi_1(X,x_0)\to \pi_1(Y,x_0)$ whose kernel is $N=\langle[\gamma_\alpha \phi_\alpha\overline{\gamma}_\alpha]\mid \alpha\in \Lambda, \gamma_\alpha:x_0\to \phi_\alpha(1)\rangle$  where the $\phi_{\alpha}$ are seen as loops.
Exercise. Let $X$ be the quotient space of $S^2$ obtained by identifying the north and south poles to a single point. Put a cell complex structure on $X$ and use this to compute $\pi_1(X)$.

Solution. We can give $S^2$ the usual cell structure with one $0-$cell $a$ and one $2-$cell attached by the only map $S^2\to \{a\}$. Instead, we can give $S^2$ another cell structure from $S^1$ pasting two $2-$cells to the $1-$cell structure of $S^1$. We can also define a $2-$cell structure on $S^2$ by starting with two points $n,s$, two $1-$cells $\gamma,\delta$ between $n$ and $s$, and two $2-$cells pasted into $\alpha\beta$. This last approach works, since then we are giving $X$ the structure of the quotient complex $S^2/\{n,s\}$. It also gives a cell complex structure to $S^1$ starting with two points and attaching two $1-$cells.

We can instead think of it geometrically. When identifying the poles of the sphere with a point the resulting figure seems like a torus with a generating circle identified to a point. This determines the cell structure shown by the next drawing:
Here we have $X$ built from $S_1=\text{im}\alpha$ by attaching a $2$-cell by the map $\phi=\alpha\overline{\alpha}$ which is nullhomotopic. So, by the lemma, the homomorphism
$$\pi_1(S^1,v_1)\to \pi_1(X,v_1)$$
induced by the inclusion is an isomorphism, and $\pi_1(X,v_1)\cong \bZ$.
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