The projective plane:

We work this example out in meticulous detail. Such details will be progressively cut down and left for the students to fill in as we go along. The projective plane $\mathbb{R}P^2$ is obtained by attaching a two cell $E^2$ to $S^1$ using the map given in complex form as $f(z) = z^2$. Let $p$ denote the center of $E^2$ and $\eta:E^2\longrightarrow \mathbb{R}P^2$ be the quotient map. Taking $U$ to be the interior of $E^2$ and $V = \mathbb{R}P^2 - \{p\}$ we apply corollary (26.7). For computing the image of $i_*$ we take a generator for the infinite cyclic group $\pi_1(U\cap V, y_0)$ with base point $y_0 = 1/2$. The generator is the equivalence class of the loop

$$\gamma(t) = \frac{1}{2}\exp(2\pi it),\quad 0\leq t \leq 1. \eqno(26.5)$$

Figure: Computing $\pi_1(\mathbb RP^2)$

[width=0.4]GKSBook/fig17/fig17.eps

We also need a base point $x_0$ sitting on the loop $\Gamma$ given by

$$\Gamma(t) = \eta(\exp(i\pi t)),\quad 0\leq t \leq 1, \eqno(26.6)$$

which generates $\pi_1(\mathbb{R}P^2 - \{p\}, x_0)$. Taking a path $\beta$ joining $y_0$ and $x_0$ we get a generator for the infinite cyclic group $\pi_1(\mathbb{R}P^2 - \{p\}, y_0)$ namely, the class of the loop $\beta *\Gamma * \beta^{-1}.$ Having set the stage we are ready to compute $i_*[\gamma]$ namely, the homotopy class of the loop $\gamma$ in $\mathbb{R}P^2 - \{p\}$. This loop $\gamma$ based at $y_0$ is homotopic to the loop

$$\beta * \Gamma * \Gamma * \beta^{-1}. \eqno(26.7)$$

The required homotopy is $\eta\circ F$ where $F$ is a map of a rectangle onto a suitable annulus (see exercise (1)). Introducing a $\beta^{-1}*\beta$ we get

$$i_*[\gamma] = [\beta *\Gamma *\beta^{-1}][\beta *\Gamma *\beta^{-1}] \eqno(26.8)$$

or in additive notation it is the map $\mathbb{Z} \longrightarrow \mathbb{Z}$ given by $n\mapsto 2n$. We conclude from corollary (26.7) that $\pi_1(\mathbb{R}P^2)$ is the cyclic group of order two.