# The Fundamental Group of the Real Projective Plane

/## An important observation

To make our application of van Kampen a little easier, we start with a simple observation:

*projective plane - disk = Möbius strip*

We can also visualize the above by putting a CW-complex structure* on the projective plane and then removing the disc:

## Applying van Kampen

*that*we must amalgamate, i.e. list the generators and relations of $\pi_1(A)$ and $\pi_1(B)$ along with

*new*relations which we find by looking at $\pi_1(A\cap B)$**:

*one*loop around $\gamma$ corresponds to

*two*loops around $b$! (This is shown near the 2:30 mark in the video above.) Hence $$i_{B_*}([\gamma])=b^2.$$ This allows us to write down the final presentation of $\pi_1(\mathbb{R}P^2)$, namely

QED!

*Footnotes:*

* with one 0-cells, one 1-cell, and one 2-cell which is glued onto the 1-cell according to the arrows indicated in the drawing.

** More generally, if we let $R_A$ denote the set of relations of the generators of $\pi_1(A)$ (and similarly for $R_B$) and if $\pi_1(A)=\langle \alpha_1,\ldots,\alpha_n|R_A\rangle$ and $\pi_1(B)=\langle \beta_1,\ldots,\beta_m|R_B\rangle$ then the amalgamation process tells us that for $X=A\cup B$, $$\pi_1(X)=\langle \alpha_1,\ldots,\alpha_n,\beta_1,\ldots,\beta_m\;|\; R_A,\: R_B,\: i_{A_*}([\gamma_1])=i_{B_*}([\gamma_1]), \ldots,i_{A_*}([\gamma_t])=i_{B_*}([\gamma_t]) \rangle$$ where $\gamma_1,\ldots,\gamma_t$ are the generators of $\pi_1(A\cap B)$ and $i_{A_*}:\pi_1(A\cap B)\to\pi_1(A)$ and $i_{B_*}:\pi_1(A\cap B)\to\pi_1(B)$ are the homomorphisms induced by the injection maps $i_A:A\cap B\hookrightarrow A$ and $i_B:A\cap B\hookrightarrow B$, respectively. (Recall, any map $\varphi:X\to Y$ induces a homomorphism $\varphi_*:\pi_1(X)\to\pi_1(Y)$ given by $\varphi_*([f])=[\varphi\circ f]$. ) The only real tricky part here is figuring out what the maps $i_{A_*}$ and $i_{B_*}$ should be.