TITLE: Determining homotopy classes [T^2, RP^2] 
QUESTION [11 upvotes]: So I've been interested in computing homotopy classes of maps $T^2=S^1\times S^1$ to $\mathbb{R}\mathbb{P}^2$. So first, we can decompose $T^2$ into a cell complex with one zero cell, $S^1\vee S^1$ and a disc $D^2$ glued down to $S^1\vee S^1$ using the commutator. Since the commutator vanishes in $\pi_1(\mathbb{R}\mathbb{P}^2)$ given any map $S^1\vee S^1\rightarrow \mathbb{R}\mathbb{P}^2$, one can extend this to a map $T^2\rightarrow \mathbb{R}\mathbb{P}^2$ by the map $D^2\rightarrow T^2$, but since the gluing map is nullhomotopic, this is the same thing as a map $S^2\rightarrow \mathbb{R}\mathbb{P}^2$ and there are $\mathbb{Z}$ homotopy classes of maps like this. Therefore we can write (as a set) $[T^2,\mathbb{R}\mathbb{P}^2] = \{a,b,c,d\}\times \mathbb{Z}$ where each of the $\{a,b,c,d\}$ is a homotopy class $S^1\vee S^1\rightarrow \mathbb{R}\mathbb{P}^2$. My question is given an arbitrary map $f:T^2\rightarrow \mathbb{R}\mathbb{P}^2$ is there some sort of computation we can do in $\mathbb{R}\mathbb{P}^2$ to determine which homotopy class $f$ belongs to or at least which of the $a,b,c,d$ it belongs to?

REPLY [12 votes]: For dimension reasons, the set of homotopy classes of maps between these spaces can be computed by using their fundamental crossed module. Despite the question has already been answered twice, let me explain this approach, just for those who may like it.
A crossed module $C_*$ is a group homomorphism $\partial\colon C_2\rightarrow C_1$ together with an action of $C_1$ on $C_2$, that we write exponentially $c_2^{c_1}$, such that 
$$\begin{array}{rl}
\partial(c_2^{c_1})&=-c_1+\partial(c_2)+c_1,\\\
c_2^{\partial(c_2')}&=-c_2'+c_2+c_2'.
\end{array}$$
Here we use additive notation despite the groups may be nonabelian.
The canonical (topological) example of a crossed module is the boundary map in the long exact sequence of a pair of spaces:
$$\partial\colon\pi_2(X,Y)\longrightarrow\pi_1(X).$$
The fundamental crossed module of CW-complex $X$ is obtained in this way for the pair formed by $X$ and its $1$-skeleton.
$$\partial\colon\pi_2(X,X^1)\longrightarrow\pi_1(X^1).$$
Notice that $\ker\partial\cong\pi_2(X)$. The image of $\partial$ is normal in any crossed module, and in this case $\operatorname{coker} \partial\cong\pi_1(X)$.
A morphism of crossed modules $f_{*}\colon C_{*}\rightarrow D_{*}$ is a commutative square
$$\begin{array}{rcccl}
&C_1&\stackrel{\partial}\longrightarrow&C_2\\\
{\scriptstyle f_2}&\downarrow&&\downarrow&{\scriptstyle f_1}\\\
&D_1&\stackrel{\partial}\longrightarrow&D_2
\end{array}$$
such that $f_2(c_2^{c_1})=f_2(c_2)^{f_1(c_1)}$. Notice that any cellular map between CW-complexes $X\rightarrow Y$ induces a morphism between their fundamental crossed modules.
Two such morphisms $f_{*},g_{*}\colon C_{*}\rightarrow D_{*}$ are homotopic if there exists a map $H\colon C_1\rightarrow D_2$ such that 
$$\begin{array}{rl}
H(c_1+c_1')&=H(c_1)^{f(c_1')}+H(c_1'),\\\
\partial H(c_1)&=-f_1(c_1)+g_1(c_1),\\\
 H\partial(c_2)&=-f_2(c_2)+g_2(c_2).
\end{array}$$
If $X$ and $Y$ are CW-complexes and $X$ is $2$-dimensional the set of homotopy classes $[X,Y]$ can be computed as the set of algebraic homotopy classes between their fundamental crossed modules.
The fundamental crossed module of $T^2$ is isomorphic to $\partial\colon G'\hookrightarrow G=\langle a,b\rangle$. Here $G$ os a free group on two generators, $G'$ is its commutator subgroup, and $\partial$ is the inclusion. The action of $G$ on $G'$ is by conjugation.
The fundamental crossed module of $\mathbb{R}P^2$ is even easier: $\partial=2\cdot\varepsilon\colon \mathbb{Z}[\mathbb{Z}/2]\rightarrow \mathbb{Z}$. Here $\varepsilon\colon \mathbb{Z}[\mathbb{Z}/2]\rightarrow \mathbb{Z}$ is the augmentation map of the group ring and $\mathbb{Z}$ acts on $\mathbb{Z}[\mathbb{Z}/2]$ via the natural projection $\mathbb{Z}\twoheadrightarrow \mathbb{Z}/2$.
It is a beautiful exercise to compute the set of homotopy classes of maps between these two crossed modules. It's easy since the second one is made of abelian groups! If you're given a map $T^2\rightarrow\mathbb{R}P^2$ and you manage to find a homotopic cellular map (e.g. using the proof of the cellular approximation theorem), then you can say what homotopy class you started with by looking at the induced morphism on the level of crossed modules.