# stable homotopy groups of the projective plane

The projective plane $\mathbb{R}P^2$ is obtained by attaching a $2$-cell to $S^1$ via a degree $2$ map $$S^1\overset{[2]}{\longrightarrow}S^1\longrightarrow\mathbb{R}P^2.$$

Question 1: the degree $2$ map $$[2]: \mathbb{R}P^2\longrightarrow\mathbb{R}P^2,$$ in the stable category, is the composite $$\mathbb{R}P^2\overset{\text{pinch}}{\longrightarrow} S^2\overset{\eta}{\longrightarrow}S^1\longrightarrow \mathbb{R}P^2.$$ Why? What does this mean?

Question 2:
$$S^1\overset{[2]}{\longrightarrow}S^1\longrightarrow\mathbb{R}P^2$$ is a cofibre sequence and there is an associated long exact sequence of stable homotopy groups. Why? What does this mean?

Question 3: apply the above to calculate the stable homotopy groups of $\mathbb{R}P^2$. Could you illustrate how to calculate?