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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?

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