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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.