TITLE: Induced maps on homotopy groups by self maps of $\mathbb{CP}^n$
QUESTION [11 upvotes]: Let $f:\mathbb{S}^2\to \mathbb{S}^2$ with degree $d$.
It is well known that the induced map $$f_\ast:\pi_3(\mathbb{S}^2)=\mathbb{Z}\to \pi_3(\mathbb{S}^2)=\mathbb{Z}$$ is given by multiplication by $d^2$.
But, $\mathbb{S}^2=\mathbb{C}P^1$ and $[\mathbb{C}P^n,\mathbb{C}P^n]=\mathbb{Z}$ for $n\geq 1$ and for any $f:\mathbb{C}P^n\to \mathbb{C}P^n$ determined by $d\in \mathbb{Z}$ the induced map $$f_\sharp: H_{2k}(\mathbb{C}P^n)=\mathbb{Z}\to H_{2k}(\mathbb{C}P^n)=\mathbb{Z}$$ is given by multiplication by $d^k$ for $1\leq k\leq n$.
Question: is it true that the map $$f_\ast : \pi_{2n+1}(\mathbb{C}P^n)=\mathbb{Z}\to \pi_{2n+1}(\mathbb{C}P^n)=\mathbb{Z}$$ is determined by multiplication by $d^{n+1}$?

REPLY [5 votes]: This is just to provide a bit of background to Dmitri's elegant answer.
You can do this by induction by recognising that $\mathbb{C}P^n$ is the $n^{th}$ projective plane of the $H$-space $S^1$, and in particular may be constructed using the Hopf construction. This iteratively produces quasi-fibrations $\gamma_n:\ast^nS^1\rightarrow \mathbb{C}P^{n-1}$, starting with $n=1$, where $\ast^nS^1$ is the $n$-fold join of $S^1$, and defines $\mathbb{C}P^n$ as the cofiber
$$\ast^nS^1\xrightarrow{\gamma_n} \mathbb{C}P^{n-1}\rightarrow\mathbb{C}P^n.$$
Note that the degree $d$ self map $d:S^1\rightarrow S^1$ is homotopic to the $d$-fold power map (defined with the Lie product), and since $S^1$ is abelian this map is an $A_{\infty}$-map. It follows that there is an induced map $\underline d_n:\mathbb{C}P^n\rightarrow \mathbb{C}P^n$ between the Hopf constructions at each stage.
In particular, at the $(n+1)^{th}$ stage we have $\ast^{(n+1)}S^1\cong \Sigma^n\bigwedge^{n+1}S^1\cong S^{2n+1}$ and $\ast^{n+1}d\cong \Sigma^n\bigwedge^{n+1}d\simeq d^{n+1}$ sitting in a diagram of cofibrations
$\require{AMScd}$
\begin{CD}
    S^{2n+1}@>\gamma_{n+1}>> \mathbb{C}P^n@>>>\mathbb{C}P^{n+1}\\
    @Vd^{n+1}  V V @VV \underline d_n  V@VV \underline d_{n+1} V\\
    S^{2n+1} @>\gamma_{n+1}>> \mathbb{C}P^n@>>>\mathbb{C}P^{n+1}.
\end{CD}
Technically, it now remains to confirm that $\underline d_n$ is indeed the map you describe, but its Friday and the pub's open, so I'm going to stop typing here.