Consider the usual cell structure on $\mathbb C \mathbb P^\infty$. The skeleta are the $\mathbb C \mathbb P^n$’s, and there is one cell in each even degree. So we have cofiber sequences $S^{2n+1} \to \mathbb C \mathbb P^n \to \mathbb C \mathbb P^{n+1}$.

Each of these attaching maps $S^{2n+1} \to \mathbb C \mathbb P^n$ is at least stably nilpotent. This follows easily from the nilpotence theorem: $MU_\ast(\mathbb C \mathbb P^n) \to MU_\ast(\mathbb C \mathbb P^{n+1})$ is injective because $MU$ is complex-oriented, so the attaching map is zero on $MU_\ast$; by the nilpotence theorem this implies that the attaching map is stably nilpotent (some suspension of some smash power of the map is zero).

Question 1: Are the attaching maps $S^{2n+1} \to \mathbb C \mathbb P^n$ unstably nilpotent?

Question 2: Is there some way to see that the attaching maps $S^{2n+1} \to \mathbb C \mathbb P^n$ are stably nilpotent without invoking the nilpotence theorem?

Question 3: How many smash powers do we need to make $S^{2n+1} \to \mathbb C \mathbb P^n$ become stably null? How many suspensions?

Notes:

• When $n = 1$, the attaching map $S^3 \to \mathbb C\mathbb P^1$ is the Hopf fibration $\eta$. It’s obvious that $\eta$ is stably nilpotent because it’s an odd-degree class in the stable homotopy groups of the sphere, which form a graded-commutative ring. (Er — maybe that’s not quite right — $\eta$ is 2-torsion after all... but at any rate this follows from Nishida nilpotence)

• I’m actually most interested in Question 2. The other questions are just there because they are more precise questions whose answer would seem to require thinking past the nilpotence theorem :)

• I should probably observe somewhere here that the map in question $S^{2n+1} \to \mathbb C \mathbb P^n$ is the quotient map usually used to define $\mathbb C \mathbb P^n$. It’s a fiber bundle with fiber $S^1$. I’m not sure if these facts are useful...