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 $\eta$. That $\eta$ is stably nilpotent follows from Nishida nilpotence because it’s a positive-degree class in the stable homotopy groups of the sphere.
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...
