# Postnikov tower for $S^2$

I am learning Postnikov towers from this lecture. Here is the first part of the proof that I am studying Why is true the marked statement?

For example, let be $$X = S^2$$.

To build $$Y_1$$ (i.e, with killed $$\pi_2$$, $$, \pi_3$$...) I add to $$X$$ a $$3$$-cell glued on a generator of $$\pi_2(S^2) = \mathbb{Z}\cdot[id]$$. So $$Y_1^{(1)} = D^3$$. We have no more homotopy groups to kill, then $$Y_1 = Y_1^{(1)} = D^3$$.

To build $$Y_2$$ I add to $$X$$ a $$4$$-cell to a generator of $$\pi_3(S^2) = \mathbb{Z}\cdot[hopf]$$ and then maybe $$5$$-cells, $$6$$-cells...

Then what is the "canonical inclusion" $$Y_2 \to Y_1$$? I had not adjoined more cells for $$Y_1$$ than for $$Y_2$$ as claimed!

• I suppose that in the construction of the spaces $Y_n^{(k)}$ you don't just attach cells for a set of generators of the homotopy group in question but rather attach a cell for each continuous map from $S^{k}$ to $Y_n^{(k-1)}$. This is the canonical thing to do (it is functorial as it involves no choices). With this construction it is indeed true that $Y_{n+1}$ is contained in $Y_n$ for the reason given. – Gustavo Granja Jan 18 at 17:40
• @Gustavo, right you are, but we (or at least I) don't have access to Lemma 2 in the source: the question has two answers, depending on that. Of course, rectification of this first step to a Postnikov tower is where the real math lies either way. – Peter May Jan 18 at 20:01
• @GustavoGranja math.ru.nl/~mgroth/teaching/htpy13/Section11.pdf – Marco Francesco Nervo Jan 18 at 20:04
• – Marco Francesco Nervo Jan 18 at 20:05

I have no idea what source you are quoting, but you are quite right that it is wrong, unless we are both screwing up. One builds $$\phi_n$$ rigorously by inducting on the stages of the construction of $$Y_{n+1}$$ from $$X$$, using null homotopies of attaching maps. The resulting tower is then corrected to a Postnikov tower. A glib outline construction is given in Section 22.4 of Concise