I am learning Postnikov towers from this lecture. Here is the first part of the proof that I am studying

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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!

  • 2
    $\begingroup$ 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. $\endgroup$ – Gustavo Granja Jan 18 at 17:40
  • $\begingroup$ @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. $\endgroup$ – Peter May Jan 18 at 20:01
  • $\begingroup$ @GustavoGranja math.ru.nl/~mgroth/teaching/htpy13/Section11.pdf $\endgroup$ – Marco Francesco Nervo Jan 18 at 20:04
  • $\begingroup$ @PeterMay math.ru.nl/~mgroth/teaching/htpy13/Section11.pdf $\endgroup$ – 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


A less concise but tediously careful proof along the lines you are considering is given in Section 3.5 of More Concise


where the construction is generalized to nilpotent spaces.


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