Postnikov tower of a CW complex is unique up to homotopy equivalence

Question

The above is extracted from Hatcher's Algebraic Topology. I have understood that every connected CW complex has a Postnikov tower, but I can't see how Corollary 4.19 implies that such a tower is unique up to homotopy equivalence.

Thanks in advance.

Answer 1

Does Hatcher say what he means by a tower being unique up to homotopy equivalence? Here is a sketch of a proof that the terms and the maps in the tower are unique up to homotopy, in case that's what you're asking for:

Let $X\langle 0, n \rangle$ be an $(n+1)$-connected CW model for the CW pair $(CX, X)$. This is one choice for the $n$th space in the Postnikov tower. By the corollary, this is unique up to homotopy; in a bit more detail, if $X'\langle 0, n\rangle$ is another CW complex together with a map from $X$ such that

• $X \to X' \langle 0, n \rangle$ induces an isomorphism in $\pi_i$ for $i \leq n$, and
• $\pi_i X' \langle 0, n \rangle = 0$ for $i > n$,

then up to homotopy equivalence, we can replace $X' \langle 0, n \rangle$ by a CW complex $X''\langle 0, n \rangle$ having $X$ as a subcomplex, where the map is the inclusion (via the mapping cylinder). Then $X'' \langle 0,n\rangle$ is also an $(n+1)$-connected cover for $(CX, X)$, so it is homotopy equivalent to $X \langle 0, n \rangle$.

Now given terms $X\langle 0, n+1 \rangle$ and $X \langle 0, n\rangle$ in the Postnikov tower, I would actually appeal to Hatcher's 4.18 rather than its corollary to deduce that there is a map, unique up to homotopy, from one to the other, compatible with the map from $X$.