Homotopy equivalence between connected n-dimensional CW complexes

Question

I have a question on the following Exercise 4.2.13 on Hatcher's book:

Show that a map $f:X\rightarrow Y$ between connected $n$-dimensional CW complexes is a homotopy equivalence if it induces an isomorphism on $\pi_i$ for $i\leq n$. [Pass to universal covers and use homology.]

Following the hint, I lift the $f$ to $\tilde{f}:\tilde{X}\rightarrow\tilde{Y}$ between the universal covering spaces $\tilde{X}$ and $\tilde{Y}$. Due to the simply-connectedness and finite dimensions of the universal coverings, it is sufficient to prove that $H_i(\tilde{X})\cong H_i(\tilde{Y})$ by $\tilde{f}_*$ for $i\leq n$. Then $\tilde{f}$ induces a homotopy equivalence between $\tilde{X}$ and $\tilde{Y}$ and $f$ induces a homotopy equivalence by the Hurewicz Theorem.

However, I have a difficulty in $\tilde{f}_*:H_n(\tilde{X})\cong H_n(\tilde{Y})$ and how to prove it?

Answer 1

I would say you have to use two things :

1- That the Hurewicz morphism is a morphism of long exact sequences for a pair $(X,A)$ of spaces (apply this to, e.g. $(M_{\tilde f},\tilde X)$, where $M_{\tilde f}$ is a mapping cylinder for $\tilde f$)

2- That the Hurewicz theorem is slightly more precise than "if all homotopy groups $\leq n-1$ vanish, then it's an iso in degree $n$" : it also says that in this case, the Hurewicz morphism is surjective in degree $n+1$.

So then you have the following commutative diagram (where I assume, wlog thanks to the cylinder, that $\tilde f : \tilde X\to \tilde Y$ is the inclusion of a subspace):

$\require{AMScd}\begin{CD}\pi_{n+1}(\tilde Y,\tilde X)@>>> \pi_n(\tilde X)@>>> \pi_n(\tilde Y) @>>> \pi_n(\tilde Y,\tilde X)\\ @VVV @VVV@VVV@VVV \\ H_{n+1}(\tilde Y,\tilde X) @>>> \tilde H_n(\tilde X)@>>> \tilde H_n(\tilde Y)@>>> H_n(\tilde Y,\tilde X)\end{CD}$

I have hidden the rest of the details so that you can try to write it down yourself.

You know that the two outermost maps in the top row are $0$ because $\tilde f$ is an iso on $\pi_n$, that the rightmost vertical map is an iso, that $\pi_n(\tilde Y,\tilde X)=0$ and that the leftmost vertical map is an epimorphism. This is enough to conclude.

${}$

Indeed it follows that $H_n(\tilde Y,\tilde X) = 0$; and then let $x\in H_{n+1}(\tilde Y,\tilde X)$, it comes from $y\in \pi_{n+1}(\tilde Y,\tilde X)$ by the epimorphism, which is sent to $0$ in $\pi_n(\tilde X)$, and then to $0$ in $\tilde H_n(\tilde X)$, so by commutativity, $x$ is also sent to $0$. Therefore $H_{n+1}(\tilde Y,\tilde X)\to \tilde H_n(\tilde X)$ is $0$, and $\tilde H_n(\tilde Y)\to H_n(\tilde Y,\tilde X)$ too, so this allows us to conclude.