I want to prove the following lemma:

Let $X$ be a n-connected space. Then there exists a CW-approximation $f:K\rightarrow X$ such that $K$ has trivial n-skeleton.

What I have done so far:

By assumption: $\pi_k(X,x_0)=\ast$ for each $k\leq n$ and all $x_0\in X$. By the CW-approximation theorem there exists $f:K\rightarrow X$ such that $\pi_k(K,k_0)\cong \pi_k(X,f(k_0))$ for each $k\in\mathbb{N}$ and each $k_0\in K$.

I think that I have to use Whitehead's Theorem to conclude the result, but i don't see how to go on. Can someone help me? Thanks a lot.

  • $\begingroup$ Maybe one have to consider the proof of CW approximation in more detail? $\endgroup$ – HubUp55 Dec 29 '14 at 12:45
  • $\begingroup$ Hurewicz theorem is probably all you need. Start with some CW-approximation $K$ and then if there is an $n-i$-cycle, it must also be a boundary by Hurewicz, so you can contract the $n-i+1$-chain that it bounds to kill off that cycle. $\endgroup$ – Dan Rust Dec 29 '14 at 14:16
  • $\begingroup$ @DanielRust: Sorry, we don't discussed this Theorem in lecture. Is there another possibility? $\endgroup$ – HubUp55 Dec 29 '14 at 14:20

Without loss of generality we may assume $X$ is path-connected (we can just work path component at a time then take the union).

Definition: A map $f:A \to B$ is an $k$-equivalence iff it induces bijections on $\pi_j$ for $j < k$ and surjection on $\pi_k$.

We build our CW-approximation $K$ inductively (induction on $m$):

For observe that the inclusion of a point $* \to X$ is $n$-connected and set the $n$-th skeleton to be $K_n := *$. Now assume that we have a CW complex $K_m$ and an m-equivalence $f_m: K_m \to X$ compatible with the previous steps. (Meaning s.t. for all $j \le m$ the skeleton inclusion $K_j \to K_m$ composed with $f_m$ is the $j$-equivalence $f_j$. with together with a $m$-equivalence $f_m :K_m \to X$).

Now consider $H:=Ker(\pi_m(K_m,*) \to \pi_m(X,*))$. Choose generators for $H$ and glue $m+1$-cells to $K_m$ along these maps. The resulting CW complex $K^{'}_{m+1}$ has $K_m$ as an $m$-skeleton and by construction the induced map $\pi_{m}(K_{m+1}^{'},*) \to \pi_{m}(X,*)$ is also injective so that $K_{m+1}^{'} \to X$ induces isomorphism on $\pi_j$ for all $j \le m$. We're almost done only we need to take care of surjectivity on $\pi_{m+1}$. For this we can just take $K_{m+1}^{'}\bigvee_{ \phi } S^{m+1}$ where $\phi$ runs over a set of generators for $\pi_{m+1}(X,*)$ (or if we want a smaller model we can take generators for $Coker(\pi_{m+1}(K_{m+1}^{'},*) \to \pi_{m+1}(X,*))$ ).

Carrying this out inductively and setting $f:K \to X$ to be the colimit of all the $f_m : K_m \to X$'s we get a tower of skeletons $K_n \subset \dots \subset K_m \subset K_{m+1} \subset \dots \subset K$. Moreover $f: K \to X$ is by construction a weak homotopy equivalence we have for the $n$-th skeleton $K_n = *$. QED.


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