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I'm trying to show that if $X$ is a CW-complex, then $$ \pi_1(X) = \pi_1(X^2)$$ where $X^2$ is the 2-skeleton.

I found the following proposition in Hatcher's book:

Proposition 1.26.

(a) If $Y$ is obtained from $X$ by attaching $2$-cells as described above, then the inclusion $X \hookrightarrow Y$ induces a surjection $\pi_1 (X, x_0) \to \pi_1 (Y,x_0)$ whose kernel is $N$. Thus $\pi_1 (Y) \approx \pi_1 (X)/N$.

(b) If $Y$ is obtained from $X$ by attaching $n$-cells for a fixed $n > 2$, then the inclusion $X \hookrightarrow Y$ induces an isomorphism $\pi_1 (X, x_0) \approx \pi_1 (Y,x_0)$.

(c) For a path-connected cell complex $X$ the inclusion of the $2$-skeleton $X^2 \hookrightarrow X$ induces an isomorphism $\pi_1 (X^2,x_0) \approx \pi_1 (X,x_0)$.

(In (a), $N$ is a normal subgroup of $\pi_1(X,x_0)$.)

I was wondering if there's a more "direct" proof, or if I should just follow this proposition. Thanks in advance!

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The CW-flavoured argument uses the relative version of CW approximation:

Let $X$ and $Y$ be CW complexes, $A\subset X$ a subcomplex. If $f\colon X \to Y$ is a continuous function that is cellular on $A$ then there is a homotopy $H\colon X\times I \to Y$ such that $H_0 = f$, $H_t(a) = f(a)$ for all $a\in A$ and $t\in I$, and $H_1$ is cellular.

Let $X$ be a CW complex, and $\iota \colon X^n \to X$ the inclusion of its $n$-skeleton (we omit the subscript on $\iota$ for the sake of notation). We want to show that $\iota_*\colon \pi_1(X^2) \to \pi_1(X)$ is an isomorphism. Suppose $S^1$ is given a CW structure so that the basepoint is a $0$-cell.

If $f\colon S^1 \to X$ is basepoint preserving, then by relative CW approximation there is a basepoint preserving homotopy between $f$ and a pointed cellular function $\tilde{f}\colon S^1 \to X$. By cellularity the image of $\tilde{f}$ is in $X^1$, so $\iota_*\colon \pi_1(X^2)\to \pi_1(X)$ is surjective.

Now suppose $f\colon S^1 \to X^2$ is a pointed map such that $[\iota\circ f] = 0 \in \pi_1(X)$, i.e. $[f]$ is in the kernel of $\iota_*$. Without loss of generality we can assume that $f$ is a cellular map. If we consider any basepoint-preserving null-homotopy $H\colon S^1 \times I \to X$ of $\iota\circ f$, then by relative CW approximation (note that $H$ is cellular on the subcomplex $(X\times \{0\}) \cup (\{x_0\} \times I) \subset X \times I$) $H$ is homotopic to a basepoint-preserving null-homotopy $\tilde{H}$ of $\iota\circ f$ which is cellular. In particular the image of this null-homotopy is in $X^2$ so in fact $[f] = 0 \in \pi_1(X^2)$, therefore $\iota_*$ is injective.


Note: an essentially identical argument shows that $\pi_n(X) \cong \pi_n(X^{n+1})$ for all $n\geq 0$, as an exercise you should write out the details.

Edit: Further note: after looking at Hatcher's proof of this proposition, it seems more elementary than the full force of CW approximation, even though I feel like CW approximation is the "conceptual" way to answer your specific question about CW complexes.

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  • $\begingroup$ Thanks for the clear answer! So basically if I want a more direct result I should use CW approximation, but if I just want to use basic constructions on CW complexes I have to make use of that proposition, for example? $\endgroup$ – SantiMontouliu May 23 at 19:35
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    $\begingroup$ I think the methods used in the proofs of that proposition are a bit more elementary than CW approximation, and they work in a more general context (I had assumed they would just be using CW approximation but it looks like they aren't). In my opinion CW approximation is the "conceptual" way of seeing the result you want, but Hatcher's proposition I think has less proof. $\endgroup$ – William May 23 at 20:11

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