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Does anyone have a good quick proof of this using the Simplicial Approximation Theorem? I'm aware that it comes out as a corollary when considering edge paths and the edge group, but this seems like quite heavy machinery for what should be a simple idea. I haven't been able to put together a convincing argument myself though!

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Simiplicial approximation should give it to you immediately. Loops are $1$-dimensional, so can be pushed into the $1$-skeleton and homotopies are $2$-dimensional, so can be pushed into the $2$-skeleton. You can also use Seifert-VanKampen's theorem. – Grumpy Parsnip Jun 2 '12 at 18:06
@JimConant: I agree that it should be immediate, but I can't see how to formalise your suggestion of pushing into. Could you possibly expand on this? Many thanks! – Edward Hughes Jun 2 '12 at 18:31
"Pushing into" means that there is a homotopy. So, every loop is homotopic to a loop in the $1$-skeleton. Then you need to argue that if two loops are homotopic then they are homotopic through the $2$-skeleton. The SAT implies that a homotopy regarded as a map from $I^2$ is itself homotopic to one that lies in the $2$-skeleton. (However the fact that the homotopy is homotopic to the new one is not relevant for the argument.) – Grumpy Parsnip Jun 2 '12 at 18:49
Okay - I'll try to work through that idea! – Edward Hughes Jun 2 '12 at 19:12
yes, that's what I was thinking. – Grumpy Parsnip Jun 3 '12 at 23:52

Let $X$ be a polyhedron, and consider the inclusion $i : \operatorname{sk}_2 X \to X$. It induces a morphism on fundamental groups $i_* : \pi_1 \operatorname{sk}_2 X \to \pi_1 X$, for some choice of base point $x_0$ in the $0$-skeleton. This morphism is:

  • Surjective: let $\alpha : I \to X$ be a representative of some class in $\pi_1 X$, $\alpha(0) = \alpha(1) = x_0$. Then by the simplicial approximation theorem, since $I$ is 1-dimensional, then $\gamma$ is homotopic to $\beta : I \to \operatorname{sk}_2 X$; since $\alpha$ was already simplicial on the 0-skeleton of $I$, we can assume the homotopy is constant on the 0-skeleton, hence $\beta(0) = \beta(1) = x_0$ too. Then $i_*[\beta] = [\alpha]$.
  • Injective: let $\alpha, \beta : I \to \operatorname{sk}_2 X$ be two loops such that $i_*[\alpha] = i_*[\beta]$, i.e. there is a homotopy $H : I \times I \to X$ such that $H(t,0) = \alpha(t)$, $H(t,1) = \beta(t)$, and $H(0,s) = H(1,s) = x_0$. Then similarly by the simplicial approximation theorem, $H$ is homotopic to some $G : I \times I \to \operatorname{sk}_2 X$ ($I \times I$ is 2-dimensional), and one can moreover take the homotopy to be constant on $\partial(I \times I)$ (on which $H$ is already simplicial). Then $G$ is a homotopy in $\operatorname{sk}_2 X$ between $\alpha$ and $\beta$, hence $[\alpha] = [\beta]$.

It follows that $i_* : \pi_1(\operatorname{sk}_2 X) \to \pi_1(X)$ is an isomorphism. More generally one can immediately adapt the arguments above to show that $\pi_n(\operatorname{sk}_{n+1} X) \cong \pi_n(X)$.

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Taking this off the unanswered list. – Najib Idrissi Nov 19 '15 at 20:33

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