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I need to prove that a smooth map $f\colon SL_2(\mathbb{R})\rightarrow S^4$ is homotopic to the constant map. I think that computing the corresponding homotopy groups may help, but I don't see how to do this.

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  • $\begingroup$ Does it help to know that $\pi_1(SL_2(\mathbb{R}))=\mathbb{Z}$ and $\pi_1(S^4)=\{1\}$? $\endgroup$ Commented May 6, 2016 at 13:09
  • $\begingroup$ $SL(2,R)$ is 3-dimensional; do you know how to show that every smooth map $f: SL(2,R)\to S^4$ is not surjective? $\endgroup$ Commented May 9, 2016 at 18:31

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$SL(2,R)$ is a deformation retract of its maximal compact subgroup which is $S^1$. So it is equivalent to show that every map $S^1\rightarrow S^4$ is homotopic to the constant map, a fact which is equivalent to the fact that $\pi_1(S^4)=1$.

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