If $m<n$ use the simplicial approximation theorem to prove that any map $f:S^m\to S^n$ is null homotopic. Deduce that $\pi_1(S^n)$ is trivial if $n>1$.
we have not covered lot on simplicial approximation. I don't have any reference book ether. If you know any book that has lot of information and solved problems about simplicial map and approximation then please give me the link.
And if you could do this for me then i would be blessed...
On
If you accept that every map from a contractible space is trivial, notice that $S^n-\{x\}$ is iso. to $\mathbb R^{n-1}$. Then $f$ can be expressed/"factored" as a composition of maps, one map with domain $\mathbb R^n$. For $\pi_1$ use the same "factorization" of f and functoriality properties of $\pi_1$. EDIT Note we use simplicial approximation , as mentioned by Mariano, to allow for the fact that the map is ( homotopic to a map that is ) not onto.
One particular consequence of simplicial approximation of maps $f:S^m\to S^n$ with $m<n$ is that any such map is homotopic to one which is not surjective: indeed, a simplicial map maps simplex to simplices of no greater dimension, and it follows that any point in the interior of an $n$-dimensional simplex of the codomain is not in the image.
Now it is very easy to show that a non-surjective map $S^n\to S^m$ (whatever the dimensions) is homotopic to a constant map.