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Suppose $M$ is a closed smooth n-manifold.

a)Does there always exist a smooth map $f:M\to S^n$ from $M$ into the n-sphere, such that $f$ is essential(i.e. $f$ is not homotopic to a constant map)? Justify your answer.

b) Same question, replacing $S^n$ by the n-torus $T^n$.

For the question a), I think since $M$ is a n-manifold, there is a neighbourhood $U$ of $M$ and $U$ is homeomorphic to $\mathbb R^n$,and so $U$ is homeomorphic to $S^n \backslash N$, $N$ is the north pole of $S^n$.And then we map $M\backslash U$ to $N$. Does the map constructed above satisfy the condition smooth map?

Could anybody give some hints on how to slove a) and b)? Thanks!

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    @AaronMazel-Gee Thank you for your advise :)2012-07-10

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