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Is it true that any continuous map $\mathbb{S}^n\to \mathbb{S}^m$ is not surjective if $n?

Thanks.

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    they can be surjective, but are homotopic to nonsurjective maps (eg hatcher section 4.1)2011-12-26

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No, it is not true. There are variations of the Peano curve which provide surjective maps $S^1\to S^n$ for all $n\geq1$.

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    @Aspirin A version of the simplicial approximation theorem remains true in the case where $K$ and $L$ are arbitrary (not necessarily finite) simplicial complexes. However, in the general case, one needs to consider subdivisions of $K$ more general than barycentric subdivision. The details underyling the ideas that I have presented here can be found in pages 79-99 of *Elements of Algebraic Topology* by James Munkres.2011-12-26