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Possible Duplicate:
Continuous map $\mathbb{S}^n\to \mathbb{S}^m$

Why is every continuous function $f:\mathbb{S}^n\to\mathbb{S}^m,$ for $n homotopic to a constant map?

Thanks.

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    That is not true. For instance, there are space filling curves $S^1\rightarrow S^2$. What is true is that any map $S^n\rightarrow S^m$ is homotopic to a map satisfying that condition.2012-01-14
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    @Aspirin Could you please look at [my comments here](http://math.stackexchange.com/questions/94131/continuous-map-mathbbsn-to-mathbbsm/94136#comment221555_94136) which were, in fact, directed to you? For your convenience, I have reproduced them as an answer below.2012-01-14
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    @Amitesh Datta: I'm sorry, I forget for this thread2012-01-14
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    Dear @Asprin: I just thought I would point that out; it is not a problem. I think that you can either change the question to "Why is every continuous function $f:S^n\to S^m$ for $n homotopic to a constant map?" or flag the question for moderator attention (because, at the moment, this question is an exact duplicate and I think should be closed).2012-01-14
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    This is NOT a duplicate of the marked question. That question was asking whether such a mapping could be surjective, while this question asks why all such maps are homotopic to constant maps.2014-01-30

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