Topology (specifically homotopy) question

[timboj2008]

Could anybody help me with this topology question?

i) Prove that every map e: X-> R^n is homotopic to a constant map.

ii) If f: X->S^n is a map that is not onto (surjective), show that f is homtopic to a constant map.

It's part of a past exam paper but it does not come with solutions. Any help on the solution would be greatly appreciated.

Thanks.

 

[quasar987]

i) Try to find a homotopy H:I x X-->R^n between e and the map that sends everything in X to 0.

ii) Hint: Use (i)

 

[Landau]

Basically you need to realize that R^n is contractible. This is pretty easy to visualize: e.g. take R^2 or R^3 and send every point to the origin along the straight line between them.

 

[mathwonk]

these are the same question.