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.

 

[blue_raver22]

Sure, I can help you with this topology question.

i) To prove that every map e: X-> R^n is homotopic to a constant map, we need to show that there exists a continuous map h: X x [0,1] -> R^n such that h(x,0) = e(x) for all x in X and h(x,1) is a constant map.

To construct such a map, let c be a point in R^n and define h: X x [0,1] -> R^n as h(x,t) = (1-t)e(x) + tc. This is a continuous map since it is a linear combination of continuous maps. Also, h(x,0) = e(x) and h(x,1) = c, which is a constant map. Therefore, h is a homotopy between e and a constant map, and thus e is homotopic to a constant map.

ii) Now, let's consider the map f: X->S^n that is not onto (surjective). This means that there exists a point y in S^n that is not in the image of f.

To show that f is homotopic to a constant map, we can again use a similar approach as in part i). Let c be a point in S^n that is not in the image of f. We can define h: X x [0,1] -> S^n as h(x,t) = (1-t)f(x) + tc. This is a continuous map since it is a linear combination of continuous maps. Also, h(x,0) = f(x) and h(x,1) = c, which is a constant map. Therefore, h is a homotopy between f and a constant map, and thus f is homotopic to a constant map.

I hope this helps you with your past exam paper. Let me know if you have any further questions or need clarification on any part of the solution. Good luck!