Show that every map(maybe continuous)

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In summary, the problem is to prove that if dimM=m<p, every map Mm -> Sp is homotopic to a constant. The speaker has already proven this for non-surjective maps, but is unsure if it holds for surjective maps as well. Another speaker suggests using Sard's Theorem to show that the map can be homotopic to a smooth map.
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Stiger
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If dimM=m<p, show that every map(maybe continuous) Mm -> Sp is homotopic to a constant.


This is the problem 5 in chap8. of 'topology from the differentiable viewpoint(Milnor)'.


I proved it when the map is not onto. But I think it can be onto.
Please help me.
 
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  • #2


It can indeed be onto. For instance, for M=S^1 and S^p=S², take f:S^1-->[0,1]² one of the infamous space-filling curve (loop) (http://en.wikipedia.org/wiki/Space-filling_curve). Then make [0,1]² into S² by identifying all the edges together. Then p o f:S^1-->S² is a continuous surjection, where p:[0,1]²-->[0,1]²/~=S² is the quotient map.
 
  • #3


Stiger said:
If dimM=m<p, show that every map(maybe continuous) Mm -> Sp is homotopic to a constant.


This is the problem 5 in chap8. of 'topology from the differentiable viewpoint(Milnor)'.


I proved it when the map is not onto. But I think it can be onto.
Please help me.

Is the map maybe supposed to be differentiable?
 
  • #4


Maybe all you need is that it's homotopic to a smooth map :)

(Then Sard's Theorem)
 
  • #5


I would first like to clarify the problem statement. It seems that the problem is asking to show that for a continuous map from a manifold M of dimension m to a sphere of dimension p, if the dimension of M is less than p, then every map is homotopic to a constant map. This is indeed a known result in topology, known as the "Hairy Ball Theorem" or the "Poincaré-Hopf Theorem". It states that for a smooth map from an n-dimensional sphere to an n-dimensional sphere, there exists at least one point where the differential of the map is zero. This implies that the map is not onto, as you have mentioned in your attempt to prove the problem.

To show that every map is homotopic to a constant map, we can use the concept of degree of a map. The degree of a continuous map from an n-dimensional sphere to an n-dimensional sphere is defined as the number of pre-images of a regular value of the map. A regular value is a point in the target sphere whose differential is non-zero at all points in the pre-image. For a map from an m-dimensional manifold to a p-dimensional sphere, the degree is defined as the number of pre-images of a regular value of the map, where the pre-image is a submanifold of dimension m-p.

In the case where the dimension of M is less than p, we have m < p, which means that the degree of the map is zero. This implies that there are no regular values for the map, and hence the map is homotopic to a constant map, since it can be continuously deformed to a constant map.

In conclusion, the result that every map from an m-dimensional manifold to a p-dimensional sphere, where m < p, is homotopic to a constant map, is a well-known result in topology. This can be proved using the concept of degree of a map and the existence of regular values.
 

1. What is the definition of a continuous map?

A continuous map is a function between two topological spaces that preserves the topological structure. This means that if a small change is made to the input of the function, the output will also have a small change.

2. How do you prove that a map is continuous?

To prove that a map is continuous, you must show that the pre-image of every open set in the output space is also open in the input space. This can be done by using the definition of continuity or by showing that the inverse image of the open set is an open set.

3. Can you give an example of a continuous map?

One example of a continuous map is the function f(x) = x^2, where the input and output spaces are both the set of real numbers with the standard topology. This can be proven by showing that the inverse image of any open interval in the output space is also an open interval in the input space.

4. What is the difference between a continuous map and a differentiable map?

A continuous map is a function that preserves the topological structure between two spaces, while a differentiable map is a function that not only preserves the topological structure, but also has a defined derivative at every point in the input space.

5. How does continuity relate to the concept of limits?

Continuity and limits are closely related concepts. A map is continuous at a point if and only if the limit of the map at that point exists and is equal to the value of the map at that point. This means that as the input of the function approaches a certain value, the output of the function also approaches a certain value.

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