Show that there is no immersion of S^n into R^n

Thread starter hadi amiri 4
Start date Jul 12, 2008

In summary, an immersion of S^n into R^n is a smooth map that embeds the n-dimensional sphere into the n-dimensional Euclidean space without any self-intersections or overlapping points. It is important to show that there is no immersion of S^n into R^n as it helps us understand the limitations of embedding objects into higher-dimensional spaces. The Whitney embedding theorem states that there is no immersion of an n-sphere into R^n for n > 1. The proof of this statement involves using algebraic topology and differential topology techniques, such as the Brouwer degree. Other spaces into which S^n cannot be immersed include spaces with non-trivial fundamental group and non-orientable manifolds or spaces with non-integer dimensions.

Jul 12, 2008

hadi amiri 4

show that there is no immersion of ~~[/n] into [R][/n]~~

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Jul 12, 2008

hadi amiri 4

show that there is no immersion of S^n into R^n

Jul 12, 2008

roam

By definition;

[tex]f : S^{n}\rightarrow R^n[/tex] is an immersion if;

[tex]D_{p}f : T_{p}S^n \rightarrow T_{f(p)}R^n[/tex]

1. What is an immersion of S^n into R^n?

An immersion of S^n into R^n is a smooth map that preserves the local structure of the n-dimensional sphere S^n in the n-dimensional Euclidean space R^n. In other words, it is a way of embedding the sphere into the space without any self-intersections or overlapping points.

2. Why is it important to show that there is no immersion of S^n into R^n?

This is an important question in differential topology, as it helps us understand the limitations of embedding objects into higher-dimensional spaces. It also has implications in other fields, such as physics and computer graphics, where the ability to embed objects into higher-dimensional spaces is crucial.

3. Can you give an example of an immersion of S^2 into R^2?

No, it is not possible to give an example of an immersion of S^2 into R^2. This is because the famous Whitney embedding theorem states that there is no immersion of an n-sphere into R^n for n > 1.

4. How can you show that there is no immersion of S^n into R^n?

The proof of this statement involves using algebraic topology and differential topology techniques. One approach is to use the Brouwer degree, which is a topological invariant that measures the number of times a map wraps around a given point. By showing that the Brouwer degree of any potential immersion is zero, we can conclude that there is no immersion of S^n into R^n.

5. Are there any other spaces into which S^n cannot be immersed?

Yes, there are other spaces into which S^n cannot be immersed. For example, there is no immersion of S^n into any space with non-trivial fundamental group, such as a torus. This is known as the Hatcher-Smale theorem. Additionally, there are spaces where the notion of immersion does not even make sense, such as non-orientable manifolds or spaces with non-integer dimensions.