# How Many Critical Points Must a Morse Function Have on a Sphere?

• pp31
In summary: However, I am not sure how to go about finding the extra critical points in a Morse function on a surface of genus g.
pp31
Suppose a Morse function $$f:S^n\rightarrow R$$ satisfies f(x) = f(-x). Show that f must have atleast 2 critical points of index j for all j = 0,...,n.

Show that any MOrse function on a compact set of genus g has at least 2g+2 critical points.

These are the questions and I have no idea how to get started. If some one could just push me in the right direction I would really appreciate it.

pp31 said:
Suppose a Morse function $$f:S^n\rightarrow R$$ satisfies f(x) = f(-x). Show that f must have atleast 2 critical points of index j for all j = 0,...,n.

Show that any MOrse function on a compact set of genus g has at least 2g+2 critical points.

These are the questions and I have no idea how to get started. If some one could just push me in the right direction I would really appreciate it.

if f(x) = f(-x) for all x then if f'(x) = 0 so is f'(-x).

Further, f projects to a function on projective space. Find a CW complex decomposition of projective space.

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So what you are saying is that not only can we find the CW complex structure of a space from a Morse function but we can use the CW structure to determine the critical points and the index of a Morse function.

Correct me if I am wrong

pp31 said:
So what you are saying is that not only can we find the CW complex structure of a space from a Morse function but we can use the CW structure to determine the critical points and the index of a Morse function.

Correct me if I am wrong

I am no expert on this. But yes. I think that is one of the results of Morse theory.

At each non-degenerate critical point a cell gets attached. So the manifold has the homotopy type of a CW complex with a k dimensional cell attached for each critical point of index k.

I would be glad to read through this with you in Milnor's Morse theory. I need to learn it.

However I think there are different decomposition of a sphere as a CW complex for instance one way is attaching a 0 cell and a n cell and the other one is 2 0 cells,..., 2 n cells. So we have different morse funcitions based on two different decompositions so I think we have to be careful when we use the cell structure to determine the morse function

pp31 said:
However I think there are different decomposition of a sphere as a CW complex for instance one way is attaching a 0 cell and a n cell and the other one is 2 0 cells,..., 2 n cells. So we have different morse funcitions based on two different decompositions so I think we have to be careful when we use the cell structure to determine the morse function

you are probably right. The thing is though that projective space has non-zero homology in every dimension so your probably OK no matter how you do it. ( the sphere only has homology in dimension zero and n).

But let's work on decompositions and compare notes.

I think we could use the fact that the boundary operators vanish when we construct the chain complex of the projective plane over Z2 which in turn implies that Hn(RP$$^{n}$$;Z2 ) =Z2 to show that we need atleast one critical point of each index and since f(x) = f(-x) we have at least two critical points of each index.
I am not sure if this works or not but just let me know what you think.

And do you have any idea of how to get started on the other question.

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pp31 said:
I think we could use the fact that the boundary operators vanish when we construct the chain complex of the projective plane over Z2 which in turn implies that Hn(RP$$^{n}$$;Z2 ) =Z2 to show that we need atleast one critical point of each index and since f(x) = f(-x) we have at least two critical points of each index.
I am not sure if this works or not but just let me know what you think.

And do you have any idea of how to get started on the other question.

I guess you mean a surface of genus g. The alternating sum of the number of cells in each dimension is the Euler characteristic of a CW complex.

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The projective plane must have at least one zero cell, one 1 cell, and one 2 cell. So a Morse function on the 2 sphere would seems to have to have at least 2 critical points in each domension sonce the sphere is a 2 fold cover of the projective plane. The projection map would cut the number of cells in half.

I guess this type of argument works for a projective space of any dimension.

lavinia said:
The projective plane must have at least one zero cell, one 1 cell, and one 2 cell. So a Morse function on the 2 sphere would seems to have to have at least 2 critical points in each domension sonce the sphere is a 2 fold cover of the projective plane. The projection map would cut the number of cells in half.

I guess this type of argument works for a projective space of any dimension.

THanks a lot for the help

## 1. What is a Morse function on a sphere?

A Morse function on a sphere is a continuous function that maps a sphere onto a real line and has only non-degenerate critical points. In other words, it is a smooth function that has a unique value at each point on the sphere and has no flat or degenerate points.

## 2. What is the significance of a Morse function on a sphere?

Morse functions on a sphere are important in topology and differential geometry because they provide a way to study the structure of a sphere and its critical points. They also have applications in physics, particularly in the study of dynamical systems and shape analysis.

## 3. How is a Morse function on a sphere different from a regular function?

Morse functions on a sphere have the additional property of being non-degenerate at all of their critical points. This means that the derivative of the function is non-zero at each critical point, and the function has a unique minimum or maximum at each critical point on the sphere.

## 4. What is the Morse index of a critical point on a sphere?

The Morse index of a critical point on a sphere is a measure of its stability and is given by the number of negative eigenvalues of the Hessian matrix at that point. A critical point with a Morse index of 0 is a non-degenerate minimum, while a critical point with a Morse index of n is a non-degenerate maximum on an n-dimensional sphere.

## 5. Are there other types of Morse functions besides those on a sphere?

Yes, there are Morse functions defined on other surfaces and manifolds, including Euclidean spaces, tori, and more complex surfaces. However, the study of Morse functions on a sphere is particularly useful because of the simple structure of the sphere and its connections to other areas of mathematics and science.

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