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

[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.

 

[lavinia]

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.

 

[pp31]

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

 

[lavinia]

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.

 

[pp31]

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

 

[lavinia]

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.

 

[pp31]

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.

 

[lavinia]

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.

 

[lavinia]

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.

 

[pp31]

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