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Morse function on a sphere

  1. Oct 4, 2010 #1
    Suppose a Morse function [tex]f:S^n\rightarrow R[/tex] 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.
     
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  3. Oct 4, 2010 #2

    lavinia

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    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.
     
    Last edited: Oct 4, 2010
  4. Oct 5, 2010 #3
    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
     
  5. Oct 5, 2010 #4

    lavinia

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    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.
     
  6. Oct 5, 2010 #5
    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
     
  7. Oct 5, 2010 #6

    lavinia

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    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.
     
  8. Oct 5, 2010 #7
    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[tex]^{n}[/tex];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.
     
    Last edited: Oct 5, 2010
  9. Oct 5, 2010 #8

    lavinia

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    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.
     
    Last edited: Oct 5, 2010
  10. Oct 5, 2010 #9

    lavinia

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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.
     
  11. Oct 5, 2010 #10
    THanks a lot for the help
     
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