Skip to main content
MathOverflow

Return to Answer

deleted 262 characters in body
Source Link
Thomas Kragh
Thomas Kragh
  • 2.6k
  • 18
  • 23

Assume dim$(M) \geq 6$ and $M$ is simply connected.

By cancelling Morse critical points for a Morse function $f$ on $M$ one can get the number of critical points equal to the minimum number of generators needed in $C_* (M)$ to generate $H_* (M)$. I.e. the sum of the Betti numbers plus 2 for each torsion generator. This is also the number of cells in a minimal cell structure (CW-complex) for $M$.

This means that if one can define Floer homology with $\mathbb{Z}$ coefficients and the PSS isomorphism works with $\mathbb{Z}$ coefficients, then the strong Arnold conjecture is true.

I believe this is the case for e.g. monotone symplectic manifolds.

It think this works for all surfaces. Indeed, all oriented surfaces has a Morse function which describe a minimal cell structure. However, here this equals the sum of the Betti numbers, since there are no torsion in the homology groups of these surfaces.

Assume dim$(M) \geq 6$ and $M$ is simply connected.

By cancelling Morse critical points for a Morse function $f$ on $M$ one can get the number of critical points equal to the minimum number of generators needed in $C_* (M)$ to generate $H_* (M)$. I.e. the sum of the Betti numbers plus 2 for each torsion generator. This is also the number of cells in a minimal cell structure (CW-complex) for $M$.

This means that if one can define Floer homology with $\mathbb{Z}$ coefficients and the PSS isomorphism works with $\mathbb{Z}$ coefficients, then the strong Arnold conjecture is true.

I believe this is the case for e.g. monotone symplectic manifolds.

It think this works for all surfaces. Indeed, all oriented surfaces has a Morse function which describe a minimal cell structure. However, here this equals the sum of the Betti numbers, since there are no torsion in the homology groups of these surfaces.

Assume dim$(M) \geq 6$ and $M$ is simply connected.

By cancelling Morse critical points for a Morse function $f$ on $M$ one can get the number of critical points equal to the minimum number of generators needed in $C_* (M)$ to generate $H_* (M)$. I.e. the sum of the Betti numbers plus 2 for each torsion generator. This is also the number of cells in a minimal cell structure (CW-complex) for $M$.

This means that if one can define Floer homology with $\mathbb{Z}$ coefficients and the PSS isomorphism works with $\mathbb{Z}$ coefficients, then the strong Arnold conjecture is true.

I believe this is the case for e.g. monotone symplectic manifolds.

Source Link
Thomas Kragh
Thomas Kragh
  • 2.6k
  • 18
  • 23

Assume dim$(M) \geq 6$ and $M$ is simply connected.

By cancelling Morse critical points for a Morse function $f$ on $M$ one can get the number of critical points equal to the minimum number of generators needed in $C_* (M)$ to generate $H_* (M)$. I.e. the sum of the Betti numbers plus 2 for each torsion generator. This is also the number of cells in a minimal cell structure (CW-complex) for $M$.

This means that if one can define Floer homology with $\mathbb{Z}$ coefficients and the PSS isomorphism works with $\mathbb{Z}$ coefficients, then the strong Arnold conjecture is true.

I believe this is the case for e.g. monotone symplectic manifolds.

It think this works for all surfaces. Indeed, all oriented surfaces has a Morse function which describe a minimal cell structure. However, here this equals the sum of the Betti numbers, since there are no torsion in the homology groups of these surfaces.