Talk:Arnold conjecture

Solved?

Is this conjecture still open? Didn't Floer solve this? 77.3.23.230 (talk) 11:25, 31 July 2023 (UTC)

Badly written

The conjecture is described in the article as follows:

"Let $(M,\omega )$ be a compact symplectic manifold. For any smooth function $H:M\to {\mathbb {R} }$, the symplectic form $\omega$ induces a Hamiltonian vector field $X_{H}$ on $M$, defined by the identity

$\omega (X_{H},\cdot )=dH.$

"The function $H$ is called a Hamiltonian function.

"Suppose there is a 1-parameter family of Hamiltonian functions $H_{t}:M\to {\mathbb {R} },0\leq t\leq 1$, inducing a 1-parameter family of Hamiltonian vector fields $X_{H_{t}}$ on $M$. The family of vector fields integrates to a 1-parameter family of diffeomorphisms $\varphi _{t}:M\to M$. Each individual $\varphi _{t}$ is a Hamiltonian diffeomorphism of $M$.

"The Arnold conjecture says that for each Hamiltonian diffeomorphism of $M$, it possesses at least as many fixed points as a smooth function on $M$ possesses critical points."

The last sentence, which finally describes the actual conjecture, make no reference to anything that came before. Surely this can be written much more clearly so that the connection of the conjecture to what preceded it is clear.