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International Mathematics Research Notices (2000) 2000:271-279, doi:10.1155/S1073792800000167
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Copyright © 2000 Hindawi Publishing Corporation. All rights reserved.

On strict category weight, gradient-like flows, and the Arnold conjecture

Yuli B. Rudyak

The Arnold conjecture claims that, for every Hamiltonian symplectomorphism {varphi} : M -> M of a closed symplectic manifold (M,{omega}), the number Fix {varphi} of fixed points of {varphi} is at least the minimal number of critical points of a smooth function on M. For every (M,{omega}) with {pi}2(M) = 0, Floer [F] and Hofer [H] performed a certain analytical reduction of the problem and used this reduction in order to estimate Fix {varphi} by the cup-length of M.

Using the Floer-Hofer reduction, Rudyak and Oprea have completed the proof of the Arnold conjecture in case {pi}2(M) = 0 by proving a certain result of Lusternik-Schnirelmann type (see [R2], [RO]). This proof used surgery and cobordism theory. Here we give a purely cohomological proof of this result.


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