Topological Method Does Not Work for Frankel-mcduff Conjecture
نویسنده
چکیده
In dealing with transformation group, topological approach is very natural. But, it is not sufficient to investigate geometric properties of transformation group and we need geometric method. Frankel-McDuff Conjecture is very interesting in the point that it shows struggling between topological method and geometric method. In this paper, the author suggest generalized FrankelMcDuff conjecture as a topological version of the conjecture and construct a counterexample for the generalized version, and from this we assert that topological method does not work for FrankelMcDuff Conjecture. 1. Frankel-McDuff Conjecture and its generalization In dealing with transformation group, topological approach is very natural, and equivariant cohomology is a main tool in this approach as cohomology in nonequivariant setting. Equivariant cohomology is too widely used to list all applications, but we want to point out that it has been successfully used to study toric variety. On the other hand, topological method is not sufficient to investigate geometric properties of transformation group. For example, it has been conceived that topological method is not sufficient in studying Hamiltonian action which is a generalization of toric variety to the symplectic category, and instead of it moment map has been widely used which is a main tool of (symplectic) geometric method. However, some mathematicians have tried to understand Hamiltonian action in topological point of view. Some of them are Allday, Hauschild, Received December 27, 2006. 2000 Mathematics Subject Classification: Primary 53D05, 53D20; Secondary 55Q05, 57R19.
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