Deformations of Whitehead Products, Symplectomorphism Groups, and Gromov-Witten Invariants

O. Buse
2010 International mathematics research notices  
We provide a new way of understanding the multiplicative structure of the rational homotopy groups π * (X λ ) ⊗ Q for a family of topological spaces, once we know enough about their additive structure. This allows us to interpret the condition of realizing as an A k map a multiple of a map f : S 1 −→ G between two topological groups in terms of the existence of a rational Whitehead product of order k. Our main example will be when the X λ are classifying spaces of symplectomorphism groups
more » ... rphism groups BSymp( g × S 2 , ω λ ) where ω λ is a symplectic deformation on the trivial ruled surface g × S 2 . Our method of detecting nontriviality is based on computations of equivariant Gromov-Witten invariants. One application gives a homotopy-theoretic counterpart to a geometric result obtained by Karshon. Another application concerns the ring structure of H * (BSymp(S 2 × S 2 , ω λ )).
doi:10.1093/imrn/rnp215 fatcat:ul4i5b7whbgufinavdsaizc6ey