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

Abstract

We provide a new way of understanding the multiplicative structure of the rational homotopy groups π∗(Xλ) ⊗ 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 Ak map a multiple of a map f : S1→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 Xλ are classifying spaces of symplectomorphism groups where ωλ is a symplectic deformation on the trivial ruled surface ⁠. 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(S2×S2,ωλ)).