Nonexistence of Symplectic Structures on Certain Family of 4-manifolds

Speaker: Jianfeng Lin (Tsinghua University)

Date: March 16 Wed. 10:00am--11:00am

Venue: Tencent Meeting Room: 374 274 206     Password:22086     

Abstract: Let Symp(X) be the group of symplectomorphisms on a symplectic 4-manifold X. It is a classical problem in symplectic topology to study the homotopy type of Symp(X) and to compare it with the group of all diffeomorphisms on X. This problem is closely related to the existence of symplectic structures on smooth families of 4-manifolds. In this talk, we will discuss the proof of following results:

(1) For any X that contains a smoothly embedded 2-sphere with self-intersection -1 or -2, there exists a loop of self-diffeomorphisms on X that is not homotopic to a loop of symplectomorphisms.

(2) Consider a family of 4-manifolds obtained by resolving an ADE singularity using a hyperkahler family of complex structures, this family never support a family symplectic structure in a constant cohomology class.

(3) For any non-minimal symplectic 4-manifold whose positive second-betti number does not equal to 3, the space of symplectic form is not simply connected. The key ingredient in the proofs is a new gluing formula for the family Seiberg-Witten invariant.