Chambers of homotopy groups stability in irrational blow up ruled surfaces

Resumen:

We will give a gentle introduction to questions about the homotopy type of  symplectomorphism groups of ruled symplectic 4- manifolds. Expanding results from the nineties on minimal rational ruled surfaces, several strides have been made in more recent years recently in the cases of blow-ups of such manifolds. We focus on understanding at large how such groups behave as we deform the symplectic forms within the cohomology cone in the case of irrational ruled surfaces with arbitrarily many blow ups. Using improved inflation techniques and a better understanding of the spaces of J-holomorphic curves, we will introduce a chamber structure on the reduced symplectic cone of such manifolds, so that the symplectomorphisms groups remain homotopically the same within the chambers. We will then discuss how an instance of such extremal ray limiting behaviour yields nontrivial symplectic isotopies, contrasting
to the minimal cases. This is joint work with Jun Li.

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