Symplectic mapping class groups of some Stein and rational surfaces

TL;DR

This paper computes the homotopy groups of symplectomorphism groups for certain blow-ups of the projective plane and related surfaces, advancing understanding of their symplectic mapping class groups and their algebraic structures.

Contribution

It provides new computations of homotopy groups for symplectomorphism groups of specific Stein and rational surfaces, including blow-ups and Milnor fibers, and explores their algebraic embeddings.

Findings

Homotopy groups of symplectomorphism groups of blow-ups are computed.

The Hamiltonian group of A_n-Milnor fibers is contractible.

Symplectic mapping class groups embed into braid groups.

Abstract

In this paper we compute the homotopy groups of the symplectomorphism groups of the 3-, 4- and 5-point blow-ups of the projective plane (considered as monotone symplectic Del Pezzo surfaces). Along the way, we need to compute the homotopy groups of the compactly supported symplectomorphism groups of the cotangent bundle of $\mathbf{RP}^{2}$ and of $\mathbf{C}^*\times\mathbf{C}$. We also make progress in the case of the $A_n$-Milnor fibres: here we can show that the (compactly supported) Hamiltonian group is contractible and that the symplectic mapping class group embeds in the braid group on $n$-strands.