# The homotopy Lie algebra of symplectomorphism groups of 3-fold blow-ups of $(S^2 \times S^2, \sigma_{std} \oplus \sigma_{std}) $

## TL;DR

This paper computes the rational homotopy Lie algebra of the symplectomorphism group of a 3-point blow-up of S^2×S^2, revealing how specific circle actions generate its full topology, extending understanding of symplectic topology.

## Contribution

It provides a detailed description of the rational homotopy Lie algebra of the symplectomorphism group for the 3-point blow-up of S^2×S^2, identifying generators via circle actions.

## Key findings

- The symplectomorphism group is generated by specific circle actions.
- The rational homotopy Lie algebra is explicitly described.
- Rank of homotopy groups for small blow-ups of CP^2 is determined.

## Abstract

We consider the 3-point blow-up of the manifold $ (S^2 \times S^2, \sigma \oplus \sigma)$ where $\sigma$ is the standard symplectic form which gives area 1 to the sphere $S^2$, and study its group of symplectomorphisms $\rm{Symp} ( S^2 \times S^2 \#\, 3\overline{\mathbb C\mathbb P}\,\!^2, \omega)$. So far, the monotone case was studied by J. Evans and he proved that this group is contractible. Moreover, J. Li, T. J. Li and W. Wu showed that the group Symp$_{h}(S^2 \times S^2 \#\, 3\overline{ \mathbb C\mathbb P}\,\!^2,\omega) $ of symplectomorphisms that act trivially on homology is always connected and recently they also computed its fundamental group. We describe, in full detail, the rational homotopy Lie algebra of this group. We show that some particular circle actions contained in the symplectomorphism group generate its full topology. More precisely, they give the generators of the homotopy graded Lie algebra of $\rm{Symp} (S^2 \times S^2 \#\, 3\overline{ \mathbb C\mathbb P}\,\!^2, \omega)$. Our study depends on Karshon's classification of Hamiltonian circle actions and the inflation technique introduced by Lalonde-McDuff. As an application, we deduce the rank of the homotopy groups of $\rm{Symp}({\mathbb C\mathbb P}^2 \#\, 5\overline{\mathbb C\mathbb P}\,\!^2, \tilde \omega)$, in the case of small blow-ups.