Rank formula conjecture for symplectic mapping groups of rational surfaces

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Let (X,ω)(X,\omega) be a rational symplectic surface with 8≤χ(X)≤118\leq\chi(X)\leq 11, and let Q(X)=12(χ(X)−2)(χ(X)−3)Q(X)=\frac{1}{2}(\chi(X)-2)(\chi(X)-3). Let r+(X,ω)r^+(X,\omega) denote the number of homology classes admitting (−2)(-2) Lagrangian spheres. Rank formula conjecture.

Q(X)=r+(X,ω)+Rank⁡[π1(Symp⁡h(X,ω))]−Rank⁡[π0(Symp⁡h(X,ω))].Q(X)=r^+(X,\omega)+\operatorname{Rank}\bigl[\pi_1(\operatorname{Symp}_h(X,\omega))\bigr]-\operatorname{Rank}\bigl[\pi_0(\operatorname{Symp}_h(X,\omega))\bigr].

This is proposed as an extension of a formula known for rational surfaces with χ(X)≤7\chi(X)\leq 7. The source presents the extension as a speculation, and its general validity remains open.

References

Primary source

Jun Li and Weiwei Wu, “Topology of symplectomorphism groups and ball-swappings”, arXiv:1910.01682 (2019).

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