Surface bundles and the section conjecture

We formulate a tropical analogue of Grothendieck’s section conjecture: that for every stable graph $$\Gamma $$ genus $$g>2$$ , and field k, the generic curve with reduction type over k satisfies conjecture. prove many cases this In so doing we show existence examples curves no rational points satisfying conjecture fields geometric interest, then p-adic number via Chebotarev argument. construct two Galois cohomology classes $$o_1$$ $$\widetilde{o_2}$$ which obstruct $$\pi _1$$ -sections hence points. The first is an abelian obstruction, closely related to period class on moduli space $${\mathscr {M}}_g$$ studied by Morita. second 2-nilpotent obstruction appears be new. study degeneration these topological techniques, produce surface bundles surfaces where sections. use constructions each obstructs Among our results are new proof $$g\ge 3$$ even divisor degree one (where genuinely non-abelian).