We compute the top-weight rational cohomology of Ag for g = 5, 6, and 7, and we give some vanishing results for the top-weight rational cohomology of A8, A9, and A10. When g = 5 and g = 7, we exhibit nonzero cohomology groups of Ag in odd degree, thus answering a question highlighted by Grushevsky. Our methods develop the relationship between the top-weight cohomology of Ag and the homology of the link of the moduli space of principally polarized tropical abelian varieties of rank g. To compute the latter we use the Voronoi complexes used by Elbaz-Vincent-Gangl-Soulé. Our computations give natural candidates for compactly supported cohomology classes of Ag in weight 0 that produce the stable cohomology classes of the Satake compactification of Ag in weight 0, under the Gysin spectral sequence for the latter space.

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In this article, we study the Bruhat-Chevalley-Renner order on the complex symplectic monoid M Sp n . After showing that this order is completely determined by the Bruhat-Chevalley-Renner order on the linear algebraic monoid of n × n matrices M n , we focus on the Borel submonoid of M Sp n . By using this submonoid, we introduce a new set of type B set partitions. We determine their count by using the "folding" and "unfolding" operators that we introduce. We show that the Borel submonoid of a rationally smooth reductive monoid with zero is rationally smooth. Finally, we analyze the nilpotent subsemigroups of the Borel semigroups of M n and M Sp n . We show that, contrary to the case of M Sp n , the nilpotent subsemigroup of the Borel submonoid of M n is irreducible.

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