TAUTOLOGICAL RINGS AND STABILISATION

We construct a ring homomorphism comparing the tautological ring, fixing a point, of a closed smooth manifold with that of its stabilisation by S 2a ×S 2b .

Socle pairings on tautological rings

We study some aspects of the $\lambda_g$ pairing on the tautological ring of $M_g^c$, the moduli space of genus $g$ stable curves of compact type. We consider pairing kappa classes with pure boundary strata, all tautological classes supported on the boundary, or the full tautological ring. We prove that the rank of this restricted pairing is equal in the first two cases and has an explicit formula in terms of partitions, while in the last case the rank increases by precisely the rank of the $\lambda_g\lambda_{g - 1}$ pairing on the tautological ring of $M_g$. Comment: 18 pages, 1 figure; v3: journal version; v2: minor revisions to sections 1.1 and 4.1, results unchanged

Tautological rings for high-dimensional manifolds

We study tautological rings for high-dimensional manifolds, that is, for each smooth manifold$M$the ring$R^{\ast }(M)$of those characteristic classes of smooth fibre bundles with fibre$M$which is generated by generalised Miller–Morita–Mumford classes. We completely describe these rings modulo nilpotent elements, when$M$is a connected sum of copies of$S^{n}\times S^{n}$for$n$odd.

Tautological rings of spaces of pointed genus two curves of compact type

We prove that the tautological ring of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$, the moduli space of $n$-pointed genus two curves of compact type, does not have Poincaré duality for any $n\geqslant 8$. This result is obtained via a more general study of the cohomology groups of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$. We explain how the cohomology can be decomposed into pieces corresponding to different local systems and how the tautological cohomology can be identified within this decomposition. Our results allow the computation of $H^{k}({\mathcal{M}}_{2,n}^{\mathsf{ct}})$ for any $k$ and $n$ considered both as $\mathbb{S}_{n}$-representation and as mixed Hodge structure/$\ell$-adic Galois representation considered up to semi-simplification. A consequence of our results is also that all even cohomology of $\overline{{\mathcal{M}}}_{2,n}$ is tautological for $n<20$, and that the tautological ring of $\overline{{\mathcal{M}}}_{2,n}$ fails to have Poincaré duality for all $n\geqslant 20$. This improves and simplifies results of the author and Orsola Tommasi.

On the independence of the generators of tautological rings

We prove that all monomials of κ-classes and ψ-classes are independent in $R^k({\overline {\cal M}}_{g,n})/R^k(\partial {\overline {\cal M}}_{g,n})$ for all $k \leqslant [g/3]$. We also give a simple argument for $\kappa _l \neq 0$ in Rl(ℳg) for $l \leqslant g-2$.