Some phenomena in tautological rings of manifolds

Abstract

We prove several basic ring-theoretic results about tautological rings of manifolds W , that is, the rings of generalised Miller-Morita-Mumford classes for fibre bundles with fibre W . Firstly we provide conditions on the rational cohomology of W which ensure that its tautological ring is finitelygenerated, and we show that these conditions cannot be completely relaxed by giving an example of a tautological ring which fails to be finitely-generated in quite a strong sense. Secondly, we provide conditions on torus actions on W which ensure that the rank of the torus gives a lower bound for the Krull dimension of the tautological ring of W . Lastly, we give extensive computations in the tautological rings of CP 2 and S 2 × S 2 . arXiv:1611.03038v2 [math.AT] 6 May 2018 1.3. Krull dimension. Our second main result is a general technique, continuing on from our work with Galatius and Grigoriev [GGRW17, §4], for estimating the Krull dimension (for which we write Kdim) of the rings R * (W ) from below in terms of torus actions on W . The general statement is Theorem 3.1, but the hypotheses of that theorem are somewhat involved: we state here one of its corollaries with hypotheses which are easy to verify. Corollary B. Let a k-torus T act effectively on W , and suppose that either

Tautological classes of definite 4–manifolds

“…The rings R .CP 2 /, R .CP 2 ; / and R .CP 2 ; D 4 / were investigated in [19], but their structure was not fully determined. By computing these rings we have settled some open problems posed in [19].…”

“…Combining the constant terms from (4-21) and (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) and equating this to v ∅ , we obtain…”