Relations among characteristic classes of manifold bundles

Abstract

We study relations among characteristic classes of smooth manifold bundles with highly-connected fibers. For bundles with fiber the connected sum of g copies of a product of spheres S d × S d and odd d , we find numerous algebraic relations among so-called "generalized Miller-Morita-Mumford classes". For all g > 1, we show that these infinitely many classes are algebraically generated by a finite subset.Our results contrast with the fact that there are no algebraic relations among these classes in a range of cohomological degrees that grows linearly with g, according to recent homological stability results. In the case of surface bundles (d = 1), our approach recovers some previously known results about the structure of the classical "tautological ring", as introduced by Mumford, using only the tools of algebraic topology. 55R40; 57R22, 55T10 IntroductionLet M be a 2d -dimensional closed oriented smooth manifold. We denote by Diff M the topological group of orientation-preserving diffeomorphisms of M . The bar construction can be used to construct the space BDiff(M) that classifies bundles with fiber M . For any characteristic class of vector bundles p ∈ H * +2d (BSO 2d ; Q), we will define a generalized Miller-Morita-Mumford class (or just kappa class) κ p ∈ H * (BDiff(M); Q). These are the simplest examples of characteristic classes of bundles 1 with fiber M and structure group Diff M .We are mainly interested in the case where the fiber is #g S d × S d , the connected sum of g copies of S d × S d . More generally, we let the fiber to be a highly-connected manifold (see Definition 2.5) of genus g and dimension 2d , denoted M 2d g or M g . Recall that H * (BSO 2d ; Q) = Q[p 1 , . . . , p d−1 , e], where p i is the Pontryagin class of degree 4i and e is the Euler class of degree 2d . Let S ⊂ H * (BSO 2d ; Q) consist of the monomials 1 A geometric example of such a bundle is a proper submersion f : E → B of smooth, oriented manifolds that has M as its fiber.arXiv:1310.6804v3 [math.AT]

Some phenomena in tautological rings of manifolds

“…The result under hypothesis (H2) generalises a theorem of Grigoriev [Gri17], and proceeds by establishing the same basic source of relations among κ-classes found by Grigoriev. In the case 2n = 2 this source of relations had been established by the author [RW12], using ideas of Morita [Mor89a,Mor89b]. As the later results of [Gri17] and the results of [GGRW17] are deduced almost entirely from this basic source of relations, the same results largely follow assuming only hypothesis (H2). For example, for g > 1, k odd, and n ≥ k, it follows that Q[κ ep1 , κ ep2 , .…”

“…(a 2 ) d+1 2 = 0. Now that we have Theorem 2.8, the entirety of Section 5 of [Gri17] goes through with only notational changes, as this only uses the statement of Grigoriev's theorem. In particular, for p ∈ H * (BSO(2n)) of even degree and χ = χ(W ) = 0, the analogue of [Gri17, Example 5.19] gives the relation…”

Tautological rings for high-dimensional manifolds

“…In proving these theorems, we obtain results about the vanishing of certain elements in for any manifold (in Theorem 2.1) and about the algebraic independence of certain elements in for even (in Theorem 4.1). We cannot obtain results as conclusive as Theorem 1.1 for even, as our argument relies on [Gri13], which does not apply in this case.…”

“…In this section, we prove that certain generalised Miller–Morita–Mumford classes are nilpotent in for any smooth even-dimensional manifold . We will later apply this together with results from [Gri13] to the manifolds in order to obtain an upper bound on .…”

“…The second author has shown (see [Gri13, Theorem 1.1]) that for odd the ring is finitely generated as a -algebra, and hence is too. (In fact, [Gri13, Theorem 1.1] only shows this for , but in Proposition 5.7 we will show that , and hence , is finitely generated for all .) Corollary 1.3 therefore implies the following result.…”