Mumford Classes Research Articles

Polynomial Relations Among Kappa Classes on the Moduli Space of Curves

Abstract We construct an infinite collection of universal—independent of $(g,n)$—polynomials in the Miller–Morita–Mumford classes $\kappa _m\in H^{2m}( \overline{\mathcal{M}}_{g,n},{\mathbb{Q}})$, defined over the moduli space of genus $g$ stable curves with $n$ labeled points. We conjecture vanishing of these polynomials in a range depending on $g$ and $n$.

Characteristic classes for families of bundles

The generalized Miller–Morita–Mumford classes of a manifold bundle with fiber M depend only on the underlying tau _M-fibration, meaning the family of vector bundles formed by the tangent bundles of the fibers. This motivates a closer study of the classifying space for tau _M-fibrations, Baut(tau _M), and its cohomology ring, i.e., the ring of characteristic classes of tau _M-fibrations. For a bundle xi over a simply connected Poincaré duality space, we construct a relative Sullivan model for the universal xi -fibration with holonomy in a given connected monoid, together with explicit cocycle representatives for the characteristic classes of the canonical bundle over its total space. This yields tools for computing the rational cohomology ring of Baut(xi ) as well as the subring generated by the generalized Miller–Morita–Mumford classes. To illustrate, we carry out sample computations for spheres and complex projective spaces. We discuss applications to tautological rings of simply connected manifolds and to the problem of deciding whether a given tau _M-fibration comes from a manifold bundle.

Characteristic classes of bundles of K3 manifolds and the Nielsen realization problem

Let $K$ be the K3 manifold. In this note, we discuss two methods to prove that certain generalized Miller--Morita--Mumford classes for smooth bundles with fiber $K$ are non-zero. As a consequence, we fill a gap in a paper of the first author, and prove that the homomorphism $Diff(K)\to \pi_0 Diff(K)$ does not split. One of the two methods of proof uses a result of Franke on the stable cohomology of arithmetic groups that strengthens work of Borel, and may be of independent interest.

Characteristic numbers of manifold bundles over surfaces with highly connected fibers

We study smooth bundles over surfaces with highly connected almost parallelizable fiber $M$ of even dimension, providing necessary conditions for a manifold to be bordant to the total space of such a bundle and showing that, in most cases, these conditions are also sufficient. Using this, we determine the characteristic numbers realized by total spaces of bundles of this type, deduce divisibility constraints on their signatures and $\hat{A}$-genera, and compute the second integral cohomology of ${\rm BDiff}^+(M)$ up to torsion in terms of generalized Miller--Morita--Mumford classes. We also prove analogous results for topological bundles over surfaces with fiber $M$ and discuss the resulting obstructions to smoothing them.

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.

Generalised Miller–Morita–Mumford classes for block bundles and topological bundles

The most basic characteristic classes of smooth fibre bundles are the generalised Miller-Morita-Mumford classes, obtained by fibre integrating characteristic classes of the vertical tangent bundle. In this note we show that they may be defined for more general families of manifolds than smooth fibre bundles: smooth block bundles and topological fibre bundles.

Tautological Classes of the Stack of Rational Nodal Curves

This is the second in a series of three papers in which we investigate the rational Chow ring of the stack 𝔐0 consisting of nodal curves of genus 0. Here we define the basic classes: the classes of strata and the Mumford classes.

Secondary characteristic classes of surface bundles

The Miller-Morita-Mumford classes associate to an oriented surface bundle $E\to B$ a class $\kappa_i(E) \in H^{2i}(B;\Z)$. In this note we define for each prime $p$ and each integer $i\geq 1$ a secondary characteristic class $\lambda_i(E) \in H^{2i(p-1)-2}(B;\Z)/\Z\kappa_{i(p-1)-1}$. The mod $p$ reduction $\lambda_i(E) \in H^*(B; \F_p)$ has zero indeterminacy and satisfies $p\lambda_i(E) = \kappa_{i(p-1)-1}(E) \in H^*(B;\Z/p^2)$.

Integral Riemann–Roch formulae for cyclic subgroups of mapping class groups

AbstractThe first author conjectured certain relations for Morita–Mumford classes and Newton classes in the integral cohomology of mapping class groups (integral Riemann–Roch formulae). In this paper, the conjecture is verified for cyclic subgroups of mapping class groups.

Graph cohomology and Kontsevich cycles

We use the duality between compactly supported cohomology of the associative graph complex and the cohomology of the mapping class group to show that the duals of the Kontsevich cycles [ W λ ] correspond to polynomials in the Miller–Morita–Mumford classes. We also compute the coefficients of the first two terms of this polynomial. This extends the results of (Combinatorial Miller–Morita–Mumford classes and Witten cycles, math.GT/0207042, 2002), giving a more detailed answer to a question of Kontsevich (Commun. Math. Phys. 147(1) (1992) 1) and verifying more of the conjectured formulas of Arbarello and Cornalba (J. Algebraic Geom. 5 (1996) 705).

Weierstrass points and Morita–Mumford classes on hyperelliptic mapping class groups

We describe the Morita–Mumford classes on the hyperelliptic mapping class group of genus g⩾2, and prove a slightly weakened version of Akita's conjectures in the hyperelliptic case. The proof involves the notion of Weierstrass points.

Periodic surface automorphisms and algebraic independence of Morita–Mumford classes

We prove a vanishing theorem for the Morita–Mumford classes on periodic surface automorphisms, and construct enough periodic automorphisms to give an alternative and elementary proof of the stable rational algebraic independence of the Morita–Mumford classes, originally shown by Miller (J. Differential Geom. 24 (1986) 1–14) and Morita (Invent. Math. 90 (1987) 551–557).

Higher Weil-Petersson volumes of moduli spaces of stablen-pointed curves

Moduli spaces of compact stable $n$-pointed curves carry a hierarchy of cohomology classes of top dimension which generalize the Weil-Petersson volume forms and constitute a version of Mumford classes. We give various new formulas for the integrals of these forms and their generating functions. We also discuss their relation to the Kuenneth formula in quantum cohomology.

Invariants of smooth 4-manifolds from topological gravity

A theory of topological gravity for smooth 4-manifolds is constructed. For a given 4-manifold, M, the measure in the functional integral over metrics on M is concentrated at metrics for which the self-dual part of the Weyl tensor vanishes, and the Ricci scalar is constant. A cohomology theory is presented, nontrivial observables are found, and identified with closed forms on the moduli space of such metrics on M. These observables are topological invariants of M, the analogues of Mumford classes for 4-manifolds admitting such metrics. An explicit analysis is provided for T 4.

Topological gravity and moduli space

We construct a theory of topological gravity in two dimensions, with a measure that is concentrated on constant curvature metrics. Following the insights of Ouvry, Stora and van Baal, we construct an equivalent cohomology theory, which we interpret as a cohomology theory for moduli space. We gauge fix our diffeomorphism-invariant action, following treatment in a previous article of the first author. We present nontrivial observables in this theory, and evaluate them in the weak coupling limit. These classical expressions are pushdowns of a curvature form (and its powers) on a line bundle over the universal curve. They bear a striking resemblance to Wolpert's expressions for the Mumford classes.