Characteristic numbers of manifold bundles over surfaces with highly connected fibers

Abstract

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 andÂ-genera, and compute the second integral cohomology of 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. By work of Chern-Hirzebruch-Serre [14], the signature of a closed oriented manifold is multiplicative in fiber bundles as long as the fundamental group of the base acts trivially on the rational cohomology of the fiber. The necessity of this assumption was illustrated by Kodaira [26], Atiyah [6], and Hirzebruch [23] who constructed manifolds of non-trivial signature fibering over surfaces, whereupon Meyer [31, 32] computed the minimal positive signature arising in this way to be 4. The divisibility of the signature σ : Ω SO * → Z by 4 is therefore a necessary condition for a manifold to fiber over a surface up to bordism, which, when combined with the vanishing of a certain Stiefel-Whitney number, is also sufficient [2, Theorem 3]. A more refined problem is to decide which manifolds fiber over a surface up to bordism with prescribed d-dimensional fiber M , or equivalently, to determine the image of the map Ω SO 2 BDiff + (M) → Ω SO d+2 , defined on the bordism group of smooth oriented M-bundles over surfaces, which assigns to a bundle its total space. The main objective of this work is to provide a solution to this problem, and its analogue for topological M-bundles, for highly connected, almost parallelizable, 2n-dimensional manifolds M such as M = g (S n × S n) that satisfy a condition on their genus g(M) = max{g 0 | there is a manifold N such that M ∼ = g (S n × S n) N }. In the first part of this work, we focus on smooth bundles and use parametrized Pontryagin-Thom theory to show that the bordism class of a manifold that smoothly fibers over a surface with fiber M as above lifts to the bordism group Ω n 2n+2 of highly connected (that is, nconnected) manifolds, and that this property, together with the divisibility of the signature by 4, detects such bordism classes for g(M) 5.

Mapping class groups of highly connected $$(4k+2)$$-manifolds

“…Remark After the completion of this work, Burklund-Hahn-Senger [11] and Burklund-Senger [12] showed that for n odd, the homotopy sphere Q ∈ 2n+1 bounds a parallelisable manifold if and only if n = 11. This implies in particular that aside from the case n = 11 (i) the canonical map coker(J ) 2n+1 → τ >n 2n+1 is an isomorphism, which extends the theorem attributed to Schultz and Wall above, (ii) the conjecture of Galatius-Randal-Williams mentioned in the third part of the previous remark holds, and (iii) the minimal signature σ n appearing in Theorem E is computable from [34,Prop. 2.15].…”

“…for n ≥ 4 even in which case the morphisms t * ⊕ p * and t * ⊕ρ * are isomorphisms for all g ≥ 1. (ii) As shown in [34,Prop. 2.15], the minimal signature σ n is nontrivial for n odd, grows very quickly with n, and can be expressed in terms of Bernoulli numbers.…”

Simply Connected Manifolds With Large Homotopy Stable Classes

“…The precise value of can be calculated, assuming knowledge of the relevant Bernoulli numbers, from [KR20, Lemma 2.7]. In particular, .…”

Simply-connected manifolds with large homotopy stable classes

“…Acknowledgements. We would like to thank Manuel Krannich for advice about the homotopy sphere Σ Q , specifically for drawing our attention to Section 2 of [KR20], and Jens Reinhold for comments on an earlier draft of this paper.…”