We prove two general results concerning spectral sequences of FI-modules. These results can be used to significantly improve stable ranges in a large portion of the stability theorems for FImodules currently in the literature. We work this out in detail for the cohomology of configuration spaces where we prove a linear stable range and the homology of congruence subgroups of general linear groups where we prove a quadratic stable range. Previously, the best stable ranges known in these examples were exponential. Up to an additive constant, our work on congruence subgroups verifies a conjecture of Djament.

We construct smooth bundles with base and fiber products of two spheres whose total spaces have nonvanishing $\hat{A}$-genus. We then use these bundles to locate nontrivial rational homotopy groups of spaces of Riemannian metrics with lower curvature bounds for all ${{\operatorname{Spin}}}$ manifolds of dimension 6 or at least 10, which admit such a metric and are a connected sum of some manifold and $S^n \times S^n$ or $S^n \times S^{n+1}$, respectively. We also construct manifolds $M$ whose spaces of Riemannian metrics of positive scalar curvature have homotopy groups that contain elements of infinite order that lie in the image of the orbit map induced by the push-forward action of the diffeomorphism group of $M$.

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.

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For a Lie group G and a smooth manifold W , we study the difference between smooth actions of G on W and fiber bundles over the classifying space of G with fiber W and structure group Diff(W ). In particular, we exhibit smooth manifold bundles over BSU (2) that are not induced by an action. The main tool for reaching this goal is a technical result that gives a constraint for the values of tautological classes of the fiber bundle associated to a group action.