Abstract
Given a closed simply connected manifold M of dimension 2nge 6, we compare the ring of characteristic classes of smooth oriented bundles with fibre M to the analogous ring resulting from replacing M by the connected sum Msharp Sigma with an exotic sphere Sigma . We show that, after inverting the order of Sigma in the group of homotopy spheres, the two rings in question are isomorphic in a range of degrees. Furthermore, we construct infinite families of examples witnessing that inverting the order of Sigma is necessary.
Highlights
The classifying space BDiff+(M) of the topological group of orientation-preserving diffeomorphisms of a closed oriented manifold M in the smooth Whitney-topology classifies smooth oriented fibre bundles with fibre M
A programme of Galatius–Randal-Williams [11,13,14] culminated in an identification of the cohomology in consideration in a range of degrees in purely homotopy theoretical terms for all connected manifolds M of dimension 2n ≥ 6
In the first part of this work, we use the work of Galatius–Randal-Williams to show that for closed connected manifolds M of dimension 2n ≥ 6, the cohomology H∗(BDiff+(M)) is insensitive to replacing M by M in a range of degrees, at least after inverting the order of
Summary
The classifying space BDiff+(M) of the topological group of orientation-preserving diffeomorphisms of a closed oriented manifold M in the smooth Whitney-topology classifies smooth oriented fibre bundles with fibre M. In the first part of this work, we use the work of Galatius–Randal-Williams to show that for closed connected manifolds M of dimension 2n ≥ 6, the cohomology H∗(BDiff+(M)) is insensitive to replacing M by M in a range of degrees, at least after inverting the order of. Remark Using recent work of Friedrich [9], one can enhance Theorem A to all oriented, closed, connected manifolds M of dimension 2n ≥ 6 whose associated group ring Z[π1 M] has finite unitary stable rank. By Theorem B, the first homology groups of BDiff+(Wg) and BDiff+(Wg ) are isomorphic after inverting 2 This holds with Wg replaced by any connected manifold M, which raises the question of whether the failure for Theorem A to hold integrally is purely a 2-primary phenomenon.
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