Exotic smooth structures on topological fibre bundles I

Sebastian Goette; Kiyoshi Igusa; Bruce Williams

Home/Archive/arXiv

arXiv · 1203.2203

Exotic smooth structures on topological fibre bundles I

Sebastian Goette·Kiyoshi Igusa·Bruce Williams

View original source ↗Read source document ↗Export metadata ↓

Abstract

When two smooth manifold bundles over the same base are fiberwise tangentially homeomorphic, the difference is measured by a homology class in the total space of the bundle. We call this the relative smooth structure class. Rationally and stably, this is a complete invariant. We give a more or less complete and self-contained exposition of this theory which is a reformulation of some of the results of [7]. An important application is the computation of the Igusa-Klein higher Reidemeister torsion invariants of these exotic smooth structures. Namely, the higher torsion invariant is equal to the Poincaré dual of the image of the smooth structure class in the homology of the base. This is proved in the companion paper [11] written by the first two authors.

Explore related subjects

math.KT math.AT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Sebastian Goette, Kiyoshi Igusa, Bruce Williams. 2012-04-07. Exotic smooth structures on topological fibre bundles I. https://arxiv.org/abs/1203.2203

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Gerstenhaber algebra structure on the Hochschild cohomology ring of the Xu--Snashall algebra

Let $A$ be a finite dimensional algebra and let $\rmHH^*(A)$ be its Hochschild cohomology ring, which is a Gerstenhaber algebra. Denote by $\calN$ (resp. $G$, $\calG$) the ideal (resp. weak Gerstenhaber ideal, Gerstenhaber ideal) generated by all homogeneous nilpotent elements. Motivated by their work on support varieties via Hochschild cohomology, Snashall and Solberg conjectured that $\rmHH^*(A)/\calN$ is a finitely generated algebra. Xu constructed a counterexample to the Snashall-Solberg conjecture over a base field of characteristic two, and Snashall generalized this example to arbitrary characteristic. Hermann further asked whether $\rmHH^*(A)/G$ is a finitely generated algebra and suggested considering first the Xu--Snashall algebra. In this paper, we answer this question for the Xu--Snashall algebra. In fact, by explicitly computing the Gerstenhaber algebra structure on the Hochschild cohomology ring, we show that $G=\calN$; hence $\rmHH^*(A)/G=\rmHH^*(A)/\calN$ is not a finitely generated algebra. Furthermore, we show that $\rmHH^*(A)/\calG\cong K$. Therefore, one may ask whether, for a finite dimensional algebra $A$, $\rmHH^*(A)/\calG$ is always a finitely generated algebra. Our main tools are two-sided Anick resolutions and weak self-homotopies.

math.KT↗

arXiv2026

K-theory of Matroids and Monoid Schemes

This paper continues the study of the $K$-theory of monoid schemes, using it to give a useful definition of the higher $K$-theory of a matroid via its Bergman fan.

math.KT↗

arXiv2026

Improved injective stability for relative $\mathrm{K_1Sp}$-groups

We prove a relative version of Vorst's theorem concerning the equality of the group of all invertible matrices and the group of all elementary matrices over $R[X]$ with respect to an ideal $I\subset R$ such that $R/I$ is regular, where $R$ is a regular $k$-spot. We then introduce a relative version of the symplectic elementary Witt group and show that it fits into a relative version of the Karoubi periodicity sequence. Combining these results, we improve the existing injective stability bounds for relative linear and symplectic $\mathrm{K_1}$-groups of smooth affine algebras over various base fields. As an application, we give a necessary and sufficient condition for the freeness of stably free modules over smooth real $4$-folds with empty real locus.

math.KT↗