Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Affine transverse foliations in sphere bundles

Ilya Alekseev, Ivan Nasonov, Gaiane Panina

Abstract

Let~$S^{n-1}\rightarrow E \rightarrow M^n$ be an oriented sphere bundle over an oriented closed manifold with amenable fundamental group. We provide an elementary proof that the Euler number of the bundle vanishes whenever the bundle admits a smooth transverse affine foliation.

Affine transverse foliations in sphere bundles

Abstract

Let~ be an oriented sphere bundle over an oriented closed manifold with amenable fundamental group. We provide an elementary proof that the Euler number of the bundle vanishes whenever the bundle admits a smooth transverse affine foliation.
Paper Structure (12 sections, 15 theorems, 23 equations, 2 figures)

This paper contains 12 sections, 15 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. How to compute the Euler number
  3. Proof of Theorem \ref{['ThmMain']} via ingredients of M. Gromov, J. Hirsh, W. Thurston, and D. Sullivan
  4. Preliminary construction. A local formula
  5. Flat quasisections
  6. Construction of a cell decomposition $\widetilde{\mathcal{B}}$ of the base. A random partial section
  7. Evaluation of $\mathbb{E}(ind_s(v))$. The local formula for the Euler class for antipodally generic quasisections
  8. The general case
  9. Proof of Theorem \ref{['ThmMain']}
  10. Appendix
  11. Preliminaries on triangulations
  12. Proof of Proposition \ref{['PropTriang']}

Key Result

Theorem 1

Let $M^n$ be a closed oriented smooth manifold whose fundamental group is amenable and let $S^{n-1}\rightarrow E\rightarrow M^n$ be an oriented sphere bundle. If $E$ admits a smooth transverse affine foliation, then the Euler number of the bundle vanishes.

Figures (2)

  • Figure 1: A fragment of the cell decompositions ${\mathcal{B}}$ (left) and the corresponding fragment of $\widetilde{\mathcal{B}}$ (right). The neighborhood of the self-intersection point of $\pi(\partial \mathcal{Q})$ is divided into $2^n$ chambers. After passing to $\widetilde{\mathcal{B}}$, the self-intersection point turns to $2^n~$ vertices. We list the sheets over each of the chambers.
  • Figure 2: A cell decomposition $\mathcal{C}$ (left), the cell decomposition $\mathcal{B}$ with $\nu$ (center), and the $2$-cube dual to the central vertex together with its symmetric triangulation (right).

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • proof
  • proof
  • proof
  • Lemma 1
  • Proposition 2
  • Definition 1
  • Definition 2
  • ...and 19 more