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Simplicial volume and isolated, closed totally geodesic submanifolds of codimension one

Chris Connell, Yuping Ruan, Shi Wang

Abstract

We show that for any closed Riemannian manifold with dimension at least two and with nonpositive curvature, if it admits an isolated, closed totally geodesic submanifold of codimension one, then its simplicial volume is positive. As a direct corollary of this, for any nonpositively curved analytic manifold with dimension at least three, if its universal cover admits a codimension one flat, then either it has non-trivial Euclidean de Rham factors, or it has positive simplicial volume.

Simplicial volume and isolated, closed totally geodesic submanifolds of codimension one

Abstract

We show that for any closed Riemannian manifold with dimension at least two and with nonpositive curvature, if it admits an isolated, closed totally geodesic submanifold of codimension one, then its simplicial volume is positive. As a direct corollary of this, for any nonpositively curved analytic manifold with dimension at least three, if its universal cover admits a codimension one flat, then either it has non-trivial Euclidean de Rham factors, or it has positive simplicial volume.

Paper Structure

This paper contains 27 sections, 53 theorems, 211 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Connections with relative hyperbolicity
  4. Connections with $4$-dimensional analytic manifolds of nonpositive curvature.
  5. Comments on graph manifolds
  6. Outline of the proof and plan of the paper
  7. Setting and basic notations
  8. Hyperbolicity away from submanifolds
  9. Barycentric simplices and almost separation
  10. Barycentric simplices and their properties
  11. Almost separation by elements in $\Gamma F$
  12. $\Omega(\cdot,\cdot)$, $\Theta(\cdot,\cdot)$ and their properties
  13. Bicombing construction inspired by Mineyev
  14. Combinatorics of simplices and $\Theta$-separations
  15. Motivations: actual separation of simplices by elements in $\Gamma F$
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $M$ be a connected, closed, oriented Riemannian manifold of nonpositive curvature with dimension at least two. If $M$ admits an isolated, codimension one, closed totally geodesic submanifold $N$, then $||M||>0$.

Figures (24)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 6:
  • Figure 7:
  • ...and 19 more figures

Theorems & Definitions (169)

  • Theorem 1.1
  • Remark
  • Corollary 1.2
  • Corollary 1.3
  • Conjecture 1.4
  • Conjecture 1.5
  • Conjecture 1.6
  • Lemma 3.1
  • proof
  • Proposition 3.2: BridsonHaefliger99
  • ...and 159 more