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Intersection pairing on hyperbolic 4-manifolds

Michael Kapovich

TL;DR

The paper proves an upper bound on the intersection pairing between two surface classes in a complete hyperbolic 4-manifold in terms of topological data of the representing surfaces, via a function of their Euler characteristics. The approach combines Margulis-thin/thick decomposition with piecewise-geodesic surface representatives, and introduces geometric tools in the thin part (Margulis cones/tubes, ruled films, and the Key Property of Index) to control intersections. The main result is an explicit exponential-type bound that holds first under a maximality condition and then in full generality, providing a 0-th order obstruction to the existence of complete hyperbolic structures on certain 4-manifolds. This connects geometric intersection data to topological invariants (via Thurston-like norms) and informs how the global hyperbolic geometry constrains homological information, with potential implications for counts of discrete Kleinian groups and related moduli questions.

Abstract

We prove an estimate on intersection pairing of homology classes in hyperbolic 4-manifolds in terms of Thurston norms of these classes.

Intersection pairing on hyperbolic 4-manifolds

TL;DR

The paper proves an upper bound on the intersection pairing between two surface classes in a complete hyperbolic 4-manifold in terms of topological data of the representing surfaces, via a function of their Euler characteristics. The approach combines Margulis-thin/thick decomposition with piecewise-geodesic surface representatives, and introduces geometric tools in the thin part (Margulis cones/tubes, ruled films, and the Key Property of Index) to control intersections. The main result is an explicit exponential-type bound that holds first under a maximality condition and then in full generality, providing a 0-th order obstruction to the existence of complete hyperbolic structures on certain 4-manifolds. This connects geometric intersection data to topological invariants (via Thurston-like norms) and informs how the global hyperbolic geometry constrains homological information, with potential implications for counts of discrete Kleinian groups and related moduli questions.

Abstract

We prove an estimate on intersection pairing of homology classes in hyperbolic 4-manifolds in terms of Thurston norms of these classes.
Paper Structure (5 sections, 25 theorems, 142 equations, 9 figures)

This paper contains 5 sections, 25 theorems, 142 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Geometry of Margulis cones
  3. Proof of Theorem \ref{['t1']} under the maximality condition
  4. Proof of Theorem \ref{['t1']} in general
  5. Appendix

Key Result

Theorem 1.2

There exists a function $C(.,.)$ such that for any complete hyperbolic 4-manifold $M$ and for any classes $[\sigma _1 ] , [ \sigma _2 ]$ in $H_2(M, \mathbb Z )$ which have incompressible representatives, we have:

Figures (9)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5:
  • ...and 4 more figures

Theorems & Definitions (42)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Conjecture 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 32 more