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Simplicial volume of open books in dimension 4

Thorben Kastenholz

TL;DR

This note proves that the simplicial volume of any 4-manifold admitting an open book decomposition vanishes, by adapting the Bucher-Neofytidis approach to the 4D setting and avoiding reliance on Geometrization. The method passes to irreducible pages via a JSJ-type decomposition, constructs arbitrarily small fundamental cycles on the pieces, and glues them along amenable boundary components to obtain a closed cycle with vanishing norm. A key consequence is that Quinn's asymmetric signature invariant cannot characterize open books in dimension four, even after connected-sum stabilizations. The results also imply the existence of 4-manifolds with vanishing asymmetric signature that do not admit open books, clarifying fundamental distinctions between 4D and higher-dimensional open book theory and expanding the class of 4-manifolds with vanishing simplicial volume in this context.

Abstract

In this short note we adapt a proof by Bucher and Neofytidis to prove that the simplicial volume of 4-manifolds admitting an open book decomposition vanishes. In particular this shows that Quinns signature invariant, which detects the existence of an open book decomposition in dimensions above 4, is insufficient to characterize open books in dimension 4, even if one allows arbitrary stabilizations via connected sums.

Simplicial volume of open books in dimension 4

TL;DR

This note proves that the simplicial volume of any 4-manifold admitting an open book decomposition vanishes, by adapting the Bucher-Neofytidis approach to the 4D setting and avoiding reliance on Geometrization. The method passes to irreducible pages via a JSJ-type decomposition, constructs arbitrarily small fundamental cycles on the pieces, and glues them along amenable boundary components to obtain a closed cycle with vanishing norm. A key consequence is that Quinn's asymmetric signature invariant cannot characterize open books in dimension four, even after connected-sum stabilizations. The results also imply the existence of 4-manifolds with vanishing asymmetric signature that do not admit open books, clarifying fundamental distinctions between 4D and higher-dimensional open book theory and expanding the class of 4-manifolds with vanishing simplicial volume in this context.

Abstract

In this short note we adapt a proof by Bucher and Neofytidis to prove that the simplicial volume of 4-manifolds admitting an open book decomposition vanishes. In particular this shows that Quinns signature invariant, which detects the existence of an open book decomposition in dimensions above 4, is insufficient to characterize open books in dimension 4, even if one allows arbitrary stabilizations via connected sums.

Paper Structure

This paper contains 4 sections, 4 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Open book decompositions in dimension $4$
  3. Reducible pages and their diffeomorphisms
  4. Irreducible pages

Key Result

Theorem A

There exists a $4$-manifold $X$, whose asymmetric signature vanishes, but which does not admit an open book decomposition. Moreover for any $4$-manifold $Y$, the connected sum $X\# Y$ does not admit an open book decomposition as well.

Theorems & Definitions (5)

  • Theorem A
  • Theorem B
  • Definition 2.1
  • Proposition 2.2: Proposition 3.2 in HatcherMcCullough
  • Corollary 2.3