Surface bundles over surfaces with arbitrarily many fiberings

TL;DR

This paper constructs examples of 4-manifolds that admit arbitrarily many distinct surface bundle structures over surfaces, demonstrating that the number of such structures can grow exponentially with the manifold's Euler characteristic.

Contribution

It provides the first known examples of surface bundles over surfaces with three or more fiberings, expanding understanding of the complexity of surface bundle structures.

Findings

Constructed 4-manifolds with at least n fiberings for any n ≥ 3

Examples where monodromy lies in the Torelli group

Number of fiberings grows exponentially with Euler characteristic

Abstract

In this paper we give the first example of a surface bundle over a surface with at least three fiberings. In fact, for each $n \ge 3$ we construct $4$-manifolds $E$ admitting at least $n$ distinct fiberings $p_i: E \to \Sigma_{g_i}$ as a surface bundle over a surface with base and fiber both closed surfaces of negative Euler characteristic. We give examples of surface bundles admitting multiple fiberings for which the monodromy representation has image in the Torelli group, showing the necessity of all of the assumptions made in the main theorem of our recent paper [arXiv:1404.0066]. Our examples show that the number of surface bundle structures that can be realized on a $4$-manifold $E$ with Euler characteristic $d$ grows exponentially with $d$.