Surface bundles over surfaces with arbitrarily many fiberings

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 $d$.Comment: This version contains the same text as the published versio

Ropes, fractions, and moduli spaces

This is an exposition of John H. Conway's tangle trick. We discuss what the trick is, how to perform it, why it works mathematically, and finally offer a conceptual explanation for why a trick like this should exist in the first place. The mathematical centerpiece is the relationship between braids on three strands and elliptic curves, and we a draw a line from the tangle trick back to work of Weierstrass, Abel, and Jacobi in the 19th century. For the most part we assume only a familiarity with the language of group actions, but some prior exposure to the fundamental group is beneficial in places.Comment: Expository article. 13 pages with 10 figures. Suitable for graduate students and advanced undergrads. Comments welcome! New in V2: error in Example 2.1 corrected, brief discussion of variants of the trick, acknowledgements adde

Stratified braid groups I: monodromy

The space of monic squarefree complex polynomials has a stratification according to the multiplicities of the critical points. This is the first of a planned series of articles on the topology of these strata. At the center of our analysis is a study of the infinite-area translation surface associated to the logarithmic derivative $df/f$ of the polynomial. Here we determine the monodromy of these strata in the braid group, thus describing which braidings of the roots are possible if the orders of the critical points are required to stay fixed. Mirroring the story for holomorphic differentials on higher-genus surfaces, we find the answer is governed by the framing of the punctured disk induced by the horizontal foliation on the translation surface.Comment: 28 pages, 9 figures. Comments welcome