Surface Bundles in Topology, Algebraic Geometry, and Group Theory Nick Salter and Bena Tshishiku Surface Bundles 푆 and whose structure group is the group Diff(푆) of dif- A surface is one of the most basic objects in topology, but feomorphisms of 푆. In particular, 퐵 is covered by open −1 the mathematics of surfaces spills out far beyond its source, sets {푈훼} on which the bundle is trivial 휋 (푈훼) ≅ 푈훼 × 푆, penetrating deeply into fields as diverse as algebraic ge- and local trivializations are glued by transition functions ometry, complex analysis, dynamics, hyperbolic geometry, 푈훼 ∩ 푈훽 → Diff(푆). geometric group theory, etc. In this article we focus on the Although the bundle is locally trivial, any nontrivial mathematics of families of surfaces: surface bundles. While bundle is globally twisted, similar in spirit to the Möbius the basics belong to the study of fiber bundles, we hope strip (Figure 1). This twisting is recorded in an invariant to illustrate how the theory of surface bundles comes into called the monodromy representation to be discussed in the close contact with a broad range of mathematical ideas. section “Monodromy.” We focus here on connections with three areas—algebraic topology, algebraic geometry, and geometric group theory—and see how the notion of a surface bundle pro- vides a meeting ground for these fields to interact in beau- tiful and unexpected ways. What is a surface bundle? A surface bundle is a fiber bun- dle 휋 ∶ 퐸 → 퐵 whose fiber is a 2-dimensional manifold Nick Salter is a Ritt Assistant Professor at Columbia University. His email address is [email protected]. Bena Tshishiku is an assistant professor at Brown University. His email address is [email protected]. Communicated by Notices Associate Editor Daniel Krashen. For permission to reprint this article, please contact: Figure 1. The Möbius strip is the total space of a bundle over [email protected]. 푆1 whose fibers are diffeomorphic to [0, 1]. DOI: https://doi.org/10.1090/noti2016 146 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 2 Then 퐸 = {(푥, 푦, 푏) ∣ (푥, 푦) ∈ 푆(푏)} is the total space of an 푆-bundle over 퐵 under the projection map (푥, 푦, 푏) ↦ 푏. Here 퐵 is the configuration space of 푛 (ordered) points in ℂ. The study of this single 푆-bundle is already incredibly rich, with connections to representations of braid groups and geometric structures on moduli spaces of Riemann sur- faces [McM13]. f Vector bundles are also a source of surface bundles: given a rank-3 real vector bundle, the associated unit- sphere bundle is an 푆2-bundle. In fact, any 푆2-bundle is obtained from this construction because, by a theorem of Smale, Diff(푆2) is homotopy equivalent to the orthogonal group 푂(3) (this homotopy equivalence implies the bun- dle statement by the theory of classifying spaces discussed in “The Classification Problem”). On the other hand, if 푆푔 Figure 2. The mapping torus 퐸푓 of a surface diffeomorphism is a closed oriented surface of genus 푔 ≥ 1, then Diff(푆푔) is 푓 ∶ 푆 → 푆. Note that the Möbius strip (Figure 1) can be not homotopy equivalent to a compact Lie group. As such, constructed in a similar way. the study of 푆푔-bundles for 푔 ≥ 1 is the first instance of a A surface bundle 퐸 → 퐵 with fiber 푆 is also called an nonlinear bundle theory. There are many analogies between 푆-bundle over 퐵, and 퐸 is called the total space. Informally, the theory of vector bundles and surface bundles, but there one thinks of 퐸 as a family of surfaces parameterized by 퐵; i.e., are also many new phenomena, connections, and open for each 푏 ∈ 퐵, there is a surface 휋−1(푏) ≅ 푆. questions. Surface bundles in nature. Surface bundles arise natu- Conventions. For the remainder of this article we assume, 푆 rally across mathematics. The most basic source of - for simplicity, that 푆 = 푆푔 is a closed, oriented surface of bundles comes from the mapping torus construction. Given genus 푔 ≥ 1 (and at times 푔 ≥ 2). Working with oriented 푓 ∈ Diff(푆), define 퐸푓 as the quotient of [0, 1] × 푆 by iden- surfaces, we consider only orientation-preserving diffeo- tifying {0} × 푆 with {1} × 푆 by 푓; then 퐸푓 is the total space morphisms; for brevity, we suppress this from the notation 1 of an 푆-bundle over the circle 퐸푓 → 푆 . See Figure 2. Sur- and will not mention it further. prisingly, this simple-minded construction is ubiquitous in the classification of 3-manifolds and in particular hy- The mapping class group. Given the wealth of examples perbolic 3-manifolds. Thurston proved that if 푓 is suffi- of surface bundles described above, we need a good way ciently complicated (pseudo-Anosov; cf. Theorem 2), then to tell different surface bundles apart. We’ll discuss two 퐸푓 admits a hyperbolic structure, i.e., a Riemannian metric approaches to this—classifying spaces and monodromy— with sectional curvature 퐾 ≡ −1. Furthermore, by work of in “The Classification Problem” and “Monodromy.” Mon- Agol, Wise, and Kahn–Markovic, every closed hyperbolic odromy is a special feature for 푆푔-bundles compared to 3-manifold 푀 has a finite cover of the form 퐸푓 → 푀 for other bundle theories, and it is where the mapping class some 푓 ∶ 푆 → 푆 [Ago13]. group plays a prominent role. Surface bundles also figure prominently in 4-manifold To explain this, consider the mapping torus construc- theory. Donaldson [Don98] proved that every symplec- tion discussed above (Figure 2). If 푓 is isotopic to the iden- tic 4-manifold 푀 admits a Lefschetz fibration 푀 → ℂ푃1, tity (i.e., there is a path from 푓 to id in Diff(푆푔)), then 퐸푓 1 which can be viewed as a surface bundle where finitely is just the product bundle 푆 × 푆푔. More generally, for any many fibers are allowed to acquire singularities of asim- 푓 ∈ Diff(푆푔), the bundle 퐸푓 is unchanged if 푓 is changed by ple form (so-called nodes). an isotopy. Therefore, if we want to understand the differ- Surface bundles appear in algebraic geometry, where ent bundles obtained as mapping tori, we should start by 1 they are more commonly known as families of curves. Spe- considering the quotient Mod(푆푔) ∶= Diff(푆푔)/ Diff0(푆푔) cial examples can be obtained by simply writing down fam- by the (normal) subgroup of diffeomorphisms isotopic to ilies of equations. For instance, let 퐵 be the space of tuples the identity. The group Mod(푆푔) is called the mapping class 푏 = (푏1, … , 푏푛) of distinct points in ℂ, fix 푑 ≥ 2, and for group. It is isomorphic to the group 휋0 Diff(푆푔) of path 푏 ∈ 퐵, consider the surface components of Diff(푆푔). For example, Mod(푇2) ≅ SL (ℤ). Any 퐴 ∈ SL (ℤ) acts 푆(푏) = {(푥, 푦) ∈ ℂ2 ∶ 푦푑 = (푥 − 푏 ) ⋯ (푥 − 푏 )}. (1) 2 2 1 푛 linearly on ℝ2 and descends to 푇2, and, conversely, up 2 1Since Riemann surfaces have complex dimension one, algebraic geometers re- to homotopy or isotopy, a diffeomorphism of 푇 is deter- 2 2 fer to them as curves. mined by its action on 휋1(푇 ) ≅ ℤ . For 푔 ≥ 1, Mod(푆푔) FEBRUARY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 147 is an infinite, finitely presented group. In “Monodromy” 휋푖(BDiff(푆푔)) ≅ 휋푖−1(Diff(푆푔)). When 푔 ≥ 2, the homotopy we explain how Mod(푆푔) plays a central role, not only for type of Diff(푆푔) is as simple as possible. 푆 -bundles over 푆1, but for 푆 -bundles over any base. 푔 푔 Theorem 1 (Earle–Eells). If 푔 ≥ 2, then the identity compo- The Classification Problem nent Diff0(푆푔) < Diff(푆푔) is contractible. Consequently, the Diff(푆 ) → Mod(푆 ) In this section we describe the basic tools and framework surjection 푔 푔 is a homotopy equivalence. from algebraic topology for studying 푆-bundles. As men- The homotopy type of Diff(푆푔) for 푔 = 0, 1 is also tioned above, we focus on the case 푆 = 푆푔. known: Diff(푆2) is homotopy equivalent to 푂(3), and ′ Two bundles 퐸 → 퐵 and 퐸 → 퐵 are isomorphic if there Diff(푇2) is homotopy equivalent to 푇2 ⋊ SL (ℤ); see e.g. ′ 2 is a diffeomorphism 퐸 → 퐸 that sends fibers to fibers and [Mor01]. Theorem 1 was originally proved using complex covers the identity map on 퐵. analysis (Teichmüller theory) and PDE; a purely topologi- Optimistically, one would like to solve the classifica- cal proof was given by Gramain (see [Hat]). tion problem: for a given 퐵, determine the set of isomor- By Theorem 1, BDiff(푆푔) is homotopy equivalent to phism classes of 푆푔-bundles 퐸 → 퐵. This problem can be BMod(푆푔) for 푔 ≥ 2. Since Mod(푆푔) is a discrete translated to a homotopy-theoretic problem via classifying group, its classifying space is an Eilenberg–Mac Lane space space theory. BMod(푆푔) ≅ 퐾(Mod(푆푔), 1). Observe that a map 푓 ∶ Usually the classification problem is too difficult to 퐵 → BDiff(푆푔) ≃ BMod(푆푔) induces a homomorphism solve completely. In practice one wants a rich collection 휋1(퐵) → Mod(푆푔). This is a fundamental invariant of the of invariants that (i) measure topological properties of 푆푔- bundle associated to 푓, known as the monodromy; we dis- bundles and (ii) enable us to distinguish 푆푔-bundles found cuss it further in “Monodromy.” in nature. In the study of vector bundles, a primary role is In practice, it can be useful to have a concrete model played by characteristic classes. Surface bundles also have a for BDiff(푆푔).