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). From the point of view of homotopy theory theory of characteristic classes, but as we explain, these are (as in [MW05, Hat]), the most useful model is the “Grass- fairly coarse invariants. mannian” of surfaces embedded in ℝ∞. Unfortunately it Classifying space for surface bundles. For a CW-complex would be too much of a detour to dwell on this further;
퐵, let Bun푆푔 (퐵) be the set of isomorphism classes of 푆푔- see [Hat]. bundles over 퐵. For each 푔 ≥ 0, there is a space BDiff(푆푔) A second model for BDiff(푆푔) is known as moduli space and a bijection ℳ푔. Using Theorem 1 it suffices to give a model for BMod(푆푔). For this we need a contractible space with Bun푆 (퐵) ≅ [퐵, BDiff(푆푔)] (2) 푔 a free, properly discontinuous action of Mod(푆푔). To where the right-hand side is the set of homotopy classes this end, consider the space ℋ of hyperbolic metrics on 푆푔. The group Diff(푆푔) acts by pullback of metrics, and of maps 퐵 → BDiff(푆푔). The space BDiff(푆푔) is called the Diff0(푆푔) acts freely. Miraculously, the Teichmüller space classifying space for 푆푔-bundles. In the language of homo- topy theory, the functor 퐵 ↦ Bun (퐵) is represented, and 풯 ∶= ℋ/ Diff0(푆푔) is finite-dimensional and contractible: 푆푔 6푔−6 풯 ≅ ℝ . There is a natural action of Mod(푆푔) on 풯, BDiff(푆푔) is the universal element. and the quotient ℳ푔 ∶= 풯/ Mod(푆푔) is the moduli space The space BDiff(푆푔) is defined uniquely up to homotopy of hyperbolic metrics on 푆푔. by the property that there is a principal Diff(푆푔)-bundle We would like to say that ℳ푔 is a model for BMod(푆푔), 푃 → BDiff(푆푔) with 푃 contractible. In the bijection (2), but this is not true because Mod(푆푔) does not act freely on given a map 퐵 → BDiff(푆푔), the corresponding 푆푔-bundle 퐸 → 퐵 is obtained by pullback: 풯. Indeed, the stabilizer of [휇] ∈ 풯 is the isometry group Isom(푆푔, 휇), which is finite but not necessarily trivial. To / 푃×푆푔 circumvent this issue, we use the fact that Mod(푆푔) contains 퐸 Diff(푆푔) many finite-index, torsion-free subgroups Γ ≤ Mod(푆푔). For such a group, 풯/Γ is a genuine 퐾(Γ, 1), and there is a finite / covering 풯/Γ → ℳ of orbifolds. For this reason, we call 퐵 BDiff(푆푔) 푔 ℳ푔 a virtual classifying space for Mod(푆푔). This is adequate The bundle on the right is known as the universal 푆푔-bundle. for many purposes; e.g., there is an isomorphism See [Mor01] for more details. ∗ ∗ 퐻 (BMod(푆푔); ℚ) ≅ 퐻 (ℳ푔; ℚ). We want to understand the homotopy type of BDiff(푆푔). As mentioned above, there is a fibration Diff(푆푔) → 푃 → The moduli space ℳ푔 is many things at once. In addi- BDiff(푆푔) where 푃 is contractible. Hence the homotopy tion to the set of hyperbolic metrics up to isometry, it is types of Diff(푆푔) and BDiff(푆푔) are closely related; in- the set of algebraic curves up to isomorphism and the set deed by the long exact sequence of homotopy groups, of Riemann surfaces up to biholomorphism. This brings
148 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 2 ∗ the study of 푆푔-bundles into close contact with hyperbolic Despite all of this progress, 퐻 (BDiff(푆푔); ℚ) is still geometry, complex analysis, and algebraic geometry. mostly unknown. By an Euler characteristic computation Characteristic classes. There are very few spaces 퐵 for for ℳ푔 by Harer–Zagier, the MMM classes account for a small fraction of the total cohomology. We have only which Bun푆푔 (퐵) ≅ [퐵, BDiff(푆푔)] has been computed com- pletely. Instead one can ask for invariants that distinguish scratched the surface. We conclude this section with a simple geometric ar- different elements of [퐵, BDiff(푆푔)]. gument that shows 퐻1(BMod(푆 ); ℤ) = 0 for 푔 ≥ 3. Re- A characteristic class for 푆푔-bundles is a function 푐 that 푔 ab assigns to each 푆푔-bundle 퐸 → 퐵 a cohomology class calling that 퐻1(퐵퐺) is the abelianization 퐺 , it suffices to ∗ ab 푐(퐸) ∈ 퐻 (퐵). In order to be useful, this function should show Mod(푆푔) = 0. Dehn proved that Mod(푆푔) is gen- be natural with respect to bundle pullbacks: given a pull- erated by mapping classes known as Dehn twists that are back square supported on an annulus in 푆푔 whose complement is con- 2 ∗ / nected. Any two such Dehn twists are conjugate. There- 휙 (퐸) 퐸 ab fore, Mod(푆푔) is a quotient of ℤ, generated by the image of any Dehn twist 퐴. There is a relation 퐴퐵퐶 = 퐷퐸퐹퐺 be- 휙 / 퐵′ 퐵 tween seven Dehn twists known as the lantern relation (Fig- ure 3). For 푔 ≥ 3 all seven annuli can be chosen to have ∗ ∗ ∗ ′ we require 푐(휙 (퐸)) = 휙 (푐(퐸)) in 퐻 (퐵 ). Equivalently, connected complement, so that the image of this relation a characteristic class is a natural transformation 푐 ∶ in Mod(푆 )ab proves 3퐴 = 4퐴 or 퐴 = 0. This concludes the ∗ 푔 Bun푆(⋅) → 퐻 (⋅). proof. See also [FM12, §5.1]. Since every 푆푔-bundle 퐸 → 퐵 is obtained by pullback from the universal 푆푔-bundle over BDiff(푆푔), any cohomol- ∗ ogy class 푐 ∈ 퐻 (BDiff(푆푔)) defines a characteristic class; conversely, every characteristic class is of this form (evalu- ∗ ate on the universal bundle). In other words, 퐻 (BDiff(푆푔)) is the set (or ring) of all characteristic classes of 푆푔-bundles. ∗ Computing 퐻 (BDiff(푆푔)) is of fundamental impor- CBE tance for studying 푆푔-bundles, but it is also of interest in other fields. By the preceding discussion, FG ∗ ∗ 퐻 (BDiff(푆푔); ℚ) ≅ 퐻 (BMod(푆푔); ℚ) ∗ ≅ 퐻 (ℳ푔; ℚ). A For our purpose, it is noteworthy that elements in the cohomology of Mod(푆푔) and ℳ푔 give characteristic classes of D 푆푔-bundles. Observe that the space BDiff(푆푔), the group Mod(푆푔), Figure 3. Dehn twists about these curves satisfy the lantern relation 퐴퐵퐶 = 퐷퐸퐹퐺. and the moduli space ℳ푔 are most naturally objects of al- gebraic topology, geometric group theory, and algebraic geometry, respectively. There has been a fertile exchange of ideas, tools, and techniques between these areas. To Monodromy show this interaction, we briefly mention some of what In “Surface Bundles” we saw that the mapping torus con- ∗ 1 is known about 퐻 (BDiff(푆푔); ℚ). Much of this is dis- struction provides a rich supply of 푆푔-bundles over 푆 , but cussed in [Mor01] and references therein. The groups the argument of the preceding paragraph shows that none ∗ 퐻 (BMod(푆푔)) satisfy homological stability, meaning that for of these bundles are distinguished by characteristic classes! 푖 each 푖 ≥ 0, 퐻 (BMod(푆푔)) is independent of 푔 when 푔 ≫ 푖. In this section we discuss the monodromy representation of This was proved by Harer in the early 1980s. Around the an 푆푔-bundle. We will see that this is a complete invariant, same time, Morita and Miller defined certain characteris- so that in some sense we face the opposite problem: the 2푖 tic classes 푒푖 ∈ 퐻 (BDiff(푆푔)), and Mumford defined anal- challenge is to distill practical, computable information ogous classes in the Chow ring of ℳ푔. Collectively these from the monodromy. are known as MMM classes or as 휅 classes. Mumford con- jectured that these classes generate the cohomology in de- 2 To see this, cut 푆푔 along either annulus. By the classification of surfaces, the grees 푖 ≪ 푔, and this was proved in 2002 by Madsen–Weiss, cut-open surfaces are seen to be homeomorphic. This homeomorphism can be who determined the homotopy type of BDiff(푆푔) “in the extended via the identity across the annuli, yielding a map of 푆푔 taking one an- limit” as 푔 → ∞ [MW05]. nulus to the other.
FEBRUARY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 149 Throughout this section we assume 푔 ≥ 2. By the bijec- invariant of the total space up to homeomorphism, since a tion (2), associated to an 푆푔-bundle 퐸 → 퐵, there is a map given 3-manifold may fiber as an 푆푔-bundle in more than 퐵 → BDiff(푆푔), unique up to homotopy. The induced map one way. See [Thu86]. on fundamental groups The monodromy-topology dictionary. Let’s think more about the bijection 휋1(퐵) → 휋1(BDiff(푆푔)) ≅ 휋0(Diff(푆푔)) ≡ Mod(푆푔)
conjugation is called the monodromy representation of 퐸 → 퐵. Bun푆푔 (퐵) ≅ Hom(휋1(퐵), Mod(푆푔))/ . (3) The monodromy representation can be described con- In the previous section we gave examples where the left- cretely as follows: given [훾] ∈ 휋1(퐵) represented by 훾 ∶ 푆1 → 퐵, consider the pullback 훾∗(퐸) → 푆1. Any bundle hand side could be explicitly computed using the right- over the circle is obtained from the mapping torus con- hand side, but usually this is an unreasonable task. Even struction (remove one fiber to get a bundle over the inter- when 퐵 = 푆ℎ is also a closed surface, there is no known classification of homomorphisms 휋1(푆ℎ) → Mod(푆푔). val, which is trivial because any map [0, 1] → BDiff(푆푔) is ∗ We would like to emphasize a different perspective on null-homotopic), so 훾 (퐸) ≅ 퐸푓 for some 푓훾 ∈ Diff(푆푔) 훾 (3) that leads to interesting problems. Observe that the whose isotopy class [푓 ] ∈ Mod(푆 ) is independent of the 훾 푔 left-hand side of (3) is topological, while the right-hand choice of representative of [훾]. The monodromy represen- side is group-theoretic. Understanding how geometric or tation is the map [훾] ↦ [푓 ]. It measures how the “picture” 훾 topological properties of 푆 -bundles translate to proper- of the fiber changes under the transition maps along the 푔 ties of the monodromy and vice versa leads to a dictionary. loop 훾. Below we mention a couple of entries of this dictionary. Monodromy as a complete invariant. By equation (2) Geometric classification of mapping tori. The precise and Theorem 1 for 푔 ≥ 2, conjugacy classification of elements of Mod(푆푔) is well
Bun푆푔 (퐵) ≅ [퐵, BDiff(푆푔)] ≅ [퐵, BMod(푆푔)]. known. According to the Nielsen–Thurston classification, there are three types of conjugacy classes: periodic, reducible, From 퐾(휋, 1)-theory, a map to BMod(푆 ) is determined 푔 and pseudo-Anosov. “Periodic” is synonymous with “finite- by the induced map on 휋 , up to based homotopy. 1 order”; a reducible element preserves (setwise) some finite Hence [퐵, BMod(푆 )] is isomorphic to the quotient of 푔 collection of curves up to isotopy. Thus a pseudo-Anosov Hom(휋 (퐵), Mod(푆 )) by the action of Mod(푆 ) by conju- 1 푔 푔 element is simply any element with neither of these special gation. properties. The miracle of the Nielsen–Thurston classifica- In summary, for 푔 ≥ 2 the isomorphism class of an 푆 - 푔 tion is that every pseudo-Anosov element nevertheless has bundle is determined uniquely by its monodromy representation. a very tightly controlled form; see [FM12, §13]. The monodromy is a complete invariant! Next we give ex- Thurston used this classification to describe the geome- amples of 퐵 where this can be used to completely deter- try of mapping tori. mine Bun푆푔 (퐵). As a trivial example, if 휋1(퐵) = 0, then the only 푆푔- Theorem 2 (Thurston). Fix 푔 ≥ 2 and fix [푓] ∈ Mod(푆푔). bundle over 퐵 is the trivial bundle 퐵×푆푔. (Here it is impor- Then [푓] is tant to remember that 푔 ≥ 2.) This illustrates a stark differ- (a) periodic if and only if 퐸 admits a Riemannian metric ence between 푆 -bundles and vector bundles; for example, 푓 푔 locally isometric to ℍ2 × ℝ; 푆푘 there are many nontrivial vector bundles over spheres (b) reducible if and only if 퐸 contains an incompressible 푘 ≥ 2 푓 with . torus; 퐵 = 푆1 As a second example, for , isomorphism classes (c) pseudo-Anosov if and only if 퐸 admits a hyperbolic 푆 푆1 푓 of 푔-bundles over are in bijection with homomor- metric. phisms ℤ → Mod(푆푔) up to conjugation, i.e., with con- jugacy classes of elements of Mod(푆푔). Here we clearly see A geometric restriction on the bundle gives an algebraic 1 why conjugation is relevant: to identify 퐸 → 푆 with 퐸푓, restriction on the monodromy and vice versa. The most we must first choose a homeomorphism between the fiber striking and difficult part of the theorem is: if [푓] is pseudo- over the basepoint and 푆푔. Different choices change 푓 by Anosov, then 퐸푓 is hyperbolic. We remark that a mapping conjugation. class can be both periodic and reducible, so (a) and (b) are The surprising part of the statement “monodromy is a not mutually exclusive. complete invariant” is that for any homomorphism 휌 ∶ Given Thurston’s theorem, it is natural to ask for condi- 휋1(퐵) → Mod(푆푔), there is a bundle 퐸(휌) → 퐵 whose tions on the monodromy of a bundle 퐸 → 퐵 with dim 퐵 ≥ monodromy is 휌. It’s not at all obvious how to explicitly 2 that guarantee that 퐸 has negative curvature. This seems construct 퐸(휌) from 휌. This is the power of Theorem 1. to be a subtle question. It is not hard to see that it is neces- We note however that the monodromy is not a complete sary for every nontrivial element of the monodromy group
150 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 2 to be pseudo-Anosov [FM02], but the converse is not gen- representation 휌 ∶ 휋1(퐵) → Mod(푆푔), there is a simple erally known. It is a well-known open question whether characterization of the homotopy classes of sections of 푝. or not there exists a homomorphism 휋1(푆ℎ) → Mod(푆푔) Such sections are in correspondence with liftings 휌̃ of 휌 as such that the image of every nontrivial element is pseudo- encoded in the diagram below: Anosov. Mod(푆 , ∗) (4) r9 푔 Complex structures on 푆푔-bundles over surfaces. When 휌̃ rr rr 퐵 = 푆ℎ is a closed surface, the total space 퐸 of any 푆푔- rr bundle over 퐵 is a compact 4-manifold and thus can po- / 휋1(퐵) Mod(푆푔) tentially be diffeomorphic to a complex surface. Further- 휌 more, it is possible for the bundle projection 퐸 → 퐵 to be Here Mod(푆푔, ∗) is the based mapping class group, defined as holomorphic with respect to some complex structure on 퐵. the group of diffeomorphisms fixing a distinguished point Since the monodromy 휌 of 퐸 → 퐵 determines the topol- ∗ ∈ 푆푔, modulo isotopies fixing ∗. ogy of 퐸, this information is encoded inside 휌, albeit in a Sections of 푆푔-bundles in algebraic geometry and num- highly nontrivial way. In “Sections of 푆푔-bundles,” we will ber theory. Before we discuss some of the tools used discuss the geometric Shafarevich problem, which shows that to construct and obstruct sections of 푆푔-bundles, it is holomorphic families are exceedingly rare. Here, we men- worthwhile to mention some applications. Sections of 푆푔- tion some entries in the monodromy-topology dictionary bundles are often of interest in problems of an algebro- concerned with the (non)existence of a complex structure geometric flavor. One notable instance of this concerns on 퐸. the geometric Mordell problem. Loosely speaking, this asks Hodge theory provides one major source of obstruc- for an enumeration of holomorphic sections of 푆푔-bundles tions. This is at its most powerful when the space under over surfaces in the case where the total space has a com- study is Kähler and not merely complex. It follows quickly plex structure. Arakelov and Parshin showed that the num- from the Enriques–Kodaira classification that if 퐸 is a com- ber of such sections is always finite. In fact, this is obtained pact complex surface that fibers over a surface, then 퐸 is from the geometric Shafarevich problem alluded to in “Mon- of general type and hence Kähler. Thus the basic “Käh- odromy.” For simplicity we state the version obtained by ler package” imposes nontrivial constraints on the coho- Parshin; Arakelov treats the more general case when 퐵 is ∗ mology algebra of 퐸. By (3), the structure of 퐻 (퐸; ℤ) (as a compact Riemann surface with finitely many points re- a ring) can be obtained from 휌. In fact, the cup prod- moved. See e.g. [McM00]. uct structure on an 푆푔-bundle is encoded as a certain fam- ily of characteristic classes with “twisted coefficients”; see Theorem 3 (Geometric Shafarevich). Let 퐵 be a compact [Sal18]. Another Hodge-theoretic obstruction is provided Riemann surface. For 푔 ≥ 2, there are only finitely many truly by Deligne’s semisimplicity theorem, which places strong re- varying families 푝 ∶ 퐸 → 퐵 of Riemann surfaces of genus 푔. strictions on how 휌 can act on the homology of the fiber A truly varying family 푝 ∶ 퐸 → 퐵 is an 푆푔-bundle where [Del87]. 퐸 has a complex structure, 푝 is holomorphic, and the To close this discussion we mention a theorem of Shiga fibers are not all biholomorphic. The geometric Mordell [Shi97] providing another constraint on the monodromy problem follows from geometric Shafarevich by way of the of a holomorphic 푆푔-bundle 퐸 → 퐵 over a compact Rie- “Parshin trick.” The idea is that each section 푠 ∶ 퐵 → 퐸 of mann surface 퐵. Shiga’s theorem asserts that in this setting, a truly varying family can be used to construct a new truly either all the fibers are biholomorphic or else the mon- varying family over 퐵 by constructing a branched cover of odromy is geometrically irreducible, meaning that there is 퐸 branched along 푠(퐵). This construction will be discussed no simple closed curve globally fixed by the monodromy. further in “Bundles and Branched Covers” in the context of To further illuminate the themes under development Atiyah–Kodaira bundles. Moreover, the genus of the fibers (especially the monodromy-topology dictionary and inter- in the new family depends only on the genus of the orig- actions with algebraic geometry), in the final two sections inal. Finiteness of families over 퐵 (Shafarevich) then im- we take a closer look at two topics: sections of 푆푔-bundles plies finiteness of sections (Mordell). and 푆푔-bundles over surfaces. As explained by McMullen in [McM00], the geometric Sections of 푆 -bundles Mordell problem is actually the complex-geometric ana- 푔 logue of Faltings’s theorem in number theory. Faltings’s A basic notion in any fiber bundle theory is that ofa section: theorem concerns Diophantine equations 퐹(푥, 푦, 푧) such 푝 ∶ 퐸 → 퐵 푠 ∶ 퐵 → 퐸 if is a bundle map, then is called a sec- as 푥푛 + 푦푛 + 푧푛 = 0 (푛 ≥ 3) whose complex points 푝∘푠 = id tion if . In other words, a section is a continuously determine a Riemann surface of genus at least 2; it as- varying choice of distinguished point in each fiber. Given serts that such an equation has only finitely many ratio- 푆 푝 ∶ 퐸 → 퐵 an 푔-bundle with corresponding monodromy nal solutions. Scheme-theoretically, one can view such a
FEBRUARY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 151 Diophantine equation as a “surface bundle” over3 Spec(ℤ), mathematical disciplines. A first question is whether a where the “fibers” consist of the reductions of 퐹 mod 푝. given bundle admits any sections at all. Unlike in the the- From this point of view, a rational solution (푥, 푦, 푧) of 퐹 ory of vector bundles, where the “zero-section” provides determines a section of this bundle by assigning the distin- a quick affirmative answer to this question, an 푆푔-bundle guished point (푥, 푦, 푧) (mod 푝) to the fiber 퐹 (mod 푝) over may or may not admit a section. This is similar to the situa- 푝 ∈ Spec(ℤ). McMullen explains how Faltings’s arguments tion one encounters when studying nowhere-vanishing sec- have direct analogues in the setting of complex geometry, tions of vector bundles. The standard machinery in the leading to the proof of the geometric Shafarevich problem latter setting is obstruction theory, which manufactures co- given by Imayoshi–Shiga. In fact, the connections between homological invariants that obstruct the existence of sec- 푆푔-bundles and number theory go beyond mere analogies. tions. However, obstruction theory breaks down when the Recently, Lawrence–Venkatesh [LV] gave a new proof of fibers are 퐾(휋, 1) spaces with 휋 a group with trivial cen- Faltings’s theorem that involves a topological analysis of ter, as is the case for 푆푔-bundles. Thus, by and large, the the monodromy of certain 푆푔-bundles over surfaces. study of sections of 푆푔-bundles takes on a quintessentially Sections of tautological bundles. Another application of geometric-group-theoretic flavor governed by the study of liftings 휌̃ as in (4). the theory of sections of 푆푔-bundles occurs in studying the existence and classification of sections of “naturally oc- Given 휌 ∶ 휋1(퐵) → Mod(푆푔), how could one obstruct or classify the lifts 휌̃ ∶ 휋 (퐵) → Mod(푆 , ∗)? The theory curring” 푆푔-bundles. The most “natural” of all such bun- 1 푔 of canonical reduction systems provides one approach. Here dles is the universal curve ℳ푔,∗ → ℳ푔 whose fiber over a we provide only a casual overview of how arguments using point 푥 ∈ ℳ푔 is the Riemann surface corresponding to 푥. The section question in this case simply asks if there these ideas work; for a more precise discussion (including is a way to continuously choose a distinguished point an actual definition of a canonical reduction system), see on all Riemann surfaces simultaneously. Unsurprisingly, e.g. [FM12, §13.2]. In keeping with the basic philosophy of geometric group theory, the method is to consider the ℳ푔,∗ → ℳ푔 does not have a section for 푔 ≥ 2. However, it is possible to choose a continuously varying family of six action of 휋1(퐵) on the set of simple closed curves on 푆푔 af- everywhere-distinct points on the universal curve in genus forded by the monodromy 휌. If one finds an ample supply 2, furnished by the so-called Weierstrass points (Figure 4). of “simple” elements in the image of 휌 (e.g. elements with Thus a more sophisticated version of the section question large centralizers in Mod(푆푔)), one can profitably under- stand the dynamics of this group action from the point of view of how 휋1(퐵) shuffles around simple closed curves on the surface. This information can be used to classify and obstruct sections: one asks where a distinguished point could be placed in relation to the simple closed curves un- der study, and in favorable circumstances one can see (e.g. by exploiting relations in 휋1(퐵) and/or Mod(푆푔)) that there Figure 4. The blue set forms the real solutions of is simply no place to put a distinguished point that is com- 푦2 = −(푥2 − 1)(푥2 − 4)(푥2 − 9), plotted in ℝ2 on the left. The red dots are the Weierstrass points. After projectivizing, the patible with the known dynamics of the action. complex solutions are homeomorphic to a surface of genus 2. One shortcoming of this approach is that current tech- niques apply only when 퐵 has a fundamental group with certain properties. In many common situations (e.g. when asks if it is possible to choose, for any 푛 ≥ 1, a “multi- 퐵 = 푆 is itself a surface), there are not enough commut- section” of 푛 everywhere-distinct points. If one restricts ℎ ing elements of 휋 (퐵) to be able to implement the above attention to holomorphic multisections, work of Hubbard 1 ideas. Our knowledge of sections of 푆 -bundles over sur- [Hub76] shows that this is impossible, but this does not 푔 faces is extremely limited—in fact, the question of Mess preclude the possibility that some merely continuous mul- [Kir78, Problem 2.17] from 1990 asking if every 푆 -bundle tisection could exist. For the universal curve ℳ , it was 푔 푔,∗ over a surface admits a multisection is still open. only recently shown that no continuous multisection ex- ists for 푔 ≥ 4 by L. Chen and the first author [CS], building Bundles and Branched Covers off ideas of Mess. The basic tool is the theory of canonical The main goal of this section is to describe a construction reduction systems, described below, which can be viewed as due to Atiyah and Kodaira of 푆푔-bundles over surfaces ob- a version of the Jordan normal form for mapping classes. tained by branched coverings. In contrast to the effortless Sections: Toolkit. The study of sections of 푆 -bundles 1 푔 way that 푆푔-bundles over 푆 are constructed (Figure 2), con- again incorporates themes and tools from a variety of structing interesting 푆푔-bundles over surfaces takes work,
3For simplicity we are ignoring issues of good/bad reduction.
152 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 2 and the branched covering constructions we discuss here replacing 퐵 with a finite cover, there is a cover 퐸′ → 퐸 have many interesting applications. branched along Σ. If 퐸 → 퐵 and Σ ⊂ 퐸 are both holomor- Before we begin, we mention that the bundle 퐸 → 퐵 phic, then the resulting bundle is also holomorphic. This over the configuration space from (1) in “Surface Bundles” is the essence of the Parshin trick discussed in “Sections of is obtained via branched covers: the map 푆(푏) ∋ (푥, 푦) ↦ 푆푔-bundles.” 푥 ∈ ℂ is a 푑-fold cover branched over 푏1, … , 푏푛. Thus 퐵 The Atiyah–Kodaira examples exhibit many interesting is parameterizing a family of branched covers of ℂ with phenomena, and they appear in surprisingly many situ- moving branched points. The Atiyah–Kodaira construc- ations. A variant of the Atiyah–Kodaira construction ap- tion works similarly. pears in the work of Lawrence–Venkatesh [LV] mentioned Atiyah–Kodaira bundles. We start with the basics of the in “Sections of 푆푔-bundles.” We close by mentioning a construction, which we explain in one of the simplest sampling of other applications of the construction. cases. Consider the surface 푆3, and let 휎 ∶ 푆3 → 푆3 be a free involution (Figure 5). The product 푆3 × 푆3 contains a (dis- Signature. The total space 퐸 of an Atiyah–Kodaira bundle connected) surface Σ, defined as the union of the graphs is a closed, oriented 4-manifold and therefore has a signa- of the identity and 휎. We would like to take a 2-fold cover ture sig(퐸), defined as the signature of the intersection form 퐸 → 푆3 × 푆3, branched over Σ. 퐻2(퐸)×퐻2(퐸) → ℤ. Under a branched cover, the signature is multiplied by the degree of the cover with a correction term that is proportional to the self-intersection number of the branching locus. Thus, although sig(푆3 × 푆129) = 0, σ(x) we have sig(퐸) = 256. These were the first examples con- structed of 푆푔-bundles over surfaces with nonzero signa- x ture. 2 Consequently, the MMM class 푒1 ∈ 퐻 (BDiff(푆푔)) is nontrivial for 푔 = 6 (and hence for 푔 ≥ 6 by Harer stabil- ity). To see this, we remark that the function that assigns to an 푆푔-bundle 퐸 → 푆ℎ the value sig(퐸) ∈ ℤ can be viewed as a characteristic class. Specifically, there is a homomor- σ phism 퐻2(BDiff(푆푔)) → ℚ that sends a cycle represented Figure 5. Free involution on surface of genus 3. Cut along the by 푆ℎ → BDiff(푆푔) to the signature of the associated bun- dotted line and double to obtain a 2-fold branched cover dle. This is well defined because signature is a cobordism 푆6 → 푆3. invariant. From the Atiyah–Kodaira construction [sig] ≠ 0 2 in 퐻 (BDiff(푆6); ℚ), and since 푒1 = 3⋅[sig] (by Hirzebruch’s Before explaining more details, let’s skip ahead to the signature theorem), we conclude 푒1 ≠ 0. In fact, the class [sig] ∈ 퐻2(BDiff(푆 ); ℚ) output: the construction produces an 푆6-bundle 퐸 → 푆129. 푔 is nontrivial and generates this 푔 ≥ 3 Where do these numbers come from? For the fiber 푆6, group when . first observe that Σ ⊂ 푆3 × 푆3 meets {푥} × 푆3 in two points (푥, 푥) and (푥, 휎(푥)), so under a double cover 퐸 → 푆3 × 푆3 Nontriviality of MMM classes. Morita generalized the pre- branched over Σ, the preimage of {푥} × 푆3 is a 2-fold cover ceding argument to prove that all the MMM classes are 푆6 → 푆3 branched over two points. nontrivial. More precisely, for each fixed 푖, there is 푔 ≫ 푖 2푖 Now we explain the base 푆129. The issue is that the so that 푒푖 ≠ 0 ∈ 퐻 (BDiff(푆푔); ℚ). He proved this by iter- branched cover 퐸 → 푆3 × 푆3 is not guaranteed to exist. ating the Atiyah–Kodaira construction: for example, given A sufficient condition for the existence is that the homol- the Atiyah–Kodaira bundle 푝 ∶ 퐸 → 푆129, consider the 4 ∗ ogy class [Σ] ∈ 퐻2(푆3 × 푆3) is even. Unfortunately, [Σ] is pullback to an 푆6-bundle 푝 (퐸) → 퐸. This bundle has not even. To fix this, we first pass to the 26-sheeted cover a tautological section over which one can branch; in this 푆129 → 푆3 with deck group 퐻1(푆3; ℤ/2ℤ). The preimage of way Morita obtained a bundle over a finite cover of 퐸 with Σ under 푆129 × 푆3 → 푆3 × 푆3 determines an even homology 푒2 ≠ 0. See [Mor01, §4.4] for more details. class, and 퐸 is defined as a branched cover of 푆129 × 푆3. In other directions, the Atiyah–Kodaira construction This construction can be done very generally: given has also been used to give examples of inequivalent a surface bundle 퐸 → 퐵 over a manifold and a symplectic structures on 4-manifolds [LeB00] and exam- multisection—viewed as a codimension-2 submanifold ples of CAT(0) metrics with no Riemannian smoothings Σ ⊂ 퐸 that projects to 퐵 as a covering space—after [Sta15]. A variant of the construction has also been used to study the geography problem for symplectic 4-manifolds 4A class 푥 is even if 푥 = 2푦 for some other class 푦. [BNOP19].
FEBRUARY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 153 Conclusion. There are many ways to arrive at the the- [LV] Lawrence B, Venkatesh A. Diophantine problems and 푝- ory of surface bundles: as a nonlinear bundle theory (al- adic period mappings, https://arxiv.org/abs/1807 gebraic topology), as a source for interesting 3- and 4- .02721, 2018. dimensional manifolds (low-dimensional topology and [LeB00] LeBrun C. Diffeomorphisms, symplectic forms and Kodaira fibrations, Geom. Topol. (4):451–456, 2000, DOI geometric group theory), or as objects naturally arising 10.2140/gt.2000.4.451. MR1796448 from moduli of Riemann surfaces (algebraic geometry). [MW05] Madsen I, Weiss M. The Stable Mapping Class Group Each area brings to surface-bundle theory its own collec- and Stable Homotopy Theory, European Congress of Mathe- tion of ideas and techniques. This leads to a rich interac- matics, 2005, 283–307 MR2185751 tion where questions in one area motivate results in an- [McM00] McMullen CT. From dynamics on surfaces to ratio- other. The interactions that we have discussed above rep- nal points on curves, Bull. Amer. Math. Soc. (N.S.), no. 2 resent only a small fraction of what is known and what is (37):119–140, 2000, DOI 10.1090/S0273-0979-99-00856- left to be discovered. 3. MR1713286 [McM13] McMullen CT. Braid groups and Hodge the- ory, Math. Ann., no. 3 (355):893–946, 2013, DOI ACKNOWLEDGMENTS. The contents of this article 10.1007/s00208-012-0804-2. MR3020148 owe a deep intellectual debt to Benson Farb, who in- [Mor01] Morita S. Geometry of Characteristic Classes, Transla- troduced both of us to the theory of surface bundles tions of Mathematical Monographs, vol. 199, American as graduate students and has continued to shape our Mathematical Society, Providence, RI, 2001. Translated perspective on the field. We would like to thank him from the 1999 Japanese original; Iwanami Series in Mod- for extensive comments on a preliminary draft; we are ern Mathematics. MR1826571 [Sal18] Salter N. Cup products in surface bundles, higher also grateful to Lei Chen, Dan Margalit, and Shigeyuki Johnson invariants, and MMM classes, Math. Z., no. 3-4 Morita for their input. (288):1377–1394, 2018, DOI 10.1007/s00209-017-1938- 4. MR3779001 [Shi97] Shiga H. On monodromies of holomorphic families References of Riemann surfaces and modular transformations, Math. [Ago13] Agol I. The virtual Haken conjecture, Doc. Math. Proc. Cambridge Philos. Soc., no. 3 (122):541–549, 1997, (18):1045–1087, 2013. With an appendix by Agol, Daniel DOI 10.1017/S0305004197001825. MR1466656 Groves, and Jason Manning. MR3104553 [Sta15] Stadler S. An obstruction to the smoothability of [BNOP19] Beyaz A, Naylor P, Onaran S, Park D. From auto- singular nonpositively curved metrics on 4-manifolds by morphisms of Riemann surfaces to smooth 4-manifolds, patterns of incompressible tori, Geom. Funct. Anal., no. 5 Math. Res. Lett., to appear, 2019. (25):1575–1587, 2015, DOI 10.1007/s00039-015-0341-8. [CS] Chen L, Salter N. The Birman exact sequence does not MR3426062 virtually split, arXiv preprint, https://arxiv.org/abs [Thu86] Thurston WP. A norm for the homology of 3- /1804.11235, 2018. manifolds, Mem. Amer. Math. Soc., no. 339 (59):i–vi and [Del87] Deligne P. Un th´eorèmede finitude pour la mon- 99–130, 1986. MR823443 odromie, Discrete Groups in Geometry and Analysis (New Haven, Conn., 1984), Birkhäuser Boston, 1987, 1–19, DOI 10.1007/978-1-4899-6664-3_1 MR900821 [Don98] Donaldson SK. Lefschetz fibrations in symplec- tic geometry, Doc. Math. (Extra Vol. II):309–314, 1998. MR1648081 [FM12] Farb B, Margalit D. A Primer on Mapping Class Groups, Princeton Mathematical Series, vol. 49, Princeton Univer- sity Press, Princeton, NJ, 2012. MR2850125 [FM02] Farb B, Mosher L. Convex cocompact subgroups of mapping class groups, Geom. Topol. (6):91–152, 2002, DOI 10.2140/gt.2002.6.91. MR1914566 Nick Salter Bena Tshishiku [Hat] Hatcher A. A short exposition of the Madsen-Weiss the- Credits orem, https://arxiv.org/abs/1103.5223, 2011. [Hub76] Hubbard JH. Sur les sections analytiques de la Figures 1–5 and opener image are courtesy of the authors. courbe universelle de Teichmüller, Mem. Amer. Math. Photo of Nick Salter is courtesy of Nick Salter. Soc., no. 166 (4):ix+137, 1976, DOI 10.1090/memo/0166. Photo of Bena Tshishiku is courtesy of Nick Dentamaro/ MR0430321 Brown University. [Kir78] Kirby R. Problems in low dimensional manifold the- ory, Algebraic and Geometric Topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Amer. Math. Soc., 1978, 273–312 MR520548
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