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Surface Bundles in Topology, Algebraic Geometry, and Group Theory

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Surface Bundles in Topology, Algebraic Geometry, and Group Theory Surface bundles in topology, algebraic geometry, and group theory Nick Salter ∗ Bena Tshishiku y October 21, 2019 1 Surface bundles A surface is one of the most basic objects in topology, but the mathematics of surfaces spills out far beyond its source, penetrating deeply into fields as diverse as algebraic geometry, complex analysis, dynamics, hyperbolic geometry, geometric group theory, etc. In this article we focus on the mathematics of families of surfaces: surface bundles. While the basics belong Figure 1: The M¨obius strip is the total space of a bun- to the study of fiber bundles, we hope to illustrate dle over S1 whose fibers are diffeomorphic to [0; 1]. how the theory of surface bundles comes into close contact with a broad range of mathematical ideas. We focus here on connections with three areas: al- A surface bundle E ! B with fiber S is also called gebraic topology, algebraic geometry, and geometric an S-bundle over B, and E is called the total space. group theory, and see how the notion of a surface Informally, one thinks of E as a family of surfaces bundle provides a meeting ground for these fields to parameterized by B, i.e. for each b 2 B, there is a interact in beautiful and unexpected ways. surface π−1(b) =∼ S. What is a surface bundle? Surface bundles in nature A surface bundle is a fiber bundle π : E ! B whose Surface bundles arise naturally across mathematics. fiber is a 2-dimensional manifold S and whose struc- The most basic source of S-bundles comes from the ture group is the group Diff(S) of diffeomorphisms of mapping torus construction. Given f 2 Diff(S), de- S. In particular, B is covered by open sets fUαg on fine Ef as the quotient of [0; 1] × S by identifying −1 ∼ which the bundle is trivial π (Uα) = Uα × S, and f0g × S with f1g × S by f; then Ef is the total space 1 local trivializations are glued by transition functions of an S-bundle over the circle Ef ! S . See Figure Uα \ Uβ ! Diff(S). 2. Surprisingly, this simple-minded construction is Although the bundle is locally trivial, any nontriv- ubiquitous in the classification of 3-manifolds, and in ial bundle is globally twisted, similar in spirit to the particular hyperbolic 3-manifolds. Thurston proved M¨obiusstrip (Figure 1). This twisting is recorded in that if f is sufficiently complicated (pseudo-Anosov, an invariant called the monodromy representation to c.f. Theorem 2), then Ef admits a hyperbolic struc- be discussed in Section 3. ture, i.e. a Riemannian metric with sectional curva- ture K ≡ −1. Furthermore, by work of Agol, Wise, ∗ Nick Salter is a Ritt Assistant Professor at Columbia Uni- Kahn-Markovic, every closed hyperbolic 3-manifold versity. His email address is [email protected]. yBena Tshishiku is an assistant professor at Brown Univer- M has a finite cover of the form Ef ! M for some sity. His email address is bena [email protected]. f : S ! S [Ago13]. 1 g ≥ 1, then Diff(Sg) is not homotopy equivalent to a compact Lie group. As such, the study of Sg-bundles for g ≥ 1 is the first instance of a nonlinear bundle f theory. There are many analogies between the theory of vector bundles and surface bundles, but there are also many new phenomena, connections, and open questions. Figure 2: The mapping torus E of a surface diffeo- f Conventions. For the remainder of this article we morphism f : S ! S. Note that the M¨obiusstrip assume, for simplicity, that S = S is a closed, ori- (Figure 1) can be constructed in a similar way. g ented surface of genus g ≥ 1 (and at times g ≥ 2). Working with oriented surfaces, we only consider Surface bundles also figure prominently in 4- orientation-preserving diffeomorphisms; for brevity, manifold theory. Donaldson [Don98] proved that ev- we suppress this from the notation and will not men- ery symplectic 4-manifold M admits a Lefschetz fi- tion it further. bration M ! CP 1, which can be viewed as a surface bundle where finitely many fibers are allowed to ac- The mapping class group quire singularities of a simple form (so-called nodes). Surface bundles appear in algebraic geometry, Given the wealth of examples of surface bundles de- where they are more commonly known as families of scribe above, we need a good way to tell different curves1. Special examples can be obtained by simply surface bundles apart. We'll discuss two approaches writing down families of equations. For instance, let to this { classifying spaces and monodromy { in Sec- B be the space of tuples b = (b1; : : : ; bn) of distinct tions 2 and 3. Monodromy is a special feature for points in C, fix d ≥ 2, and for b 2 B, consider the Sg-bundles compared to other bundle theories, and surface it is where the mapping class group plays a promi- nent role. 2 d S(b) = f(x; y) 2 C : y = (x − b1) ··· (x − bn)g: (1) To explain this, consider the mapping torus con- struction discussed above (Figure 2). If f is isotopic Then E = f(x; y; b) j (x; y) 2 S(b)g is the total to the identity (i.e. there is a path from f to id in space of a S-bundle over B under the projection map 1 Diff(Sg)), then Ef is just the product bundle S ×Sg. (x; y; b) 7! b. Here B is the configuration space of More generally, for any f 2 Diff(Sg), the bundle Ef n (ordered) points in C. The study of this single S- is unchanged if f is changed by an isotopy. There- bundle is already incredibly rich, with connections to fore, if we want to understand the different bundles representations of braid groups and geometric struc- obtained as mapping tori, we should start by con- tures on moduli spaces of Riemann surfaces [McM13]. sidering the quotient Mod(Sg) := Diff(Sg)= Diff0(Sg) Vector bundles are also a source of surface bun- by the (normal) subgroup of diffeomorphisms isotopic dles: given a rank-3 real vector bundle, the associated 2 2 to the identity. The group Mod(Sg) is called the unit-sphere bundle is an S -bundle. In fact, any S - mapping class group. It is isomorphic to the group bundle is obtained from this construction because, by π0 Diff(Sg) of path components of Diff(Sg). a theorem of Smale, Diff(S2) is homotopy equivalent For example Mod(T 2) =∼ SL ( ). Any A 2 SL ( ) to the orthogonal group O(3) (this homotopy equiv- 2 Z 2 Z acts linearly on R2 and descends to T 2, and con- alence implies the bundle statement by the theory versely, up to homotopy or isotopy, a diffeomorphism of classifying spaces discussed in Section 2). On the 2 2 ∼ 2 of T is determined by its action on π1(T ) = Z . other hand, if Sg is a closed oriented surface of genus For g ≥ 1, Mod(Sg) is an infinite, finitely-presented 1Since Riemann surfaces have complex dimension one, al- group. In Section 3 we explain how Mod(Sg) plays a 1 gebraic geometers refer to them as curves. central role, not only for Sg-bundles over S , but for 2 Sg-bundles over any base. pullback: P ×Sg E / Diff(Sg ) 2 The classification problem B / BDiff(Sg) In this section we describe the basic tools and frame- work from algebraic topology for studying S-bundles. The bundle on the right is known as the universal As mentioned above, we focus on the case S = S . Sg-bundle. See [Mor01] for more details. g We want to understand the homotopy type of Two bundles E ! B and E0 ! B are isomorphic BDiff(S ). As mentioned above, there is a fibration if there is a diffeomorphism E ! E0 that sends fibers g Diff(S ) ! P ! BDiff(S ) where P is contractible. to fibers and covers the identity map on B. g g Hence the homotopy types of Diff(S ) and BDiff(S ) Optimistically, one would like to solve the classi- g g are closely related; indeed by the long exact sequence fication problem: for a given B, determine the set of homotopy groups, π (BDiff(S )) =∼ π (Diff(S )). of isomorphism classes of S -bundles E ! B. This i g i−1 g g When g ≥ 2, the homotopy type of Diff(S ) is as sim- problem can be translated to a homotopy-theoretic g ple as possible. problem via classifying space theory. Usually the classification problem is too difficult to Theorem 1 (Earle-Eells). If g ≥ 2, then the iden- solve completely. In practice one wants a rich collec- tity component Diff0(Sg) < Diff(Sg) is contractible. tion of invariants that (i) measure topological prop- Consequently, the surjection Diff(Sg) ! Mod(Sg) is erties of Sg-bundles, and (ii) enable us to distinguish a homotopy equivalence. S -bundles found in nature. In the study of vector g The homotopy type of Diff(S ) for g = 0; 1 is also bundles, a primary role is played by characteristic g known: Diff(S2) is homotopy equivalent to O(3), and classes. Surface bundles also have a theory of char- Diff(T 2) is homotopy equivalent to T 2 SL ( ); see acteristic classes, but as we explain, these are fairly o 2 Z e.g. [Mor01]. Theorem 1 was originally proved using coarse invariants. complex analysis (Teichm¨ullertheory) and PDE; a purely topological proof was given by Gramain; see Classifying space for surface bundles [Hat].
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