29th Workshop in Geometric Topology, Oregon State University, June 29, 2012
Throughout: Σg will be a connected oriented g≥2 surface, will look at Σg- bundles given by Σg→Eπ→B
(Fiber bundle where the base and total space will usually be compact manifolds, fiber is a surface)
For any f∈Diff(Σg), take cylinder on Σg and glue ends via f to obtain a 3-manifold and the surface bundle Σg→M3fπ→S1
Every 3-manifold that fibers over the circle is of this form! Why? Can remove a point from S1 to get a trivial bundle.
Another way to build bundles: take a representation ρ:π1B→Diff(Σg), let E=˜B×Σg/(x,y)∼(g.x,ρ(g)(y)) (standard construction) where ˜B is the universal cover of B.
Generalizes previous example: take real line cross surface and mod out by deck transformations.
Such bundles are called flat Σg bundles, exactly those which admit flat connections.
Note on connections: there’s a map TEπ→TB, can look at kerπ which is a 2-plane bundle over E. Take the kernel of Ef→B is obtain a “vertical” 2-plane bundle, horizontal is not well-defined and so is a choice of a 2-plane field. Flat means curvature vanishes is equivalent to Frobenius integrability theorem – the 2-plane is integrable. So curvature measures how integrable the 2-plane field is, rephrases “when is a 2-plane tangent to a foliation”.
One organizing problem: classify surface bundles over a fixed closed base manifold up to
B.F. proved algorithmic computability for bundle isomorphisms, but homeomorphisms and even fiberwise homeomorphisms are generally unsolvable.
Why study Σg bundles?
The simplest nonlinear bundle theory (i.e. structure group is not just GL(n), it is S1)
For 3-manifolds: almost all (e.g. all hyperbolic) closed 3-manifolds are finitely covered by some M3f surface bundle
Famous conjecture of Thurston! Very recent. His fields medal was finding a single hyperbolic structure on a closed 3-manifold fibering over the circle.
For 4-manifolds: huge class of symplectic manifolds.
AG: M moduli spaces of Riemann surfaces, decompose problems about varieties to problems about families of algebraic curves (i.e. what we call surface bundles)
The main invariant for these bundles: The Monodromy Representation
Let Mod(Σg) be the mapping class group of the surface, i.e. π0(Diff+Σg)) (group of homotopy/isotopy classes of diffeos on the surface).
Representation from earlier can be projected: π1B→Diff+Σgπ→Mod(Σg)
Take a homeomorphism to its homotopy class, Mod is finitely generated.
In AG, the mapping class group is the orbifold fundamental group of M, and this is a K(π1) space in the appropriate category.
So any bundle Σg→E→B yields a corresponding monodromy representation ρ:π1B→ModΣg
Look at LES in homotopy, the map π2B→π1F has image in the center, but πaΣg is centerless, yielding a SES.
For any SES of groups, you can get a representation π1B→Outπ1F
Can produce a map ModΣg→Outπ1Σg by applying a homeomorphism to π1, might move the basepoint, but these are isomorphic groups (classical theorem).
Question: when does the representation lift?
Equivalently, is every bundle flat? No, but this is an open question when the base is a surface (i.e. for all we know, every representation could lift). Conjectured that for the Kodaira manifold, it is not flat. Big problem, because this is perhaps the simplest nonlinear connection.
Classifying space theory: there exists a (crazy) space BDiff+Σg such that {iso classes Σg→E→B}⟺[B,BDiff+Σg]
Model: general construction, just find any contractible space on which Diff acts freely and take the quotient. One that works: Emb(Σg,R∞).
Not useful yet, because we don’t know what BDiff is.
Some serious math, theorem of Eels-Earle 1969:
Corollary: we care about Bdiff, classifying space theory doesn’t really see contractible stuff. So: Bdiff+Σg≃BModΣg=K(ModΣg,1)
Note that the last equality follows because there’s no topology on the mapping class group of a discrete group.
Huge conclusion:
{iso classes of Σg-bundles over B}⟺{conjugacy classes of representations ρ:πb→ModΣg}
Why? For any X=K(π,1) space, [⋅,X]≅ the conjugacy classes of [π1⋅,π1X](?)
Somehow, this all comes down to uniformization.
Note: Reduces problem of classification up to bundle isomorphism (solved in general by B.F.) to a group theory problem immediately.
How are you given the bundle? Can be given as triangulation, in which case one can produce the monodromy map (huge number of steps though! eee⋯). Or can be given the monodromy map; this determines the bundle.
Theorem: The conjugacy problem in ModΣg is solvable.
But this does not solve the problem for homeo Σg→M3→S1.
Why? Relates back to a paper of Thurston’s, theory of the Thurston Norm, from paper “A norm on the Homology of 3-manifolds”: there exist many 3-manifolds fibering over the circle (as long as β1(M3)>2) such that M3 fibers Σh→M3→S1 for infinitely many h, but finitely many for any fixed h.
How do you get an invariant out of this? Given two 3-manifolds that fiber over the circle, each with a monodromy in the mapping class group. Are they conjugate in the mapping class group? If so, then the manifolds are the same. If not, the bundles are not isomorphic, but they could be homeomorphic in some accidental different way.
How to solve: the Thurston norm is computable. Need to enumerate all the ways of fibering, and find the minimal genus fibering. (Should be the same!) Check how many ways there are of fibering. Then check, for each fibering, are the monodromies the same? At least one needs to be the same to be homeomorphic.
For suitably nice spaces, given by a mapping Σg→Eπ→B⟺c(Eπ→B)∈Hi(B)
that is natural with respect to pullbacks; i.e. take
and require that ~f∗(c(Eπ→B)=c(~f∗(Eπ→B)).
Any characteristic class is just an element of H∗(BDiff+Σg)=H∗(ModΣg), since every bundle will be a pullback of the universal bundle.
Lots of papers about the “stable cohomology” of the mapping class group, given by taking g big enough. We know χ of the mapping class group, it grows superexponentially and we know polynomially-many. No known unstable classes in genus 5 or higher!!
What do we know?
Any 2-cycle (for any space) will be homologous to the image of a map f of a surface into that space. Given blah, pullback the bundle over BDiff:
Take the signature of M4 (where you use the intersection pairing on H2, take the signature of that form). Using Novikov additivity for signature, this satisfies the cocycle condition and finally yields a number!
Look at the vertical bundle over T, you get R2→E→T.
Note: can see this bundle by looking at moduli space of Riemann surfaces Mg, covered by M∗g with fiber Σg. There is a vertical bundle over this, everything that is tangent to the fiber, which is a 2-plane bundle over the covering space.
So look at the Euler class e∈H2(T), take so-called “MMM class”. How to get a cocycle? Can try integrating over fiber, so ei=∫Σgei+1∈H2i(ModΣg;Q).
Note: the moduli space is a Kahler manifold, so you get the Weil-Peterson 2-form which is the Kahler class in H2.
Theorem (Harrer, 1980s): H2(Mod(Σ)g;R)=R, which is one-dimensional! So all of these classes are scalar multiples of each other - and in fact, sometimes not rational multiples, so sometimes you get interesting number-theoretic quantities like π26.
See book with Dan Margalit: all of this lives in the group ModΣg↪Homeo+(S1)