Minicourse: Topological aspects of diffeomorphism groups. Abstract. The of the diffeomorphism Diff(M) of a M and its classifying BDiff(M) are important to the study of fiber bundles with fiber M. In particular, we can learn a lot about M bundles by (1) finding nonzero elements of H∗(BDiff(M)) and (2) relating these classes to the / of individual bundles. A good start to (1) is to understand the topology of Diff(M), and this has been done in low (by Smale, Hatcher, Earle-Eells, Gabai, and others). An example of (2) is the study of fiber bundles admitting a flat connection (as pioneered by Milnor and Morita). This course will discuss (1) and (2) through a few rich examples and in connection to major areas of current research. Our discussion will include (a) the type of Diff(M) when dim(M) < 4; (b) circle bundles, the Euler class, and the Milnor-Wood inequality; and (c) bundles, the Miller- Morita-Mumford classes, and Nielsen realization problems.

Disclaimer: These are notes are from lectures given at a graduate student workshop on diffeomorphism groups at UC Berkeley in June 2015. They are not meant as a source for details or as a primary reference. Unless otherwise noted, all objects considered below are oriented (, diffeomorphisms, bundles, etc). Usually this is omitted from the notation. For example, Diff(M) always means orientation-preserving + diffeomorphisms, and I write GLn(R) when sometimes I mean GLn (R).

1 Lecture 1. The Euler

I. Introduction. Fiber bundles M → E → B where M,B smooth manifolds.

S Locally trivial, globally interesting, E = Uα × M/ ∼

φαβ : Uα ∩ Uβ → Homeo(M). Scanned by CamScanner Examples Covering , vector bundles, circle bundles, surface bundles

Problem Distinguish bundles.

Characteristic classes ( global nontriviality)

• M manifold

• G ⊂ Homeo(M), e.g. Diff(M), Symp(M, ω), Cont(M, α),

Definition A c is an assignment

M → E → B 7→ c(E) ∈ Hi(B) that is natural, i.e. for a pullback bundle f ∗E / E

 f  B0 / B f ∗(c(E)) = c(f ∗E).

Theorem There exists a BG and a

 Fiber bundles     continuous maps  M → E → B ←→ B → BG  with structure group G  /htpy /iso

For nice B.

Corollary Characteristic classes are elements of H∗(BG).

Problem0 Compute H∗(BG)?

2 II. One rich example

A. Nonvanishing sections and the Euler class.

2 + Setup M = R , G = GL2 (R), B = Sg closed, oriented surface g.

2 π Definition A of R → E −→ Sg is a σ : Sg → E such that π ◦ σ = 1.

σ is nonvanishing if σ(S) ∩ 0-section = ∅. Informally write |σ| > 0.

Remark. A section of TS → S is a vector field.

2 Question Does R → E → S admit |σ| > 0?

Obstruction to |σ| > 0

Definition The Euler class is defined as

h X i 2 e(E) = PD π∗ index(xi)[xi] ∈ H (S), where π∗ : H0(TS) → H0(S).

Scanned by CamScanner • e(E) under isotopy

• e(E) = 0 if and only if ∃ |σ| > 0.

P (Poincare-Hopf) For E = TS, index(xi) = χ(S).

B. Circular reasoning 2 1 0 • fiberwise on R → E → S induces S → E → S E → S has nonvanishing section ←→ E0 → S has a section ←→ E ' S × S1 (Exercise)

Examples

3 1 (1) tangent bundle T Sg → Sg

3 2 3 2 2 3 (2) Hopf bundle S → S . U(1) acts on S ⊂ C and S = S /U(1). 1 2 (3) Heisenberg bundle S → E → T  1 a c   1 a c  H3(R) = { 0 1 b  : a, b, c ∈ R} G = { 0 1 b  : a, b ∈ Z, c ∈ R} 0 0 1 0 0 1

Note that H3(Z) ⊂ G ⊂ H3(R). Define bundle

G/H3(Z) → H3(R)/H3(Z) → H3(R)/G. This can be obtained from

 1 0  by identifying the top/bottom and left/right as usual and the front/back by . 1 1

1 Definition The Euler class of S → E → Sg

Scanned by CamScanner • Triangulate Sg. • Choose σ on 0-skeleton

• Interpolate on 1-skeleton

• For 2-cell c Scanned by CamScanner1 σ ∂c : ∂c → S

Define φ : C2(S) → Z by c 7→ deg(σ ∂c)

This is a cocycle independent of choice of σ on 1-skeleton. e(E) := [φ] ∈ H2(S).

2 Exercise Compute e(E) for H3(R)/H3(Z) → T .

III. Central extensions

1 Observe If g ≥ 1 then S → E → Sg induces central

0 → Z → π1(E) → π1(S) → 1

4 Definition A section of π1(E) → π1(S) is a s : π1(S) → π1(E) such that p ◦ s = 1.

• E → Sg has a section if and only if π1(E) → π1(S) has a section if and only if π1(E) ' π1(S) × Z.

2 Exercise 0 → Z → H3(Z) → Z → 0 does not split.

An invariant of 0 → Z → Γ → G → 1 central extension.

• pick -theoretic section s : G → Γ (normalize s(e) = e)

−1 •∀ g, h ∈ G define φ : G × G → Z by φ(g, h) = s(g)s(h)s(gh) ∈ Z ≤ Γ

Facts

(1) φ satisfies ∀g, h, k ∈ G

0 = φ(h, k) − φ(gh, k) + φ(g, hk) − φ(g, h) = δφ

So φ is a 2-cocycle.

0 0 (2) s : G → Γ induces φ : G × G → Z and ∃ β : G → Z and ∀g, h

φ(g, h) = φ0(g, h) + β(g) + β(h) − β(gh)

So [φ] well-defined.

2 Definition The Euler class e(Γ) = [φ] ∈ H (G; Z).

(3) Any 2-cocycle ξ : G × G → Z induced by a group Γ = G × Z with

(g, a)(h, b) = (gh, a + b + ξ(g, h))

Multiplication is associative because ξ cocycle.

Theorem Fix G.  central extensions  elements of ←→ 2 1 → Z → Γ → G → 1 H (G; Z)

2 2 Exercise Determine the of H (Z ; Z) corresponding to

2 0 → Z → H3(Z) → Z → 0.

IV. Examples

(i) Topology. S closed, genus(S) ≥ 2

1 1 → Z → π1(T S) → π1(S) → 1

2 Euler class 0 6= e ∈ H (π1(S); Z) ' Z.

5 + 2 1 2 (ii) Geometry / Lie groups. G = PSL2(R) = SL2(R)/{±1} = Isom (H ) ' T H Since π1(G) = Z, 1 → Z → Ge → G → 1. 2 Gives e ∈ H (PSL2(R); Z) ()

Remark A hyperbolic metric on S determines

2 π1(S) → Γ ⊂ PSL2(R) discrete, faithful and S ' H /Γ.

e = ρ∗(e).

(iii) Dynamics / Topology. G = Homeo(S1)

1 Homeo(^ S ) '{f : R → R | f(x + 1) = f(x) + 1} ⊂ Homeo(R)

Have SES 1 1 1 → Z → Homeo(^ S ) → Homeo(S ) → 1 2 1 gives e ∈ H (Homeo(S ); Z).

Remarks

1 ∗ (a) α : PSL2(R) → Homeo(S ). e = α (e). 1 1 (b) Given ρ : π1(M) → Homeo(S ) get an S bundle.

Eρ → M

where Mf × S1 Mf × S1 Eρ = = . π1(M) (x, t) ∼ (g.x, ρ(g)(t)) ∗ Euler class e(Eρ) := ρ (e).

(iv) Mapping class groups. Mod(S, ∗) := π0Homeo(S, ∗)

• (Nielsen) Mod(S, ∗) ,→ Homeo(S1)

Homeo(S1) ? _?  ??  ??  ??  ? PSL2(R) Mod(S, ∗) _? ? ??  ??  ??  ?  Push π1(S) – e ∈ H2(Mod(S, ∗)) leads to characteristic classes of surface bundles: MMM classes (more tomorrow)

6 – e ∈ PSL2(R) comes from elementary :

PSL2(R) ⊃ SO(2) ⊃ Z/10 corresponding 0 → Z → Z → Z/10 → 0  1 a + b > 9 The restriction of the Euler class is the carrying cocycle. φ(a, b) = 0 else

7 Diffeomorphism Groups Workshop Minicourse: Topology of Diff(M)

Problem Set 1: The Euler class.

1 2 1. (a) Use the cellular cocycle definition to compute the Euler class of S → H3(R)/H3(Z) → T . 2 (b) Use the definition to compute the Euler class for 0 → Z → H3(Z) → Z → 0. 2 2 2 2. (a) Give a group extension 0 → Z → Γ → Z → 1 with Euler class n ∈ Z ' H (Z ; Z). 1 (b) Show that for every n ∈ Z and every g ≥ 0 there exists S → E → Sg with e(E) = n. (Use bundle pullbacks.)

2 3. (a) Show that a R → E → B is trivial if and only if it has a section. 3 (b) Give an example of a bundle R → E → B that has a section but is not trivial.

4. Use the Euler class to show that every hyperbolic representation ρ : π1(S) → PSL2(R) lifts to SL2(R) → PSL2(R).

4 1 1 (Challenge) Show that e ∪ e = 0 in H (PSL2(R); Z). Determine if e ∪ e = 0 in Diff(S ) or Homeo(S ). If e ∪ e 6= 0 find a bundle that exhibits this.

Further reading.

• Differential forms in , Bott-Tu. Pages 122 –129 discuss the Euler class using a section with isolated zeros.

• The topology of fibre bundles, Steenrod. The beginning of Part III describes the obstruction version of the Euler class presented in the lecture. If you can find it, Scorpan’s The wild world of 4-manifolds also has a nice explanation pg. 197.

• A primer on mapping class groups, Farb-Margalit. Section 5.5 has some discussion of central extensions and the Euler class.

8 Lecture 2. Surface bundles and characteristic classes

I. Characteristic classes of surface bundles

• S connected, oriented surface, genus ≥ 2

• S → E → B surface bundle

Examples

M×[0,1] 1) Given f ∈ Diff(S). Define Mf (mapping ) = (x,0)∼(f(x),1) .

1 S → Mf → S

2) B space, ρ : π1(B) → Diff(S). Define Scanned by CamScanner Be × S E := ∀g ∈ π (B) ρ (x, y) ∼ (g(x), ρ(g)(y)) 1

inducing S-bundle Eρ → B.

Bundles constructed like this are called flat.

1 Remark For B = S , ρ : Z → Diff(S), f := ρ(1), Eρ ' Mf .

E.g. ρ : π1(Sh) → Fh → Diff(Sg).

Organizing problem Classify S → E → B

• bundle

• fiberwise diffeomorphism

• diffeomorphism / / homotopy equivalence

/ contactomorphism

• biholomorphism

Characteristic classes

9 • Mod(S) = π0Diff(S)  marked hyperbolic   marked complex  • Teich(S) = = complex manifold ' N structures on S structures on S C

• Mod(S) y Teich(S). Define M = Teich/Mod moduli space (complex )

Theorem With Q-coefficients

H∗(M(S)) ' H∗(Mod(S)) ' H∗(BDiff(S)) | {z } | {z } | {z } C-/alg geo topology/ topology/fiber bundles

Elements of H∗(Mod(S))

ab Theorem (Mumford, Birman, Harer) For g ≥ 3, H1(Mod(Sg); Z) = Mod(Sg) = 0.

2i i 1 ∗ 2 (1) MMM class ei ∈ H (M(Sg); Q). Last Mod(Sg, ∗) −→ Homeo(S ) gives i (e) ∈ H (Mod(Sg, ∗)).

• Birman sequence 1 → π1(S) → Mod(S, ∗) → Mod(S) → 1. • Topologically S → M(S, ∗) → M(S) universal surface bundle. • Gysin homomorphism 2i+2 2i H (M(S, ∗); Q) → H (M(S); Q) i+1 e 7→ ei 2i Given [B] ∈ H (M(S); Q) E / M(S, ∗)

  B / M(S)

i+1 ei([B]) := e ([E]).

2  (2) Signature cocycle σ ∈ Z BDiff(S), Q . 4 Cycle represented by Sh → BDiff(Sg), which determines Sg → M → Sh.

σ([Sh]) = sig(M)

σ is a cocycle by Novikov additivity.

(3) Teich(S) ⊂ Hom(π1(S), PSL2(R))/PSL2(R)

(Goldman) Hom(π1(S),G)/G symplectic. ωWP real 2-form on M(S) Weyl-Petersson form, gives

2 [ωWP ] ∈ H (M(S); R).

(4) Period mapping. A complex has a Jacobian

 Moduli space of principal  J : M(S ) → A = = Sp ( )\Sp ( )/U(g) g g polarized abelian varieties 2g Z 2g R

g 2  Determines C → E → Mg, gives c1(E) ∈ H M(Sg) .

10 Theorem (Harer) For g ≥ 4, H2(Mod(Sg); R) ' R.

1 (Atiyah 1969, Wolpert 1990) e1 = 3σ = 2π2 ωWP II. Classifying spaces

A. Introduction Let G be a .

Definition A principal G-bundle is a fiber bundle P −→π B with a continuous right G action P ×G → P −1 such that for all x ∈ B, G y Px := π (x) freely and transitively (so Px ' G).

Examples

1. Fr(M) → M frame bundle is a principal GLn(R)-bundle.

Fr(M) = {(x, ξ): x ∈ M and ξ ⊂ TxM basis}

2. A regular Y → X is a principal G = π1(X)/π1(Y ) bundle.

Definition A principal G-bundle EG → BG is universal if EG is contractible. BG is called a classifying space for G.

Universal property For each B sufficiently nice (paracompact)

 principal G bundles   continuous  ←→ . P → B f : B → BG /iso /htpy

E.g. given f : B → BG, pullback of gives P → B.

B. Properties of BG

1. Unique up to homotopy. Moreover, since G → EG → BG, have πi(BG) ' πi−1(G). 2. Functorial. A homomorphism φ : H → G induces continuous Bφ : BH → BG. Moreover,

Proposition 1. If φ is a homotopy equivalence, then Bφ is too.

Proposition 2. A short 1 → N → G → Q induces a fibration BN → BG → BQ.

3. If G y M, then BG also classifies M-bundles with structure group G. Principal G-bundles −→ M-bundles with structure group G P ×M G → P → B 7→ M → G → B

C. Examples

(a) G discrete. If X = K(G, 1), then Xe → X is a universal principal G-bundle, so BG ∼ K(G, 1).

11 (b) G = SO(2) = U(1) = S1.

U(1) S2n−1 ⊂ n freely U(1) S∞ = lim S2n−1. y C y n ∞ ∞ ∞ S contractible implies S → CP universal bundle.  S1 bundles  ↔ [B, P ∞] = [B,K( , 2)] ' H2(B; ). E → B C Z Z

So S1 bundles are determined (up to isomorphism) by their Euler class.

∞ (c) G = Diff(M). EG = Emb(M, R ). EG × G → EG by (φ, g) 7→ φ ◦ g. ∞ ∞ BG = Sub(M, R ) = {images of M → R }.

III. Characteristic classes Definition A characteristic class is an element c ∈ Hi(BG). Given f : B → BG classifying E → B, get invariant c(E) := f ∗(c) ∈ Hi(B).

Examples

∗ (i) G = SO(2) = U(1) = T. H (BT) = Q[x], |x| = 2.

(ii) G = Diff(S1). Homotopy equivalence SO(2) ,→ Diff(S1).

∗ 1 ∗ H (BDiff(S )) ' H (BSO(2)) ' Q[e] where |e| = 2 Euler class.

(iii) G = Diff(S) where S surface, χ(S) < 0.

1 → Diff0(S) → Diff(S) → Mod(S) → 1.

(Earle-Eells) Diff0(S) ∼ ∗.

Thus BDiff(S) ∼ BMod(S).

Theorem (Harer, Ivanov, Boldsen, Randall-Williams)

1  1  Hk Mod(Sg ); Q → Hk Mod(Sg+1); Q isomorphism for g  k.

Theorem (Morita, Miller, Madsen-Weiss)

∼ ∗ 1  [e1, e2,...] −→ lim H Mod(S ); . Q g→∞ g Q

IV. Flat connections on fiber bundles (Did not ) Definition An M bundle E → B is flat if E has a F whose leaves project to B as covering spaces.

12 Central example Fix ρ : π1(B) → Diff(M). Define

Be × M q : Be × M → =: Eρ π1(B) inducing M-bundle Eρ → B.

Eρ has foliation wtih leaves q(Be × {x}) and for each x ∈ M,

Be × M Be × {x} → → B π1(B) is the universal cover.

Proposition M → E → B flat if and only if E ' Eρ for some ρ : π1(B) → Diff(M).

Proof. (⇒) A flat bundle induces a ρ : π1(B) → Diff(M) and E ' Eρ.

 flat M bundles     continuous maps  ←→ ←→ E → B π (B) → Diff(M) f : B → K(Diff(M), 1) /iso 1 /conj /htpy

Thus BDiff(M)δ classifies flat M bundles (here Gδ is G with discrete topology)

Note H∗(BDiff(M)δ) ' H∗(Diff(M)) group cohomology.

13 Diffeomorphism Groups Workshop Minicourse: Topology of Diff(M)

Problem Set 2: Surface bundles and characteristic classes.

1. Classify genus-g surface bundles over S2 for g = 0 and g ≥ 2.

2. (a) Compute the signature of Sg × Sh. (There are two ways to do this.) (b) Use the fact that signature is multiplicative under taking covers to show that the signature of a surface bundle over a torus is zero. Hint: What is the maximum value of the signature of a 2 Sg bundle over T ?(∗) ab 3. Use the following facts to deduce that Mod(S) = H1(Mod(S); Z) = 0.

(a) Mod(S) is generated by Dehn twists Tc about simple closed c ⊂ S. (b) Dehn twists about separating curves are conjugate (show this).

(c) There is a relation TaTbTc = TxTyTzTw between 7 Dehn twists called the lantern relation.

4. (a) Let E → Sh be an Sg bundle. Show that E is a symplectic 4-manifold. (See Thurston’s Some simple examples of symplectic manifolds) (**)

(b) For g ≥ 4, construct an Sg bundle E → Sh with nonzero signature. (See Endo-Korkmaz- Kotschick-Ozbagci-Stipsicz , Lefschetz fibrations and the signatures of surface bundles) (**)

(Challenge) Classify surface bundles E → S1 up to various notions of equivalence (bundle isomorphism, fiberwise diffeomorphism, diffeomorphism, cobordism, contactomorphism).

Further reading.

• For characteristic classes of vector bundles: Vector bundles and K-theory, Hatcher. In particular, you may want to browse Section 1.2 and the introduction to Chapter 3.

• For characteristic classes of surface bundles: Geometry of characteristic classes, Morita. See Chap- ter 4.

• For some relations between different cohomology associated to a topological group: Con- tinuous cohomology of groups and classifying spaces, Stasheff.

• For mapping class groups and connections: Problems on mapping class groups and related topics, Farb (editor). There’s a lot here—find something that interests you!

14 Lecture 3. Homotopy type of diffeomorphism groups I

Two great , two great proofs. Theorem (Smale, 1958) SO(3) ,→ Diff(S2) is a homotopy equivalence.

Originally proved using algebraic topology (fiber bundles) and differential topology (transversality)

Theorem (Earle-Eells, 1969) If χ(S) < 0, then Diff0(S) is contractible.

Originally proved using geometry (complex/Riemannian) and analysis (PDE)

Goal. Prove Earle-Eells using algebraic/differential topology.

I. Informal Warmup

Key facts on topology of Diff(M)

(a) C1 diffeomorphisms Diff1(M) metrizable , locally modeled on C1 vector fields on M

(b) Diffk(M) ⊂ Diff1(M) homotopy equivalence for all k

(c) (Palais) Metrizable Banach manifolds have the homotopy type of CW complex.

Recall (Whitehead) If f : X → Y weak homotopy equivalence of CW complexes (f∗ : πi(X) → πi(Y ) isomorphism ∀i), then f is a homotopy equivalence.

Thus to show G → Diff(M) is a homotopy equivalence it suffices to show πi(G) →  πi Diff(M) .

Strategy Use fiber bundles M → E → B and associated LES

πi(M) → πi(E) → πi(B) → πi−1(M) as bootstrapping tool.

n n n n Example Emb(R , R ) space of smooth embeddings f : R → R .

n n Proposition GLn(R) ,→ Emb(R , R ) homotopy equivalence.

Proof. Evaluation f 7→ f(0) defines fiber bundle

n n  n n n Emb R , R , 0 → Emb(R , R ) → R n n  n n ⇒ Emb R , R , 0 ' Emb(R , R )

n n  Claim. f 7→ (df)0 is homotopy equivalence Emb (R , 0), (R , 0) → GLn(R). Pf. Define deformation retract ( f(tx)/t t 6= 0 ft(x) = (df)0(x) t = 0 n n (T0R ' R )

15 Exercise 1 Extend this argument to show

n Emb(D ,M) → Fr(M)  f 7→ f(0), df0(e1), . . . , df0(en) is a homotopy equivalence.

II. Formal warmup

Theorem(1) (M, ∗) closed manifold. Assume M aspherical and Z(π1(M)) = 1. Then Diff(M, ∗) ,→ Diff(M) induces an isomorphism on πi for i ≥ 1.

Proof. Step 1 The evaluation map η(ϕ) = ϕ(∗) defines a fiber bundle

Diff(M, ∗) → Diff(M) → M.

(Exercise) Need to show, for U neighborhood of x ∈ M, can define a ξ : U → Diff(M) such that ξ(u)(x) = u. Then U × Diff(M, x) → η−1(U) (u, ϕ) 7→ ξ(u) ◦ ϕ

Step 2 LES in homotopy   πk+1(M) → πk Diff(M, ∗) → πk Diff(M) → πk(M).   πk Diff(M, ∗) ' πk Diff(M) for k ≥ 2 because πk(M) = 0 for k ≥ 2.

For k = 1,   δ  0 → π1 Diff(M, ∗) → π1 Diff(M) → π1(M) −→ π0 Diff(M, ∗)

Step 3 We’ll show δ is injective.

• The connecting homomorphism. Given γ : [0, 1] → M representing [γ] ∈ π1(M),

Diff(M) ? γe  η    [0, 1] / M γ

Choose γe so that γe(0) = Id. Then δ([γ]) = component of γe(1) in Diff(M, ∗).

• (Exercise) γe(1) acts on π1(M, ∗) by conjugation. δ  ρ : π1(M) −→ π0Diff(M, ∗) → Aut π1(M, ∗) .

ρ injective since Z(π1(M)) = 1. Thus δ injective.

16   Thus π1 Diff(M, ∗) ' π1 Diff(M) .

Extension of these ideas

Theorem(2) Fix ( , 0) ,→ (S, ∗). Let Diff(S, ) = {ϕ : S → S s.t. ϕ = Id}. Then Diff(S, ) ,→ D D D D Diff(S, ∗) induces isomorphism on πi for i ≥ 1.

Proof. Exercise (challenging).

III. Earle-Eells theorem (topological proof)

Theorem (Earle-Eells) Let S be a compact surface χ(S) < 0. Then Diff0(S, ∂) ∼ ∗.

Will present proof due to Cerf, Gramain

Theorem(3) Let S surface with b = |π0(∂S)| ≥ 1. Fix p, q ∈ ∂S and α : [0, 1], 0, 1) → (S, p, q). Then

A(S, α) = {arcs from p to q homotopic to α} is contractible.

 Proof of Earle-Eells. Want to show πi Diff(S) = 0 for i ≥ 1. Step 1 (Reduction to the case ∂S 6= ∅)

χ(S) < 0 implies Z(π1(S)) = 1. By Theorems (1) and (2) Diff(S, D) ,→ Diff(S, ∗) ,→ Diff(S) isomorphism on πi for i ≥ 1.

Step 2 (Reducing complexity) Diff(S, α) → Diff(S) → A(S, α). By Theorem (3), for i ≥ 1,

  0  πi Diff(S) ' πi Diff(S, α) ' πi Diff(S ) where S0 compact obtained by cutting α.

2 By cutting along arcs we may reduce to the case S = D .

17 Scanned by CamScanner Step 3 Diff(D, ∂) ∼ ∗. (Tomorrow)

IV. An application: extending group actions

1 Open Question (e.g. M = S ) Do there exist isotopies ft, gt to Id so that [ft, gt] = Id for all t?

2 If so, there is ρ : Z → Diff(M × I) such that

2 2 ρ(Z ) M×0 = hf, gi and ρ(Z ) M×1 = Id. (1) Scanned by CamScanner 2 Question For W = M × I and Z y ∂W as in (1), does ρe exist?

A more general setup

• Γ countable group

• W manifold, ∂W 6= ∅

• ρ :Γ → Diff0(∂W )

Question Does there exist ρe :Γ → Diff0(W ) such that

Diff0(W ) r9 r r r r r  Γ / Diff0(∂W ) commutes?

Example ρe always exists for ρ : Z → Diff0(∂W ).

18 Proposition Let W be a compact surface with ∂W = S1. Let S be a closed surface of genus g ≥ 2 and 1 set Γ = π1(S). Consider ρ : π1(S) → PSL2(R) → Diff(S ) (induced by hyperbolic structure). Then ρ does not lift to Diff0(W ).

Proof. If ρe exists, obtain ∗  H BDiff0(W ) (ρ)∗ mm O e mm ∗ mmm ∂ mmm vmmm H∗(S) o H∗BDiff(S1) ρ∗

∗ ∗ ∗ ∗ ∗ such that (ρe) ∂ = ρ . But ∂ = 0 since Diff0(W ) ∼ ∗. Since ρ (e) 6= 0, this means ρe cannot exist.

2 1 Remark ρ does extend to D and S × [0, 1]. What about the M¨obiusband?

Theorem (Mann) Let V = S1 ∪ · · · ∪ S1. There is a finitely-generated - Γ and a representation Γ → Diff0(V ) that does not extend to action Γ → Diff0(W ) for any W with ∂W = V .

19 Diffeomorphism Groups Workshop Minicourse: Topology of Diff(M)

Problem Set 3: Homotopy type of diffeomorphism groups I.

1. (a) Show SO(2) ,→ Homeo(S1) is a homotopy equivalence. n (b) Show that Homeo(D , ∂) is contractible. 1 2. Prove that Diff(M) ∼ Diffµ(M) where µ is any . (Note the application to Diff(S ).) 3. Complete the exercises from the lecture.

(a) Show that Diff(M) → M is a fiber bundle. n (b) Show Emb(D ,M) ∼ Fr(M). 2 4. Determine the homotopy type of Diff(T ) using the techniques from today. (**) 1 5. Does π1(S) → Diff(S ) coming from a hyperbolic structure π1(S) → PSL2(R) extend to the Mobius band? (**)

n n (Challenge) For n ≥ 2, give an example of a Γ y V := S ∪ · · · ∪ S such that

• Γ is countable and torsion free

• There exists W with ∂W = V for which the Γ-action does not extend.

Further reading.

• Ivanov, Mapping class groups. In particular, see Section 2.6 for some discussion on diffeomorphism groups and spaces.

• Hatcher, A Short Exposition of the Madsen-Weiss Theorem. The proof of Earle-Eells presented in the lecture is taken from the appendix.

20 Lecture 4. Homotopy type of diffeomorphism groups II

Goal Use geometry/analysis to prove Smale’s theorem:

Theorem (Smale, 1958) SO(3) ,→ Diff(S2) is a homotopy equivalence.

I. The Smale

Some generalities Let M compact manifold.

1 → Diff0(M) → Diff(M) → π0Diff(M) → 1

Topologically Diff(M) ' ` Diff (M). (Breaks up main problem.) π0Diff(M) 0

Low dimensions

• M = S1. 1 1 π0Diff(S ) = 1 and SO(2) ,→ Diff(S ) h.e.

• M = S2. 2 2 π0Diff(S ) = 1 and SO(3) ,→ Diff(S ) h.e.

2 • M = T . 2 2 2 2 π0Diff(T ) = Out(Z ) ' SL2(Z) and T ,→ Diff0(T ) h.e.

• M = Sg for g ≥ 2.

π0Diff(S) = Out(π1(S)) ' Mod(S) and 1 ,→ Diff0(S) h.e.

Na¨ıve general guess

(i) Diff(M) acts on π1(M).

∼ π0Diff(M) = Diff(M)/Diff0(M) −→ Out(π1(M)).

(ii) If g Riemannian metric on M with “maximal ”, then

0 Isom(M, g) ,→ Diff0(M) h.e.

Counterexamples

• (Hatcher) Diff(S1 × S2) ∼ SO(2) × SO(3) × ΩSO(3) (not unexpected because this is the space of bundle )

n n • (Milnor-Kervaire) π0Diff(S ) ' π0Diff(D , ∂) ' Θn+1 group of exotic (n + 1)- (under con- nected sum)

21 Conjecture (Smale) Na¨ıve guess correct for constant curvature 3-dimensional . Known true for

• S3 (Hatcher)

• Lens spaces (McCullough, et al)

• hyperbolic 3-manifolds (Gabai)

II. Complex and conformal structures on surfaces (setup for proof of Smale’s theorem)

A. Local picture 2k 2 Definition A complex structure on V ' R is a J : V → V such that J = −Id. 2 M2k = {J ∈ M2k(R): J = −Id}/∼.

Proposition M2k ' GL2k(R)/GLk(C).

Proof. (-stabilizer for Lie groups) GL2k(R) acts on M2k by conjugation. The action is transitive  0 −Id  (change basis). Stabilizer of J = is GL ( ). Id 0 k C

t n Definition Sym(n) = {S ∈ GLn(R) | S = S} space of inner products on R . t hu, viS = u Sv.

Proposition Sym(n) ' GLn(R)/SO(n).

Proof. Exercise.

0 × 0 Definition S ∼ S conformally equivalent if ∃ λ ∈ R such that S = λS.

Space of conformal classes of metrics:

× × Cn := Sym(n)/R ' GLn(R)/R × SO(n)

Miracle 1 For n = 2

+ × + C2 ' GL2 (R)/R × SO(2) ' GL2 (R)/GL1(C) ' M2.

Exercise M2 '{z ∈ C : |z| < 1}.

B. Global picture

2 • S surface (like C or S ) • Frame bundle Fr(S) → S • Bundle of fiberwise complex structures Fr(S) × π : M −→ S GL2(R)

22 Definition A section of π is called an almost complex structure on S. Define M(S) space of section.

Miracle 2 In 2, M(S) = {complex structures on S}

(A complex structure on S is an atlas {φα : Uα → C} with holomorphic transitions.)

C. Beltrami equation

• Ω ⊂ C (open, connected) ∞ • C (Ω, ∆) = {smooth µ :Ω → C}' M(Ω)

Definition Fix µ ∈ C∞(Ω, ∆). The Beltrami equation is the PDE

∂f ∂f ∂f ∂f  − i = µ + i . (2) ∂x ∂y ∂x ∂y

Facts

(i) f : (Ω, µ) → (C, std) holomorphic if and only if f satisfies (2) (ii) |µ| < 1 implies that (2) is elliptic PDE

∞ Miracle 3 If Ω = C for every µ ∈ C (Ω, ∆), there exists unique fµ : C → C solution to (2) that is a diffeomorphism fixing 0,1. Moreover, µ 7→ fµ is continuous.

III. Proof of Smale’s theorem Step 1 Extract a Lie group

2 Any diffeomorphism f ∈ Diff(S ) can be written uniquely as f = A ◦ g, where A ∈ Aut(Cb) = PGL2(C) and g fixes 0, 1, ∞. A(z) = f(z) for z = 0, 1, ∞ and g = A−1f.

2 2 Topologically, Diff(S ) ' PGL2(C) × Diff(S , 0, 1, ∞).

Step 2 Homotopy type of Lie groups

(Exercise) Maximal compact SO(3) ,→ PGL2(C) is homotopy equivalence

Step 3 Apply Miracle 3 There is a map 2 2 M(S ) → Diff(S ; 0, 1, ∞) µ 7→ fµ that is a homeomorphism.

2 (S2) = space of sections of Fr(S )×M → S2 (this bundle has fiber ' ∆). Since ∆ convex, the space M GL2(R) M of sections is contractible.

Finishing the proof of Earle-Eells.

23 2 Corollary Diff(D , ∂) contractible.

Proof. (Using Smale’s theorem) There is a fiber bundle 2 2 η 2 2 Diff(D , ∂) → Diff(S ) −→ Emb(D ,S ) So suffices to show η is a homotopy equivalence.

It’s obvious that

∼ 2 η 2 2 2 1 2 φ : SO(3) −→ Diff(S ) −→ Emb(D ,S ) ∼ Fr(S ) ∼ T (S ) ∼ SO(3) is the identity.

24 Diffeomorphism Groups Workshop Minicourse: Topology of Diff(M)

Problem Set 4: Homotopy type of diffeomorphism groups II.

1. Check that M(S) (the space of complex structures on the surface S) is contractible. 2. Use the Smale conjecture (Hatcher’s theorem) to show that every diffeomorphism of S3 extends to 4 a diffeomorphism of D . n 3. Show that every manifold M admits a Riemannian metric by showing that the GLn(R)/O(n) bundle associated to the frame bundle F (M) → M admits a section.

(Challenge) Let Sg → E → Sh be a surface bundle over a surface g, h ≥ 2. Is Diff0(E) homotopy equivalent to the subgroup of fiberwise-preserving diffeomorphisms?

Further reading.

• A 50-Year View of Diffeomorphism Groups, Hatcher.

• A fibre bundle description of Teichm¨uler theory, Earle-Eells. The material from the lecture is taken from Sections 1, 2, 3, and 9.

• Three-dimensional geometry and topology, Thurston. Another proof of Smale’s theorem is given (Theorem 3.10.11), which conceptually similar to Smale’s original argument.

25 Lecture 5. Application: realizing mapping classes by diffeomorphisms

I. Nielsen realization problem for point-pushing

Setup

• M manifold, ∗ ∈ M basepoint

• Diff(M, ∗)(C1, orientation preserving) diffeomorphismsScanned by CamScanner fixing ∗

• Mod(M, ∗) := π0Diff(M, ∗) isotopy classes

Push homomorphism Push : π1(M, ∗) → Mod(M, ∗)

• (γ loop at ∗) P (γ) ∈ Diff(M, ∗)

• Push([γ]) := [P (γ)]

Question(1) Does there exist ϕ : π1(M) → Diff(M, ∗) so that

Diff(M, ∗) 7 ϕ ooo ooo ooo ooo  π1(M)/ Mod(M, ∗) Push commutes?

If ϕ exists, say Push is realized by diffeomorphisms.

Significant case M = Γ\G/K locally symmetric manifold, noncompact type

Scanned byn CamScanner • G real semisimple Lie group with no compact factors (e.g. Isom(H ), SLn(R), (8)) • K ⊂ G maximal compact

26 • Γ ⊂ G torsion-free e.g. G = PSL2(R), K = SO(2), Γ = π1(Sg) for g ≥ 2 M= hyperbolic surface

Theorem (Bestvina-Church-Souto 2009, Tshishiku 2014 ) If M = Sg closed surface g ≥ 2 or a locally symmetric manifold such that (∗ ∗ ∗), then Push is not realized by diffeomorphisms.

Rough idea (BCS) Use Euler class and Milnor-Wood inequalities as obstruction to existence of ϕ.

II. Flat connections on fiber bundles

Definition An M bundle E → B is flat if E has a foliation F whose leaves project to B as covering spaces.

Central example Fix ρ : π1(B) → Diff(M). Define

Be × M q : Be × M → =: Eρ π1(B) inducing M-bundle Eρ → B.

Eρ has foliation wtih leaves q(Be × {x}) and for each x ∈ M,

Be × M Be × {x} → → B π1(B) is the universal cover.

Proposition M → E → B flat if and only if E ' Eρ for some ρ : π1(B) → Diff(M).

Proof. (⇒) A flat bundle induces a monodromy ρ : π1(B) → Diff(M) and E ' Eρ.

 flat M bundles   homomorphisms   continuous maps  ←→ ←→ E → B π (B) → Diff(M) f : B → K(Diff(M), 1) /iso 1 /conj /htpy

Thus BDiff(M)δ classifies flat M bundles (here Gδ is G with discrete topology)

Flat connections on surface bundles

6 • (Morita) ∃ Sg → E → M not flat

1 • Rmk Every Sg → E → S is flat

• Open question: Is every Sg → E → Sh flat?

Question 2

• M with π1(M) 6= 1 • M × M → M to 1st factor

27 • ∆ : M → M × M

Does M × M → M admit a flat connection for which ∆ is parallel?

Monodromy and flat connections

• M → E → B monodromy µ : π1(M) → Mod(F ) • E → B flat ⇒ Diff(M) ϕ 6 m m m m m m  π (B)/ Mod(M) 1 µ

• M × M → M monodromy Push : π1(M) → Mod(M, ∗) • M × M → M flat w.r.t. ∆ ⇒ Push realized by diffeomorphisms

Remarks on converse

• False for M with π1(M) = 1 (the only flat bundles on simply connected M are trivial) • False in higher dimensions: generally BDiff(M) 6∼ BMod(M). A realization defines a bundle with section but may not be (M × M → M, ∆).

III. Characteristic classes of flat bundles Trend Characteristic classes of flat bundles are often restricted.

Examples

n n (1) R → E → M

4i • (E) ∈ H (M) i-th Pontryagin class

• Chern-Weil theory: E → M flat implies pi(E) = 0 for i > 0 2 2 2 2 2 2 • Ex: M = CP , T CP → CP , p1(T CP ) 6= 0 so T CP → CP not flat

(2) Euler class

2 • R → E → Sg vector bundle, g ≥ 1 2 • e(E) ∈ H (Sg) ' Z Euler class

Theorem (Milnor, 1958) If E → Sg is flat, then 1 − g ≤ e(E) ≤ g − 1.

Corollary TSg → Sg does not have a flat GL2(R) connection.

Importance of structure group

• Any S1-bundle has structure group reducing to SO(2) since SO(2) ,→ Homeo(S1) is a homotopy equivalence.

28 • For flat bundles,

structure group restriction on e(E) SO(2) e(E) = 0 (Chern-Weil) 1 SL2(R), GL2(R) |e(E)| ≤ − 2 χ(Sg) (Milnor) 1 PSL2(R), Homeo(S ) |e(E)| ≤ −χ(Sg) (Wood)

1 1 Sharpness ρhyp : π1(Sg) → PSL2(R) → Diff(S ) induces Eρ with e(Eρ) = e(T Sg) = χ(Sg).

IV. Main theorem

• M n = Γ\G/K

Theorem (Tshishiku) Suppose one of the following.

(i) M product of closed surfaces, genus ≥ 2

(ii) pi(TM) 6= 0 for some i > 0

(iii) R-rank(G) ≥ 2, Γ irreducible and nonuniform, e.g. Γ = SLn(Z), G = SLn(R)

Then Push is not realized by diffeomorphisms

Proof outline  TM → M has same char classes  Step 1 (Push realized) + (M nonpositively curved) ⇒ as flat GLn(R) bundle

Diff(M, ∗) 7 ϕ ooo ooo ooo ooo  π1(M)/ Mod(M, ∗) Push

Induces n−1 ρ1 : π1(M) → GL(T∗Mf) → Homeo(S ) n−1 ρ2 : π1(M) → G → Homeo(S )

ρ1, ρ2 induce flat bundles E1,E2 such that Scanned by CamScanner 1 • E2 ' T M

• E1 has flat GLn(R) connection

E1 and E2 have the same characteristic classes because they are fiberwise bordant.

Step 2 Show TM → M does not have same char classes as flat GLn(R) bundle (using Milnor-Wood, Chern-Weil, or superrigidity)

29 Q: Which examples does theorem apply to?

Pontryagin classes Is pi(Γ\G/K) 6= 0 for some i > 0?

• (Borel-Hirzebruch, 1958) (answer depends only on G for Γ cocompact)

• (Tshishiku) implement algorithm for every G

Some nonzero Pontryagin classes All Pontryagin classes zero

SU(p, q) p, q ≥ 1 and p + q ≥ 2 SL(n, R) for n ≥ 2 Sp(2n, R) n ≥ 2 SO(n, 1) for n ≥ 2 SO(p, q) p, q ≥ 2 and (p, q) 6= (2, 2) or (3, 3) SU∗(2n) n ≥ 2

Sp(p, q) p, q ≥ 1 (−26) ∗ SO (2n) n ≥ 3 SL(n, C) for n ≥ 2

G2(2) SO(n, C) for n ≥ 2

F4(4) Sp(2n, C) for n ≥ 2

F4(−20) (C)

E6(6) F4(C)

E6(2) E6(C)

E6(−14) (C)

E7(7) E8(C)

E7(−5)

E7(−25)

E8(8)

E8(−24)

Superrigid case WTS TM → M not flat. Suppose it is flat.

K YY YYYY, G oo7 OOO ooo OOO oo OO' ooo n−1 Γ O Homeo(S ) OOO 7 OO oo OO'  ooo GLn(R)

Commutes on H∗(B−) ⇒

isotropy rep K → Aut(TeK G/K) extends to a representation of G (3)

Use to show (3) false.

30 Diffeomorphism Groups Workshop Minicourse: Topology of Diff(M)

Problem Set 5: Realizing mapping classes by diffeomorphisms.

n 1. Compute the point-pushing homomorphism for tori M = T .

ρhyp 1 2. Show that the bundle induced by π1(S) −−−→ PSL2(R) → Homeo(S ) is the unit tangent bundle. 3. Show that fiberwise bordant circle bundles have the same Euler class. (Either use the cocycle definition given in Lecture 1, or reduce it to a question about the diffeomorphism group of an annulus.)

4. Use the fact that Diff0(S) is perfect to show that every surface bundle Sg → E → Sh is stably flat, 0 i.e. there exists Sh → Sh so that the pullback bundle is flat. (**)

Challenge problems:

1. Let Ta,Tb ∈ Mod(S) be Dehn twists about simple closed curves a, b with a single transverse inter- section. The subgroup Γ = hTa,Tbi ⊂ Mod(S) is isomorphic to the on 3-strands. Is Γ realized by diffeomorphisms? ?

2. Let S be a closed surface with an embedded disk D ⊂ S. 1 (a) Show that Diff(S) → Emb(D,S) determines a disk-pushing homomorphism π1(T S) → π0Diff(S, D). (b) Is disk-pushing realized by diffeomorphism?

Further reading.

• Some groups of mapping classes not realized by diffeomorphisms, Bestvina-Church-Souto.

• Cohomological obstructions to Nielsen realization, Tshishiku.

• On the non-realizability of braid groups by diffeomorphisms, Salter-Tshishiku.

31