Minicourse: Topological aspects of diﬀeomorphism groups. Abstract. The cohomology of the diﬀeomorphism group Diﬀ(M) of a manifold M and its classifying space 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 topology/geometry of individual bundles. A good start to (1) is to understand the topology of Diﬀ(M), and this has been done in low dimensions (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 homotopy type of Diﬀ(M) when dim(M) < 4; (b) circle bundles, the Euler class, and the Milnor-Wood inequality; and (c) surface 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 (manifolds, 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 class

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 spaces, vector bundles, circle bundles, surface bundles

Problem Distinguish bundles.

Characteristic classes (measure global nontriviality)

• M manifold

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

Deﬁnition A characteristic class 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 classifying space BG and a bijection

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 genus g.

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

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

Remark. A section of tangent bundle 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) invariant under isotopy

• e(E) = 0 if and only if ∃ |σ| > 0.

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

B. Circular reasoning 2 1 0 • ﬁberwise metric 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) Unit 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 extension

0 → Z → π1(E) → π1(S) → 1

4 Deﬁnition A section of π1(E) → π1(S) is a homomorphism 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 set-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 multiplication

(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 element 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) (group cohomology)

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 arithmetic:

PSL2(R) ⊃ SO(2) ⊃ Z/10 corresponding sequence 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 group extension 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 vector bundle 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 algebraic topology, 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 theory 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 torus) = (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 up to

• bundle isomorphism

• ﬁberwise diﬀeomorphism

• diﬀeomorphism / homeomorphism / homotopy equivalence

• symplectomorphism / 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 orbifold)

Theorem With Q-coeﬃcients

H∗(M(S)) ' H∗(Mod(S)) ' H∗(BDiﬀ(S)) | {z } | {z } | {z } C-analysis/alg geo topology/group theory 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 time 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) component

(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 curve 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 topological group.

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 covering space 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 universal bundle 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 exact sequence 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 embeddings 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 cover) Deﬁnition An M bundle E → B is ﬂat if E has a foliation 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 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)

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 curves 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 Commutators, 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 theories 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 theorems, 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 Banach manifold, 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 isomorphisms 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 modulo

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 torsion-free group 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 volume form. (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 group action 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 embedding 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 conjecture

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 symmetry”, 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 automorphisms)

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

21 Conjecture (Smale) Na¨ıve guess correct for constant curvature 3-dimensional geometries. 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 linear map J : V → V such that J = −Id. 2 M2k = {J ∈ M2k(R): J = −Id}/∼.

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

Proof. (Orbit-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 dimension 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 domain (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 subgroup 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), E8(8)) • K ⊂ G maximal compact

26 • Γ ⊂ G torsion-free lattice 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 projection 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 • pi(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 sphere 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) algorithm (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 E6(−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) G2(C)

E6(6) F4(C)

E6(2) E6(C)

E6(−14) E7(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 representation theory 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 braid group on 3-strands. Is Γ realized by diﬀeomorphisms? Homeomorphisms?

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