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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; !), Cont(M; α), Lie group Definition A characteristic class c is an assignment M ! E ! B 7! c(E) 2 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 8 Fiber bundles 9 < = 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 jσj > 0. Remark. A section of tangent bundle TS ! S is a vector field. 2 Question Does R ! E ! S admit jσj > 0? Obstruction to jσj > 0 Definition The Euler class is defined as h X i 2 e(E) = PD π∗ index(xi)[xi] 2 H (S); where π∗ : H0(TS) ! H0(S). Scanned by CamScanner • e(E) invariant under isotopy • e(E) = 0 if and only if 9 jσj > 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 0 1 a c 1 0 1 a c 1 H3(R) = f@ 0 1 b A : a; b; c 2 Rg G = f@ 0 1 b A : a; b 2 Z; c 2 Rg 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] 2 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 •8 g; h 2 G define φ : G × G ! Z by φ(g; h) = s(g)s(h)s(gh) 2 Z ≤ Γ Facts (1) φ satisfies 8g; h; k 2 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 9 β : G ! Z and 8g; 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] 2 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 2 H (π1(S); Z) ' Z. 5 + 2 1 2 (ii) Geometry / Lie groups. G = PSL2(R) = SL2(R)={±1g = Isom (H ) ' T H Since π1(G) = Z, 1 ! Z ! Ge ! G ! 1: 2 Gives e 2 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 ) ' ff : R ! R j f(x + 1) = f(x) + 1g ⊂ Homeo(R) Have SES 1 1 1 ! Z ! Homeo(^ S ) ! Homeo(S ) ! 1 2 1 gives e 2 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 2 H2(Mod(S; ∗)) leads to characteristic classes of surface bundles: MMM classes (more tomorrow) 6 { e 2 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 2 Z ' H (Z ; Z). 1 (b) Show that for every n 2 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 2 Diff(S).
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