Symplectic 4-manifolds, mapping class group factorizations, and - PDF document

Symplectic 4-manifolds, mapping class group factorizations, and fiber sums of Lefschetz fibrations Denis AUROUX Massachusetts Institute of Technology

Symplectic 4-manifolds A (compact) symplectic 4-manifold ( M 4 , ω ) is a smooth 4- manifold with a symplectic form ω ∈ Ω 2 ( M ), closed ( dω = 0) and non-degenerate ( ω ∧ ω > 0 everywhere). Local model (Darboux): R 4 , ω 0 = dx 1 ∧ dy 1 + dx 2 ∧ dy 2 . E.g.: ( CP n , ω 0 = i∂ ¯ ∂ log � z � 2 ) ⊃ complex projective surfaces. The symplectic category is strictly larger (Thurston 1976, Gompf 1994). Symplectic manifolds are not always complex, but they are almost-complex, i.e. there exists J ∈ End( TM ) such that J 2 = − Id , g ( u, v ) := ω ( u, Jv ) Riemannian metric. At any given point ( M, ω, J ) looks like ( C n , ω 0 , i ), but J is ∂ 2 � = 0; no holomorphic coordinates). not integrable ( ∇ J � = 0; ¯ Hierarchy of compact oriented 4-manifolds: COMPLEX PROJ. � SYMPLECTIC � SMOOTH ⇒ Classification problems. Symplectic manifolds retain some (not all!) features of com- plex proj. manifolds; yet (almost) every smooth 4-manifold admits a “near-symplectic” structure (sympl. outside circles). 1

Lefschetz fibrations A Lefschetz fibration is a C ∞ map f : M 4 → S 2 with iso- lated non-degenerate crit. pts, where (in oriented coordinates) f ( z 1 , z 2 ) ∼ z 2 1 + z 2 2 . ( ⇒ sing. fibers are nodal) vanishing cycle s s M f × × S 2 s Monodromy around sing. fiber = Dehn twist Gompf: Assuming [fiber] non-torsion in H 2 ( M ), M carries a symplectic form s.t. ω | fiber > 0, unique up to deformation. (extends Thurston’s result on symplectic fibrations) Donaldson: Any compact symplectic ( X 4 , ω ) admits a sym- plectic Lefschetz pencil f : X \ { base } → CP 1 ; blowing up X → S 2 with base points, get a sympl. Lefschetz fibration ˆ f : ˆ distinguished − 1-sections. (extends classical alg. geometry (Lefschetz); uses “approx. hol. geometry”) ( f = s 0 /s 1 , s i ∈ C ∞ ( X, L ⊗ k ), L “ample”, sup | ¯ ∂s i | ≪ sup | ∂s i | ) 2

Monodromy vanishing cycle r r M f γ r S 2 × × × r γ 1 Monodromy around sing. fiber = Dehn twist Monodromy : ψ : π 1 ( S 2 \ { p 1 , . . . , p r } ) → Map g Map g = π 0 Diff + (Σ g ) is the genus g mapping class group. Map g is generated by Dehn twists. � 1 0 � 1 1 � � E.g. for T 2 = R 2 / Z 2 : Map 1 = SL (2 , Z ); τ a = , τ b = 0 1 − 1 1 Choose an ordered basis � γ 1 , . . . , γ r � for π 1 ( S 2 \ { p i } ) ⇒ factorization of Id as product of positive Dehn twists: � τ i = 1 . ( τ 1 , . . . , τ r ) ∈ Map g , τ i = ψ ( γ i ) , If g ≥ 2 then the factorization τ 1 · . . . · τ r = 1 determines the fibration f up to isotopy. ψ : π 1 ( R 2 \ { p i } ) → Map g,n • With n distinguished sections: ˆ Map g,n = π 0 Diff + (Σ , ∂ Σ) genus g with n boundaries. ⇒ τ 1 · . . . · τ r = δ (monodromy at ∞ = boundary twist). 3

Factorizations Two natural equivalence relations on factorizations: 1. Global conjugation (change of trivialization of reference fiber) ( τ 1 , . . . , τ r ) ∼ ( φτ 1 φ − 1 , . . . , φτ r φ − 1 ) ∀ φ ∈ Map g 2. Hurwitz equivalence (change of ordered basis � γ 1 , . . . , γ r � ) ( τ 1 , . . . , τ i , τ i +1 , . . . τ r ) ∼ ( τ 1 , . . . , τ i +1 , τ − 1 i +1 τ i τ i +1 , . . . , τ r ) ∼ ( τ 1 , . . . , τ i τ i +1 τ − 1 i , τ i , . . . , τ r ) (generates braid group action on r -tuples) s s γ 1 γ r γ 1 γ r γ i +1 ∼ γ i γ i +1 × . . . × × . . . × × . . . × × . . . × γ − 1 i +1 γ i γ i +1 { genus g Lefschetz fibrations with n sections } / isotopy ↑ ↓ 1-1 � factorizations in Map g,n � � Hurwitz equiv. δ = � (pos. Dehn twists) + global conj. � Lefschetz fibrations ? ⇒ Classification of Map g,n factorizations ? 4

Branched covers of CP 2 (D.A. ’99, D.A.-Katzarkov ’00–’02) (extends work of Zariski, Moishezon-Teicher, . . . on alg. surfaces) Alternative description of symplectic 4-manifolds: f : X → CP 2 branched covering, with crit. pts. modelled on • simple branching: ( x, y ) �→ ( x 2 , y ). • cusp: ( x, y ) �→ ( x 3 − xy, y ). Branch curve: D = crit( f ) ⊂ CP 2 symplectic curve with (complex) cusp and (+/ − ) node singularities. L CP 2 D γ i deg D = d n :1 − → r r X r π : ( x 0 : x 1 : x 2 ) �→ ( x 0 : x 1 ) CP 1 ❄ r r r ⇒ another combinatorial description of sympl. 4-manifolds: 1) Branch curve: D ⊂ CP 2 Braid monodromy = ρ : π 1 ( C −{ pts } ) → B d (braid group) ⇒ D is described by a (liftable) braid group factorization (involving cusps, nodes, tangencies) 2) Monodromy: θ : π 1 ( CP 2 − D ) → S n ( n = deg f ) (surjective, maps γ i to transpositions) 5

Classification of Lefschetz fibrations • g = 0 , 1: classical (genus 1: Moishezon-Livne). These are always isotopic to holomorphic fibrations. � 1 0 ( τ a · τ b ) 6 k = 1 � 1 1 � � In Map 1 : τ a = , τ b = 0 1 − 1 1 • g = 2, assuming no reducible sing. fibers: s s irreducible reducible Conj.: always isotopic to holomorphic fibrations, i.e. one of: ( τ 1 · τ 2 · τ 3 · τ 4 · τ 5 · τ 5 · τ 4 · τ 3 · τ 2 · τ 1 ) 2 k = 1 ( τ 1 · τ 2 · τ 3 · τ 4 · τ 5 ) 6 k = 1 τ 2 τ 4 ( τ 1 · τ 2 · τ 3 · τ 4 ) 10 k = 1 τ 3 τ 1 τ 5 Proved by Siebert-Tian (2003) under a technical assumption. (Method: pseudo-holomorphic curves) • g ≥ 3 (or g = 2 with reducible sing. fibers): Various infinite families of Lefschetz fibrations not isotopic to any holomorphic fibration! (Ozbagci-Stipsicz, Smith, Fintushel-Stern, Korkmaz, ...) Can we understand anything? 6

Fiber sums f : M → S 2 , f ′ : M ′ → S 2 genus g Lefschetz fibrations. Fix a diffeomorphism between smooth fibers. ⇒ fiber sum f # f ′ (fiberwise connected sum) r r M M ′ f f ′ × × S 2 S 2 For factorizations: ( τ 1 , . . . , τ r ), ( τ ′ 1 , . . . , τ ′ s ) �→ ( τ 1 , . . . , τ r , τ ′ 1 , . . . , τ ′ s ). Classification up to fiber sums: (D.A., ’04) ∀ g there is a genus g Lefschetz fibration f 0 g such that: ∀ f 1 : M 1 → S 2 , f 2 : M 2 → S 2 genus g Lefschetz fibrations,  χ ( M 1 ) = χ ( M 2 ) , σ ( M 1 ) = σ ( M 2 )   if f 1 , f 2 have same #’s of reducible fibers of each type  f 1 , f 2 have sections of same self-intersection  then ∀ n ≫ 0, f 1 # n f 0 g ≃ f 2 # n f 0 g . 7

Positive factorizations The proof relies on the following result: Let G = � g 1 , . . . , g k | r 1 , . . . , r l � finitely presented group, and δ ∈ G a central element. Assume there exist factorizations F 1 , . . . , F m of δ such that: • all factors in F i are in { g 1 , . . . , g k } ; • every generator g i appears at least once; • every relation can be written as an equality of positive words, w = w ′ where, viewing w, w ′ as factorizations: – either w, w ′ are Hurwitz equivalent – or w = F i and w ′ = F j for some i, j. Then, given F ′ , F ′′ factorizations of a same element in G s.t. the factors of F ′ are conjugated to those of F ′′ (up to permutation), m m i ∈ N s.t. F ′ · n ′ Hurwitz F ′′ · n ′′ ∃ n ′ i , n ′′ � � F i ∼ F i i i 1 1 We apply this result (+ some topology) to G = Map g, 1 . (There we have 4 factorizations. Relate n ′ i − n ′′ i to change in χ ( M ), σ ( M ) i . Finally, take F 0 = � F i ) ⇒ if preserved then n ′ i = n ′′ 8

Factorizations in Map g, 1 c 0 c 1 c 3 c 5 c 2 c 4 c 6 c 2 g Generators: τ 0 , . . . , τ 2 g . Relations: ( i ) τ i τ j = τ j τ i if c i ∩ c j = ∅ , τ i τ j τ i = τ j τ i τ j if c i ∩ c j � = ∅ ( ii ) for g ≥ 2 : ( τ 0 τ 2 τ 3 τ 4 ) 10 = ( τ 0 τ 1 τ 2 τ 3 τ 4 ) 6 ( iii ) for g ≥ 3 : ( τ 0 τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 ) 9 = ( τ 0 τ 2 τ 3 τ 4 τ 5 τ 6 ) 12 ( i ): Hurwitz equivalences; ( ii ) , ( iii ): both sides can be completed to factorizations of δ . Corollary: ( M 1 , ω 1 ), ( M 2 , ω 2 ) compact sympl. 4-manifolds, [ ω i ] ∈ H 2 ( M i , Z ), with same ( c 2 1 , c 2 , c 1 · [ ω ] , [ ω ] 2 ). ⇒ M 1 , M 2 become symplectomorphic after (same) blow- ups and fiber sums. Question: can M 2 be obtained from M 1 by a sequence of surgeries on Lagrangian tori? Or: given f 1 , f 2 as in main theorem, are their factorizations equivalent under Hurwitz moves + partial conjugations? ( τ 1 , . . . , τ i , τ i +1 , . . . , τ r ) ∼ ( φτ 1 φ − 1 , . . . , φτ i φ − 1 , τ i +1 , . . . , τ r ) if [ φ, τ 1 . . . τ i ] = 1. 9