Alexandru Oancea (Strasbourg) joint with Gwénaël Massuyeau (Strasbourg) and Dietmar Salamon (Zurich) Conference Braids in Paris, September 2008

Transcription

1 A Outline Part I: A of the braid group Part II: Morse theory Alexru Oancea (Strasbourg) Gwénaël Massuyeau (Strasbourg) Dietmar Salamon (Zurich) Conference Braids in Paris, September 2008

2 Outline of Part I Outline Part I: A of the braid group Part II: Morse theory 1 on the framed braid group 2 Group-ring valued cocycles 3 Linear s from cocycles

3 Outline of Part II Outline Part I: A of the braid group Part II: Morse theory 4 vanishing cycles 5 The Picard formula 6 Morse theory: Real vs. Complex

4 Linear s Part I A Representation of the Braid Group

5 Linear s The (framed) braid group Z finite set D = {z C : z 1}, z 0 D, m = card(z) Definition (braid group) B := π 1 (Sym m ( D) \,[Z]) Choose nonzero tangent vectors v z T z D, z Z Definition (framed braid group) B = π 1 (Sym m (T D \ 0) \,[Z, {v z }])

6 Mapping class groups G := {φ Diff 0 (D,z 0 ) : φ(z) = Z } G 0 := {φ G : φ t G s.t. φ 0 = Id, φ 1 = φ} Linear s Proposition There is an isomorphism G/G 0 B. G := {φ Diff 0 (D,z 0 ) : φ(z) = Z, dφ(z)v z = v φ(z), z Z } G 0 := {φ G : φ t G s.t. φ 0 = Id, φ 1 = φ} Proposition There is an isomorphism G/ G 0 B.

7 Linear s Distinguished configurations Recall the choice of Z, z 0, {v z } z Z, m := card(z) Definition A (marked) distinguished configuration is an m-tuple c = (c 1,...,c m ) of smooth embedded paths c i : [0,1] D s.t. i. c i (0) = z 0, c i intersects c j only at z 0 for all i j; ii. {c 1 (1),...,c m (1)} = Z; iii. the vectors ċ 1 (0),...,ċ m (0) are pairwise non-collinear ordered clockwise at T z0 D; iv. (marked) ċ i (1) = v ci (1). c c 2 3 c 1 c 4 z 0

8 Linear s Distinguished configurations C = { homotopy classes of distinguished configurations } Proposition The braid group G/G 0 acts freely transitively on C. C = { homotopy classes of marked distinguished configurations } Proposition The framed braid group G/ G 0 acts freely transitively on C. c i 1 c i σ i c j ε j z 0 z 0 z z 0 0

9 B m := Generators relations σ 2,...,σ m σ i σ i+1 σ i = σ i+1 σ i σ i+1 σ i σ j = σ j σ i, i j 2 Linear s B m := σ 2,...,σ m ε 1,...,ε m σ i σ i+1 σ i = σ i+1 σ i σ i+1 ε i σ i = σ i ε i 1 ε i 1 σ i = σ i ε i other generators commute Proposition B m B, Bm B. The above isomorphisms depend on the choice of a (marked) distinguished configuration c!

10 Linear s The Monodromy cocycle Fix c C ordering Z = {z 1,...,z m } g i,c ci 1 c i z 0 distinguished loops g i,c Γ = π 1 (D \ Z;z 0 ) σ B permutation π σ,c S m via σ(z i ) = z πσ,c(i) Definition (Massuyeau, O., Salamon) Define S c : B GL m (Z[Γ]) by { c 1 S c (σ) ij := i σ c j, i = π σ,c (j), 0, otherwise.

11 The Monodromy cocycle Linear s c i 1 c i σ i σ i (c i 1 ) σ i (c i ) c 1 i σ i (c i 1 ) = g 1 i 1,c c 1 i 1 σ i (c i ) = 1 z 0 z 0 z 0 ǫ i (c i ) c i ǫ i c 1 i ǫ i c i = g i,c z 0 z 0 z 0

12 Linear s Definition Point of view: non-abelian cohomology Let G A be (non-abelian) groups, such that G acts on A on the left (g,a) g a. A map s : G A is a 1-cocycle if s(gh) = s(g)g s(h). Two cocycles s 0,s 1 : G A are cohomologous if a A s.t. s 1 (g) = a 1 s 0 (g)g a. Example: B acts on Γ, hence on GLm (Z[Γ]) componentwise Proposition (i) Each map S c : B GL m (Z[Γ]) is a 1-cocycle. (ii) Any two cocycles S c S τ c are cohomologous: S τ c(σ) = S c (τ) 1 S c (σ)σ S c (τ), σ B. [S c ] is canonically defined: the monodromy class

13 Linear s Proof of the cocycle properties (i) We prove S c (στ) = S c (σ)σ S c (τ). Set j := π τ,c (k), i := π σ,c (j) = π στ,c (k). Then c 1 i σ c j σ (c 1 j τ c k ) = c 1 i σ τ c k = c 1 i (ii) We prove S c (τ)s τ c(σ) = S c (στ) = S c (σ)σ S c (τ). Set l := π σ,τ c(k) i := π τ,c (l) = π στ,c (k). Then c 1 i τ c l (τ c) 1 l σ (τ c) k = c 1 i σ τ c k = c 1 i (στ) c k (στ) c k Proposition For any c C, the map S c : B GL m (Z[Γ]) is injective. Proof. B acts on C freely transitively.

14 Linear s Digression: the Magnus cocycle B acts on Γ = π 1 (D \ Z;z 0 ) free on m generators g 1,c,...,g m,c Definition Given c C, the Magnus cocycle M c : B GL m (Z[Γ]) is ( ) σ g j,c M c (σ) := g i,c i,j=1,...,m Fox calculus for a free group Γ with basis g 1,...,g m derivation d : Z[Γ] Z[Γ] is additive homomorphism s.t. d(gh) = d(g) ε(h)+g d(h), ε : Z[Γ] Z augmentation d(1) = 0, d(g 1 ) = g 1 d(g)! derivation g i such that g j = δ j g i i example: (g i 1g i g 1 i 1 ) g i 1 = 1 + g (g i g 1 i 1 ) i 1 = 1 g g i 1 g i g 1 i 1 i 1

15 Linear s The Magnus cocycle continued Proposition Any two cocycles M c M τ c are cohomologous: M τ c(σ) = M c (τ) 1 M c (σ)σ M c (τ), σ B. Proof. [M c (τ)m τ c(σ)] ik = j τ g j,c g i,c σ τ g k,c τ g j,c = (στ) g k,c g i,c =M c (στ) ik Interesting (?) question: study H 1 ( B,GL m (Z[Γ])).

16 Linear s M c (σ i,c ) = ( = Magnus vs. monodromy cocycles - first comparison ( 1li l m i 1 g i 1,c g 1 i,c g 1 i 1,c 1 g 1 i 1,c 0 ) ( 1li 2 0 0, S c (σ i,c ) = 0 0 ) (, = M c (ε i,c ) = 1l m, S c (ε i,c ) = 0 0 1l m i 0 1 g 1 i 1,c 0 ) ( 1li g i,c l m i ) ) View B B using framing determined by trivialization of TD restriction of the Magnus cocycle M c B is injective

17 Linear s The Burau Recall action of B g i 1 g i 1 g i g 1 i 1, on Γ: (σ i ) : g i g i 1, g j g j, j i 1,i (ε i ) = Id Definition For c C, the Burau M c : B GL m (Z[t,t 1 ]) is obtained by reducing the Magnus cocycle M c (σ) := M c (σ) g1,c = =g m,c=t 1 Remarks on M c : independent of c C up to conjugation reducible 1 + (m 1), with eigenvector (1,t,t 2,...,t m 1 ) defines polynomial link invariant: Alexer polynomial (invariance under Markov moves)

18 Linear s The Monodromy Definition (MOS) For c C, the monodromy S c : B GL m (Z[t,t 1 ]) is obtained by reducing the monodromy cocycle S c (σ) := S c (σ) g1,c = =g m,c=t 1 Note: independent of c C up to conjugation does not define polynomial link invariant

19 Linear s M c (σ i,c ) = Comparison of the Burau Monodromy s ( 1li l m i ( 1 t 1 = t 0 ) ( 1li 2 0 0, S c (σ i,c ) = 0 0 ), = M c (ε i,c ) = 1l m, S c (ε i,c ) = 0 0 1l m i ( ) 0 1 t 0 ( 1li t l m i ) ) m m 2 m 1 t m tr(id) tr(s c (σ i )) tr(m c (σ i )) Corollary The Magnus class [M c ] the monodromy class [S c ] are non-trivial distinct.

20 Linear s Proposition Irreducibility of the monodromy The S c is irreducible. Proof. ( Step 1. S)( c cannot ) fix a( subspace ) { of dimension 1. { 0 1 a a b = λa ta = λ = λ t 0 b b ta = λb 2 a a,b = 0 Step 2. S c cannot be reducible m = k + l, k,l 2. Assuming this to be true, we further specialize t = 1, S c descends to the canonical S m GL m (Z) which is reducible m = 1 + (m 1). The (m 1)-dim. factor is the restriction to x x m = 0 is irreducible, a contradiction.

21 Linear s The monodromy linking numbers View B B via trivialization of TD S c : B GL m (Z[Γ]) Proposition Let σ PB B be a pure braid. Then S c (σ) = Diag(t l 1,...,t lm ), l i = j>i lk(i,j) + j<i lk(j,i), lk(i,j) = linking number of components i,j of the closed braid. Proof: direct computation using presentation of PB S c (σi,c 2 ) = Diag(1,..., t,t,...,1) i 1 i

22 Picard formula Part II Morse theory Morse theory: R vs. C

23 - local model f : C n+1 C, f (z) = z z2 n+1 Picard formula Morse theory: R vs. C f 1 (1) = {x + iy : x 2 y 2 = 1, x,y = 0} (x, y) (x/ x, x y) symplectic T S n = {(ξ,η) R n+1 R n+1 : ξ = 1, ξ,η = 0}

24 - local model Picard formula f 1 (1) T S 1 f 1 (0) L Morse theory: R vs. C f (z 1, z 2 ) = z z2 2 Dehn twist along L Monodromy around t e 2πit = Dehn twist. Vanishing cycle at 0 in the direction R 0 is the 0-section S n self-intersection number is 2( 1) n/2 for even n, 0 for odd n

25 Picard formula Morse theory: R vs. C Definition A fibration over the disc D C is a holomorphic map f : X D with nondegenerate critical points, which correspond to distinct critical values in D. X Kähler, dim X = n + 1, m := card(z = Crit.val.(f )) Fix z 0 D {v z T z D} z Z. Choice of c C vanishing cycles L 1,c,...,L m,c M := f 1 (z 0 ) (local model + parallel transport by canonical connection) Orientations for L i,c monodromy character N X c : Γ = π 1 (D \ Z;z 0 ) Z m m N X c (g) ij := L i,c,g L j,c

26 Picard formula Morse theory: R vs. C The monodromy cocycle Proposition (MOS) Given σ B we have N X σ c = S c (σ) t N X c S c (σ). (conjugate transpose + convolution product) Proof. We have L i,σ c = (c 1 i σ c i ) 1 L i,c, i = π σ,c (i) etc. L i 1, σ c L i, σ c c i 1 c i L i 1,c L i,c σ i L i, σ c z 0 z 0 L i,c c j ε j z z 0 0

27 Picard formula Morse theory: R vs. C The Picard formula Recall f : X D fibration, z 0 D, M := f 1 (z 0 ) c C loops g 1,c,...,g m,c Γ (g i,c ) Aut(H n (M)) Proposition (Picard formula) Setting ε = ( 1) n(n+1)/2, we have (g i,c ) α = α ε L i,c,α L i,c, α H n (M). N X c (gg i,c h) = N X c (gh) εn X c (g)e i N X c (h) since Γ is free on the g i,c s, the map N X c : Γ Z m m is uniquely ( explicitly) determined by the matrix N X c := N X c (1)

28 Picard formula Morse theory: R vs. C Nonlinear monodromy cocycles The matrix N c = (n ij ) = N X c has the following properties. Nc T = ( 1) n N c { 0, n odd, (anti-symmetric) n ii = 2 ( 1) n/2, n even. (symmetric) Let S m = S m (n) = the set of such matrices Mat m (Z). Proposition (Bondal,MOS) For each c C there is a map S c : B S m GL m (Z) such that N σ c = S c (σ,n c ) T N c S c (σ,n c ), σ B, moreover S c (στ,n c ) = S c (τ,n c )S c (σ,n τ c). Conclusion: the orbit of N c under conjugation with S c (,N c ) is an invariant of f.

29 Picard formula Morse theory: R vs. C Straight lines Previous procedure: consider intersection numbers of vanishing cycles in the distinguished fiber M = f 1 (z 0 ). Alternative procedure: consider intersection number of two vanishing cycles L i, L j along some arbitrary path γ ij joining the corresponding critical values collection of numbers m ij Z Particular case: γ ij = straight line segment numbers l ij Z.

30 Picard formula Morse theory: R vs. C Proposition (MOS) Straight lines - continued The collection (l ij ) of intersection numbers along straight lines determines uniquely the matrices N c, c C via the Picard formula. Proof. In the figure, we have l 13 = L 1,(g 2 ) L 3 = L 1,L 3 ε L 2,L 3 L 2 = n 13 εn 12 n induction argument z 1 z 3 z 2 c 1 c 2 c 3 z 0

31 Picard formula Morse theory: R vs. C A dictionary R C Morse function f : M R fibration f : X C {f c ρ} {f c + ρ} monodromy of loop c + ρe 2πit unstable mfd. W u (x, f ) unstable mfd. W u (x, Re(e iθ f )) connecting trajectories connecting trajectories M(x, y; f ) M ( x, y; Re(e iθ f ) ) θ = Arg(f (y) f (x)) #M(x, y; f ) L x L y along segment [f (x), f (y)] Morse differential monodromy character x = y #M(x, y)y N N(1) collection {l xy } = 0 Picard- formula l 13 = n 13 εn 12 n 23

32 Picard formula Morse theory: R vs. C R Morse homology H (X) invariant under deformations f t through smooth functions z 1 Homology C B-orbit of N invariant underdeformationsf t through z 3 z 2 c 1 c 2 c 3 z 0 Example of an invariant of the B-orbit of N: (Z m /ker N,N) free abelian group + nondeg. bilinear form

33 Picard formula Our hope of a dictionary R exact mfd. (M, ω = dλ) C exact mfd. (X, I, ω θ = dλ) Floer theory = Morse th. Floer theory = Picard th. action functional on loop space holomorphic action functional Morse theory: R vs. C The End

34 Picard formula Morse theory: R vs. C Thank you for your attention patience!