An extension of the LMO functor

Transcription

1 An extension of the LMO functor Yuta Nozaki The Univ. of Tokyo December 23, 2014 VII Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

2 Introduction Contents 1 Introduction Examples of calculation Background and motivation 2 Definitions Domain and codomain of Z Extension of the LMO functor 3 Results and applications Main results Milnor invariants Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

3 Introduction Examples of calculation Examples of calculation Figure 1 : A cobordism ψ, introduced in [CHM08] Figure 2 : A new cobordism ψ, log Z(ψ, ) = log Z(ψ, ) = (i-deg > 2), (i-deg > 2). Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

4 Introduction Background and motivation References [BGRT02a] D. Bar-Natan, S. Garoufalidis, L. Rozansky and D. P. Thurston, The Århus integral of rational homology 3-spheres. I. A highly non trivial flat connection on S 3, Selecta Math. (N.S) 8 (2002), no. 3, [BGRT02b] D. Bar-Natan, S. Garoufalidis, L. Rozansky and D. P. Thurston, The Århus integral. II. Invariance and universality, Selecta Math. (N.S.) 8 (2002), no. 3, [CHM08] D. Cheptea, K. Habiro and G. Massuyeau, A functorial LMO invariant for Lagrangian cobordisms, Geom. Topol. 12 (2008), no. 2, [LMO98] T. T. Q. Le, J. Murakami and T. Ohtsuki, On a universal perturbative invariant of 3-manifolds, Topology 37 (1998), no. 3, Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

5 Introduction Background and motivation History of LMO Le, Murakami and Ohtsuki introduced the LMO invariant of connected, oriented, closed 3-manifolds [LMO98]. Bar-Natan, Garoufalidis, Rozansky and Thurston gave an alternative construction of the LMO invariant (of QHS s) by introducing the Århus integral, which is also called the formal Gaussian integral [BGRT02a, BGRT02b]. Cheptea, Habiro and Massuyeau constructed the LMO functor Z by using the formal Gaussian integral. Z is defined on a certain category of cobordisms [CHM08]. Theorem ([CHM08, Theorem 4.13]) Z : LCob q ts A is a tensor-preserving functor. Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

6 Introduction Background and motivation What kind of extension? Roughly speaking, the objects and morphisms of the domain are extended as follows: Σ g,1 Σ g,b+1 (b 0) and. Convention Notation and terminology are almost the same as in [CHM08], but their definitions are extended. The main differences will be emphasized in red (e.g. Σ g,b+1 ). Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

7 Introduction Background and motivation Properties of Z Z has important properties proved in [CHM08]: Z(M, σ, m) = exp (Lk(M)/2) Z Y (M, σ, m) (matrix (a ij ) identify linear combination of struts i,j a ij Z is universal among rational-valued finite-type invariants. i j ), Aim of today s talk Construct an extension of Z for which the above theorem and properties are true. Show a relation with the Milnor invariants of string links. Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

8 Definitions Contents 1 Introduction Examples of calculation Background and motivation 2 Definitions Domain and codomain of Z Extension of the LMO functor 3 Results and applications Main results Milnor invariants Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

9 Definitions Domain and codomain of Z Notation Mon(, ) := (free monoid generated by letters and ). w Mon(, ). The compact surface F w is defined as follows: β1 α1 δ1 β2 α2 δ2 F = R 3. w ± Mon(, ) ( w + = w = b), σ S b. The closed surface R w + w,σ is defined as follows: R,σ = / 1 0, _ σ(1) 0 σ(2) 0 Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

10 Definitions Domain and codomain of Z Domain of Z Definition (cobordism) A cobordism from F w+ to F w ( w + = w = b) is an equivalence class of triples (M, σ, m), where M is a connected, oriented, compact 3-manifold with M = Σ w + + w +b, σ S b, m : R w + w,σ M is an orientation-preserving homeomorphism, (M, σ, m) (N, τ, n) if σ = τ and there is an ori.-pres. homeo. f : M N s.t. M m f R w + w,σ n N. Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

11 Definitions Domain and codomain of Z Definition (strict monoidal category Cob) Obj(Cob) := Mon(, ). Cob(w +, w ) := {cobordisms from F w+ to F w } (or ). (M, σ, m) (N, τ, n) := (M m+ n 1 N, στ, m n + ). Id w := (F w [ 1, 1], Id S w, Id ). (M, σ, m) (N, τ, n) := (horizontal juxtaposition of M and N). m ± is the restriction of m to the top/bottom of the reference surface. Id = Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

12 Definitions Domain and codomain of Z Definition (Lagrangian cobordism) A cobordism (M, σ, m) Cob(w +, w ) is Lagrangian if the following two conditions are satisfied: 1. H 1 (M) = m, (A w ) + m +, (H 1 (F w+ )), 2. m +, (A w+ ) m, (A w ) + m +, (D w+ ). A w := α 1,..., α w, Dw := δ 1,..., δ w H1 (F w ). The strict monoidal subcategory LCob whose morphisms are Lagrangian cobordisms is obtained. Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

13 Definitions Domain and codomain of Z Bottom-top tangles Translating cobordisms into bottom-top tangles is necessary to define Z. In fact, there is a 1-1 correspondence by digging a bottom-top tangle (B, γ) along its framed oriented tangle γ. dig pull up rotate by π/2 Figure 3 : (B, γ) 1:1 (M, σ, m) Cob(w +, w ) Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

14 Definitions Domain and codomain of Z Codomain of Z Definition (space of Jacobi diagrams) X : an oriented compact 1-manifold, C: a finite set. A(X, C) := Q{Jacobi diagrams based on (X, C)}/AS, IHX, STU STU 3 = 1 2 AS 1 3 Figure 4 : X =, C = {1, 2, 3}, deg = 10/2 = 5 Remark Take the degree completion of A(X, C) and denote it in the same way. Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

15 Definitions Domain and codomain of Z The graded Q-linear map χ S : A(X, C S) A(X S, C) is defined as follows. c χ S 1 2! 1! 0! c + c Figure 5 : X =, S = {1, 2, 3} χ S is an isomorphism and plays an important role when dealing with Jacobi diagrams. However, χ S is not enough to construct an extension of the LMO functor. Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

16 Definitions Domain and codomain of Z S : a copy of S. Introduce the graded Q-linear map χ S,S : A(X, C S S ) A(X S, C) defined as follows. ' 2 ' χ S,S' 1 2! 1! 0! 2! Figure 6 : X = C =, S = {1, 2}, S = {1, 2 } A similar idea appears in [HM]. [HM] K. Habiro and G. Massuyeau, Symplectic Jacobi diagrams and the Lie algebra of homology cylinders, J. Topol. 2 (2009), no. 3, Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

17 Definitions Domain and codomain of Z Definition (strict monoidal category ts A) Obj( ts A) := Z 2 0. ts A((g, b), (f, b)) := {x A(, g + f b 0 ) x is a series of top-substantial Jacobi diagrams}. x y := χ 1 χ b 0 b 0, b (x/i + i ), (y/i i, i 0 i 0 0 ). g ( ) g i + Id (g,b) := exp. x y := x y. i=1 n := {1, 2,..., n }. i A Jacobi diagram D is top-substantial if D contains no struts ts A((g, b), (f, b)) ts A((h, b), (g, b)), g A(, h + f b 0 b 0 ) χ b 0,b 0 A( b0, h + f ) χ 1 b 0 A(, h + f b 0 ). Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27 i + j +.

18 Definitions Extension of the LMO functor The Kontsevich(-LMO) invariant Remark The Kontsevich invariant of tangles depends a parenthesizing of their boundaries. So, Mag(, ), Cob q and LCob q are needed instead of Mon(, ), Cob and LCob respectively. Mag(, ) := (free magma generated by letters and ). For example, Mon(, ) ( ( )) Mag(, ). Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

19 Definitions Extension of the LMO functor The Kontsevich-LMO invariant of a bottom-top q-tangle (B, γ) is defined as follows: 1. Find a surgery link L satisfying ([ 1, 1] 3 L, γ) = (B, γ); 2. Take the formal Gaussian integral of Z K (L γ) ν π0(l) A(L γ, ) along the set π 0 (L); 3. Normalize this last result in accordance with the signature of the matrix Lk(L); Z K-LMO (B, γ) A(γ, ) is obtained. Definition (extension of the LMO functor) Let (M, σ, m) LCob q (w +, w ). Z(M, σ, m) := χ 1 π 0 (γ) Z K-LMO (B, γ) ts A T g ts A((g, b), (f, b)). g = w +, f = w, b = w +. The element T g A(, g + g ) is the same as defined in [CHM08]. Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

20 Results and applications Contents 1 Introduction Examples of calculation Background and motivation 2 Definitions Domain and codomain of Z Extension of the LMO functor 3 Results and applications Main results Milnor invariants Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

21 Results and applications Main results Main results Theorem (N.) Z is a tensor-preserving functor having the following properties: Z(M, σ, m) = exp (Lk B (γ)/2) Z Y (M, σ, m), Remark Z is universal among rational-valued finite-type invariants. Lk B (γ) is suitably defined and belongs to Sym π0 (γ)( 1 2 Z). ( ) log Z = (i-deg > 2), ( ) 0 1/2 Lk [ 1,1] 3(γ) = =. 1/ Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

22 Results and applications Milnor invariants String links A string link (B, σ) on l strands is, for example, the figure on the right. MJ Figure 7 : An extension of the Milnor-Johnson correspondence S fr l := {string link (B, σ) on l strands H (B) = H ([ 1, 1] 3 )}. Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

23 Results and applications Milnor invariants Milnor invariants (B, σ) S := B \ N(σ) and s : (D l [ 1, 1]) S. D l := x 1 x l. ϖ 1 = ϖ := π 1 (D l, ), ϖ n := [ϖ n 1, ϖ]. The monoid homomorphism A n : Sl fr Aut(ϖ/ϖ n+1 ) defined by (B, σ) s, 1 s +, is called the nth Artin representation (n 1). Definition (Milnor invariant) The nth Milnor invariant is the monoid homomorphism µ n : Sl fr[n](:= Ker A n) ϖ/ϖ 2 Z ϖ n /ϖ n+1 defined by µ n (B, σ) := l x i s, (λ 1 i ), where λ i is the longitude determined by the framing of σ. i=1 Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

24 Results and applications Milnor invariants µ n (B, σ) (n 2) is regarded as a linear combination µ A n (B, σ) of connected tree Jacobi diagrams via the following commutative diagram. Im µ n (ϖ/ϖ 2 ϖ n /ϖ n+1 ) Q 0 A t,c n 1 ( l ) η n 1 Q l Q Lie n ( l ) [, ] Lie n+1 ( l ) 0 η n 1 is defined by D v (color of v) comm(d v), where v runs over all external vertices in D. Example (l = 3, n = 2) D 1 := = = 1 η 1 x1 [x 2, x 3 ] + x 2 [x 3, x 1 ] + x 3 [x 1, x 2 ]. Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

25 Results and applications Milnor invariants The first non-vanishing Milnor invariant of ([ 1, 1] 3, σ) is determined by the first non-trivial term of (χ 1 π 0 (σ) Z K (σ)) Y,t and vice versa (Habegger-Masbaum, 00). The same holds for (χ 1 π 0 (σ) Z K-LMO (B, σ)) Y,t (Moffatt, 06). The same is true for Z Y,t (MJ 1 (B, σ)) [CHM08]. Theorem (N.) Let w Mon(, ) and (B, σ) S fr 2 w + w. Then, the first non-vanishing Milnor invariant of (B, σ) is determined by the first non-trivial term of Z Y,t (MJ 1 w (B, σ)) and vice versa. Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

26 Results and applications Milnor invariants Example (Borromean rings) σ = ([ 1, 1] 3, σ): the string link illustrated in Figure 7 MJ 1 (σ) = ψ, ψ,. Using the functoriality of Z, we have log Z Y (MJ 1 (σ)) = (i-deg > 1). 1 By the above theorem, we conclude µ A 2 (σ) = D 1, and its image by Id ϖ/ϖ2 θ 2 is x 1 ( (1 + X 3 )(1 + X 2 )(1 + X 3 ) 1 (1 + X 2 ) 1) deg=2 + (cyclic...) = x 1 (X 3 X 2 X 2 X 3 ) + (cyclic permutations). Therefore, µ σ (j 1, j 2 ; i) = { sgn(j 1 j 2 i) if {j 1, j 2, i} = {1, 2, 3}, 0 otherwise. Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27

27 Results and applications Milnor invariants Future research I would like to investigate the functor Z and find applications of it. 1 Introduction Examples of calculation Background and motivation 2 Definitions Domain and codomain of Z Extension of the LMO functor 3 Results and applications Main results Milnor invariants Y. Nozaki (The Univ. of Tokyo) An extension of the LMO functor December 23, / 27