An extension of the LMO functor and Milnor invariants

Transcription

1 An extension of the LMO functor and Milnor invariants Yuta Nozaki The University of Tokyo October 27, 205 Topology and Geometry of Low-dimensional Manifolds Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, 205 / 3

2 Introduction Contents Introduction Background Aim of today s talk 2 Definitions and results Domain and codomain of Z Main result 3 Relation with Milnor invariants of string links Review of Milnor invariants Result and example of calculation Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

3 Introduction Background History of LMO. Kontsevich [Kon93] constructed the Kontsevich invariant Z K of (unframed) oriented links in S 3, which was extended to an invariant of (framed) q-tangles. Z K (unknot) = 24 (deg 3) A( )/FI. 2. Using Z K, Le, Murakami and Ohtsuki [LMO98] introduced the LMO invariant of connected, oriented, closed 3-manifolds. Z LMO (L(p, )) = p 2 (p )(p 2) 24p A( ). Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

4 Introduction Background 3. Cheptea, Habiro and Massuyeau [CHM08] constructed the LMO functor (defined on a certain category of cobordisms) by using formal Gaussian integrals. log Z(ψ, ) = (i-deg > 2), where ψ, :=. M: a QHS M \ Int[, ] 3 is regarded as a cobordism between disks [, ] 2 and [, ] 2 ( ). Then Z LMO (M) = Z(M \ Int[, ] 3 ). Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

5 Introduction Background What kind of extension? Roughly speaking, the objects and morphisms of the domain are extended as follows: Σ g, Σ g,n (n 0) and. Namely, we allow a cobordim to have vertical tubes. log Z(ψ, ) = (i-deg > 2). 0 Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

6 Introduction Background Remarks 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,n ). Remark Related researches are found in [ABMP0], [Kat4]. (References are listed at the end.) Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

7 Introduction Aim of today s talk More precisely, the LMO functor is defined as a tensor-preserving functor Z : LCob q ts A between the monoidal categories [CHM08]. Z has important properties: Z(M) = exp (Lk(M)/2) Z Y (M) for M = (M, σ, m). Z is universal among rational-valued finite-type invariants of certain 3-manifolds. Z is related with Milnor invariants of string links. Aim of today s talk Construct an extension of Z with the above properties. Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

8 Definitions and results Contents Introduction Background Aim of today s talk 2 Definitions and results Domain and codomain of Z Main result 3 Relation with Milnor invariants of string links Review of Milnor invariants Result and example of calculation Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

9 Definitions and results Domain and codomain of Z Notation Mon(, ) := (free monoid generated by letters and ). w Mon(, ). The compact surface F w is defined as follows: F = R 3. w ± Mon(, ) ( w = w =: n), σ S n. The closed surface R w w,σ is defined as follows: R,σ = / 0, _ 2 0 σ() 0 σ(2) 0 Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

10 Definitions and results Domain and codomain of Z Domain of Z Definition (cobordism) A cobordism from F w to F w ( w = w =: n) is an equivalence class of triples (M, σ, m), where M is a connected, oriented, compact 3-manifold s.t. M = Σ w w n, σ S n, 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) Ext. of the LMO functor & Milnor inv s October 27, / 3

11 Definitions and results 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 N, στ, m n ). Id w := (F w [, ], 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 surface R w w,σ. For a technical reason, we only consider Lagrangian cobordisms that satisfy almost the same homological conditions as in [CHM08]. The strict monoidal subcategory LCob. Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, 205 / 3

12 Definitions and results Domain and codomain of Z Bottom-top tangles Translating cobordisms into bottom-top tangles is necessary to define Z. In fact, there is a - correspondence by digging a bottom-top tangle (B, γ) along its framed oriented tangle γ. dig pull up rotate by π/2 Figure : (B, γ) : (M, σ, m) Cob(, ) Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

13 Definitions and results Domain and codomain of Z Under the previous correspondence, (M, σ, m) is Lagrangian iff H (B) = H ([, ] 3 ) & Lk B (γ ) = O. Here, Lk B (γ) := Lk B( γ) O gg σ Cr(β) 2 Sym π 0 γ(z), where Lk B( γ) is the usual linking matrix of the (framed) link γ := γ (arcs and braid in S 3 \ Int[, ] 3 ) in the homology sphere B := B (S 3 \ Int[, ] 3 ). (B, γ) β Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

14 Definitions and results Domain and codomain of Z In the case of B = [, ] 3, it suffices to count the crossings of a projection of γ. Example ((B, γ) = ([, ] 3, figure below)) _ /2 0 0 /2 Moreover, (the corresponding cobordism of) (B, γ) is Lagrangian since Lk B (γ ) = (0). Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

15 Definitions and results Domain and codomain of Z Codomain of Z Definition (space of Jacobi diagrams) X : an oriented compact -manifold, C: a finite set. A(X, C) := Q{Jacobi diagrams based on (X, C)}/AS, IHX, STU Figure : X =, C = {, 2, 3}, deg = 2/2 = 6 Remark Take the degree completion of A(X, C) and denote it by A(X, C) again. Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

16 Definitions and results 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 2 χ S 2!! 0! c c Figure : X =, c C, S = {, 2, 3} χ S is an isomorphism and plays an important role when dealing with Jacobi diagrams. Moreover, we need a graded Q-linear map χ S,S : A(X, C S S ) A(X S, C), where S is a copy of S. Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

17 Definitions and results Domain and codomain of Z Definition (strict monoidal category ts A) Obj( ts A) := Z 2 0. ts A((g, n), (f, n)) := {x A(, g f n 0 ) x is a series of top-substantial Jacobi diagrams}. x y := χ χ n 0 n 0, n (x/i i ), (y/i i, i 0 i 0 0 ). g ( g ) i Id (g,n) := exp. x y := x y. i= k := {, 2,..., k }. i A Jacobi diagram is top-substantial if it contains no strut (x, y) ts A((g, n), (f, n)) ts A((h, n), (g, n)), g A(, h f n 0 n 0 ) χ n 0,n 0 A( n0, h f ) χ n 0 A(, h f n 0 ). i j. Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

18 Definitions and results Domain and codomain of Z Example (f = g = n = ) ( 0, 0 ) 2 2 χ 0' _ 0 0' _ _ χ 0 0, ' where the last step follows from = 0 0 _ 2! 2 _ χ 0. Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

19 Definitions and results Main result Before stating the main result... Remark The Kontsevich invariant of tangles depends on a parenthesizing of their boundaries. (In other words, it depends on the choice of an associator.) Therefore, we have to refine as follows. until now from now on Mon(, ) Mag(, ) Cob Cob q LCob LCob q There is a canonical surjection Mon(, ) forget Mag(, ), e.g., ( ( )). Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

20 Definitions and results Main result Let (M, σ, m) LCob q (w, w ) and g = w, f = w, n = w. Definition (extension of the LMO functor) Z(M, σ, m) := χ π 0 γz K-LMO (B, γ) ts A T g ts A((g, n), (f, n)). (M, σ, m) : (B, γ) surgery presentation B =[,] 3 L (L, γ) Z K Z K (L ν γ) Z(M, σ, m) T g π 0 L Z K-LMO (B, γ) π 0 L is the formal Gaussian integral along π 0L introduced in [BNGRT02a, BNGRT02b, BNGRT04]. Z K-LMO (B, γ) A( γ, ) = A(, π 0 γ). T g A(, g g ) is the same as defined in [CHM08]. Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

21 Definitions and results Main result Theorem (N. 5) Z : LCob q ts A is a tensor-preserving functor that splits as Z(M, σ, m) = exp (Lk B (γ)/2) Z Y (M, σ, m). Advantage The above Z is a non-trivial extension of the LMO functor, indeed, Z reflects interaction between the top/bottom components and the vertical components: log Z = (i-deg > 2). 0 Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

22 Relation with Milnor invariants of string links Contents Introduction Background Aim of today s talk 2 Definitions and results Domain and codomain of Z Main result 3 Relation with Milnor invariants of string links Review of Milnor invariants Result and example of calculation Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

23 Relation with Milnor invariants of string links Review of Milnor invariants String links A string link (B, σ) on l strands is, for example, the figure in the middle. 0 2* 3 * * MJ closure 2 3 Figure : A bottom-top tangle, the corresponding string link and its closure. Here, MJ is an extension of the Milnor-Johnson correspondence defined in [CHM08]. S l := {string link (B, σ) on l strands H (B) = H ([, ] 3 )}. Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

24 Relation with Milnor invariants of string links Review of Milnor invariants Milnor invariants (B, σ) S := B \ N(σ) and s : (D l [, ]) = S S. D l := x x l. ϖ := π (D l, ), ϖ = ϖ, ϖ k = [ϖ k, ϖ]. The monoid anti-homomorphism A k : S l Aut(ϖ/ϖ k ) defined by A k (B, σ) := s, s, is called the kth Artin representation (k ). Definition (Milnor invariant) The kth Milnor invariant is the monoid homomorphism µ k : S l [k](:= Ker A k ) ϖ/ϖ 2 Z ϖ k /ϖ k defined by µ k (B, σ) := l x i s, (λ i ), where λ i is the longitude of σ. i= Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

25 Relation with Milnor invariants of string links Review of Milnor invariants µ k (B, σ) (k 2) is regarded as a linear combination µ A k (B, σ) of connected tree Jacobi diagrams via the following commutative diagram. S l [k] µ k (ϖ/ϖ 2 ϖ k /ϖ k ) Q µ A k 0 A t,c k ( l ) η k Q l Q Lie k ( l ) [, ] Lie k ( l ) 0 η k (D) := v (color of v) comm(d v), where v runs over all external (i.e., trivalent) vertices in D and c c c c comm := [c, [[c 2, c 3 ], c 4 ]]. v = Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

26 Relation with Milnor invariants of string links Result and example of calculation Previous studies and our extension. Habegger and Masbaum [HM00] proved that the first non-vanishing Milnor invariant of ([, ] 3, σ) is determined by the first non-trivial term of (χ π 0 (σ) Z K (σ)) Y,t, and vice versa. 2. Moffatt [Mof06] showed that the same holds for (χ π 0 (σ) Z K-LMO (B, σ)) Y,t. 3. The same is true for Z Y,t (MJ (B, σ)) [CHM08]. Let R w : A(X, g g n 0 ) = A(X, l ) be a color-replacement map for w Mag(, ). Theorem (N. 5) Y,t If (B, σ) S l [k], then Z <k (MJ w (B, σ)) = Rw Y,t Conversely, if Z <k (MJ w (B, σ)) is written as x (i-deg x = k ), then µ A k (B, σ) = R w(x). (µ A k (B, σ)). Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

27 Relation with Milnor invariants of string links Result and example of calculation Example (l = 3, k = 2, σ = (example in p. 23)) Check MJ (σ) = ψ, ψ,. Using the functoriality of Z and Z Y ( ) = Z Y ( ) = exp ( 2 0 ) (i-deg > 2), we have Z Y (MJ (σ)) = χ 0 χ 0, 0 ( 2 = 0 By the previous theorem, µ A 2 (σ) = 0 2 (i-deg > ) (i-deg > ) ) Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

28 Relation with Milnor invariants of string links Result and example of calculation Continuation of the previous example Using the previous result µ A 2 (σ) = 2 3, we have σ µ 2 2 µ A 2 x [x 2, x 3 ] = 3 η [2, 3 ] [, ] [, [2, 3 ]] = 0 where denotes the cyclic permutations. It follows that µ 2 (σ) = x [x 2, x 3 ] x 2 [x 3, x ] x 3 [x, x 2 ] (ϖ/ϖ 2 ϖ 2 /ϖ 3 ) Q. Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

29 Relation with Milnor invariants of string links Result and example of calculation The Milnor µ-invariants (of length 3) of the closure of σ is computed as follows:. The Magnus expansion of µ 2 (σ) ϖ 2 /ϖ 3 is x (X 2 X 3 X 3 X 2 (deg > 2)) (cyclic permutations). 2. Reading the coefficient of x X 2 X 3 etc., { sgn(j j 2 i) if {j, j 2, i} = {, 2, 3}, µ σ (j, j 2 ; i) = 0 otherwise. * 2* 3* 0 MJ closure 2 3 Z <2 (MJ (σ)) µ 2 (σ) µ σ (j, j 2 ; i) Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

30 Relation with Milnor invariants of string links Result and example of calculation Future research I would like to investigate the functor Z and find some applications for it. Introduction Background Aim of today s talk 2 Definitions and results Domain and codomain of Z Main result 3 Relation with Milnor invariants of string links Review of Milnor invariants Result and example of calculation Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

31 Relation with Milnor invariants of string links Result and example of calculation [ABMP0] Jørgen Ellegaard Andersen, Alex James Bene, Jean-Baptiste Meilhan, and R. C. Penner. Finite type invariants and fatgraphs. Adv. Math., 225(4):27 26, 200. [BNGRT02a] Dror Bar-Natan, Stavros Garoufalidis, Lev Rozansky, and Dylan 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(3):35 339, [BNGRT02b] Dror Bar-Natan, Stavros Garoufalidis, Lev Rozansky, and Dylan P. Thurston. The århus integral of rational homology 3-spheres. II. Invariance and universality. Selecta Math. (N.S.), 8(3):34 37, [BNGRT04] [CHM08] [HM00] Dror Bar-Natan, Stavros Garoufalidis, Lev Rozansky, and Dylan P. Thurston. The århus integral of rational homology 3-spheres. III. Relation with the Le-Murakami-Ohtsuki invariant. Selecta Math. (N.S.), 0(3): , Dorin Cheptea, Kazuo Habiro, and Gwénaël Massuyeau. A functorial LMO invariant for Lagrangian cobordisms. Geom. Topol., 2(2):09 70, Nathan Habegger and Gregor Masbaum. The Kontsevich integral and Milnor s invariants. Topology, 39(6): , Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

32 Relation with Milnor invariants of string links Result and example of calculation [Kat4] [Kon93] [LMO98] [Mof06] [Noz5] Ronen Katz. Elliptic associators and the LMO functor. arxiv:math.gt/ v2, 204. Maxim Kontsevich. Vassiliev s knot invariants. In I. M. Gel fand Seminar, volume 6 of Adv. Soviet Math., pages Amer. Math. Soc., Providence, RI, 993. Thang T. Q. Le, Jun Murakami, and Tomotada Ohtsuki. On a universal perturbative invariant of 3-manifolds. Topology, 37(3): , 998. Iain Moffatt. The århus integral and the µ-invariants. J. Knot Theory Ramifications, 5(3):36 377, Yuta Nozaki. An extension of the LMO functor. arxiv:math.gt/ , 205. Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3

33 Relation with Milnor invariants of string links Result and example of calculation MJw Figure : An extension of the Milnor-Johnson correspondence Y. Nozaki (The Univ. of Tokyo) Ext. of the LMO functor & Milnor inv s October 27, / 3