Higher order generalization of Fukaya's Morse homotopy invariant of 3-manifolds II. Invariants of 3-manifolds with b 1 =1
aa r X i v : . [ m a t h . G T ] M a y Preprint (2016)
HIGHER ORDER GENERALIZATION OF FUKAYA’S MORSEHOMOTOPY INVARIANT OF 3-MANIFOLDS II. INVARIANTSOF 3-MANIFOLDS WITH b = 1 TADAYUKI WATANABE
Abstract.
In this paper, it is explained that a topological invariant for 3-manifold M with b ( M ) = 1 can be constructed by applying Fukaya’s Morsehomotopy theoretic approach for Chern–Simons perturbation theory to a localcoefficient system on M of rational functions associated to the maximal freeabelian covering of M . Our invariant takes values in Garoufalidis–Rozansky’sspace of Jacobi diagrams whose edges are colored by rational functions. It isexpected that our invariant gives a lot of nontrivial finite type invariants of3-manifolds. Introduction
By using an idea of his Morse homotopy theory, K. Fukaya constructed in [Fu] atopological invariant of 3-manifolds with flat bundles on them that is analogous toChern–Simons perturbation theory ([AS, Ko1]). He considered a flat Lie algebrabundle over a 3-manifold M and several Morse functions on M , and defined hisinvariant as the sum of the weights of some graphs (flow-graphs) in M whose edgesfollow the gradients of the Morse functions. The weight is given by contractingthe holonomies taken along the edges of a flow-graph by some tensor. AlthoughFukaya’s construction was given for the 2-loop graphs, his construction also worksfor general 3-valent graphs at least when M is a homology sphere with trivialconnection ([Wa1]).In this paper, we construct a topological invariant e z k for closed oriented 3-manifolds M with b ( M ) = 1 by applying Fukaya’s construction to a local coefficientsystem of rational functions associated to the maximal free abelian covering of M . There are fundamental results of Lescop about an equivariant 2-loop invariantfor closed oriented 3-manifolds with b ( M ) = 1 ([Les1, Les2]), by which we weresignificantly influenced in the construction of e z k . The construction of e z k wouldbe rewritten by equivariant intersections in configuration spaces as given in [Les1,Les2]. Prior to Lescop’s works, Ohtsuki had given in [Oh1, Oh2] a considerablerefinement of the LMO invariant for 3-manifolds M with b ( M ) = 1, which isimportant in the study of equivariant perturbative invariant for non homologyspheres. It is known that the LMO invariant ([LMO]) is very strong for homologyspheres whereas it is rather weaker for non homology spheres. It is remarkable Date : September 18, 2018.2000
Mathematics Subject Classification. that Ohtsuki’s refined LMO invariant is also very strong for 3-manifolds M with b ( M ) = 1, and moreover his equivariant invariant is computable for some examplesand yields some beautiful formulas. We expect that our invariant agrees withOhtsuki’s refined LMO invariant.In [Wa2], we construct an invariant of some degree 1 maps from 3-manifoldsto the 3-torus by a method similar to the construction of this paper and applyit to study finite type invariants. It will follow from a result of [Wa2] that thevalue of the 2-loop part e z for some 3-manifolds with b = 1 can be computedby clasper calculus of Goussarov and Habiro. In [Wa3], we define an invariant offiberwise Morse functions on surface bundles over S , which can be considered asan analogue of the construction of the present paper for S -valued Morse theory.In the case where a knot in M is present, a construction similar to that of thispaper gives a knot invariant and gives many non-trivial finite type invariants ofknots. We will explain this in a subsequent paper.Most of the construction in [Wa1] is valid for the setting of this paper. We willonly refer for what can be done by the same argument as [Wa1].2. Preliminaries
Acyclic Morse complex.
For simplicity, we assume that M is an oriented,connected, closed 3-manifold with b ( M ) = 1. Let f : M → R be a Morse functionand let Σ ⊂ M be an oriented 2-submanifold that generates the oriented bordismgroup Ω ( M ) ∼ = H ( M ; Z ) = Hom Z ( H ( M ; Z ) , Z ) ∼ = Z . By cutting M along Σand by pasting its copies, the infinite cyclic covering π : f M → M is obtained. Wedenote by t the generator of the group of covering transformation that shifts f M in the direction of the positive normal vector to Σ. Let e f : f M → R be the Morsefunction that is the pullback e f = f ◦ π and let e µ be the Riemannian metric on f M that is the pullback of the Riemannian metric µ on M .The pair ( e f , e µ ) gives the gradient e ξ on f M . Let P i be the set of critical pointsof f of index i and we identify P i with the set of critical points of e f of index i in a fundamental domain of f M between two successive components in π − Σ. Let( C ∗ ( e f ) , ∂ ) be the Morse complex for e ξ with Q -coefficients. Namely, C i ( e f ) = Λ P i (Λ = Q [ t, t − ]) and the boundary ∂ : C i ( e f ) → C i − ( e f ) is given as follows. ∂ ( p ) = X q ∈ P i − X k ∈ Z n ( e ξ ; p, t k q ) t k q, where the coefficient n ( e ξ ; p, t k q ) ∈ Z is the number of the flow-lines of − e ξ in f M thatflow from p to t k q counted with signs. In other words, the count of the flow-linesin M from p to q whose intersection number with Σ is k . The definition of the signis given in [Wa1]. We remark that the sum in the right hand side is finite. We put ∂ pq = P k ∈ Z n ( e ξ ; p, t k q ) t k . It can be checked that ( C ∗ ( e f ) , ∂ ) is a chain complex,namely, ∂ = 0. The homology of the complex is identified with H ∗ ( f M ; Q ) as aΛ-module.Let b Λ be the field of rational functions P ( t ) Q ( t ) ( P ( t ) , Q ( t ) ∈ Λ , Q ( t ) = 0) and put C i ( f ; b Λ) = C i ( e f ) ⊗ Λ b Λ = b Λ P i . GENERALIZATION OF FUKAYA’S INVARIANT OF 3-MANIFOLDS II 3
Figure 1.
A trivalent graph (left) and a ~C -graph (right)This together with the boundary ∂ ⊗ b Λ is a flat Λ-module, we have, as Λ-modules, H i ( C ∗ ( f ; b Λ)) ∼ = H i ( f M ; Q ) ⊗ Λ b Λby the universal coefficient theorem (e.g., [CE, Theorem VI.3.3]). From the factthat rank Λ H ( f M ; Q ) = 0 (e.g., [Les1, Lemma 2.2]) and by the Poincar´e duality, itfollows that ( C ∗ ( f ; b Λ) , ∂ ⊗
1) is acyclic.2.2.
Combinatorial propagators. C = C ∗ ( f ; b Λ) is a chain complex of based free b Λ-modules. Let End b Λ ( C ) = Hom b Λ ( C, C ) and let End b Λ ( C ) k be its degree k part.We define δ : End b Λ ( C ) k → End b Λ ( C ) k − by the following formula. δg = ∂ ◦ g − ( − k g ◦ ∂. This satisfies δ = 0. By the acyclicity of C and by the K¨unneth theorem (e.g.,[CE, Theorem VI.3.1a]), it follows that (End b Λ ( C ) , δ ) is acyclic, too.For example, 1 ∈ End b Λ ( C ) is a δ -cycle. Thus there exists g ∈ End b Λ ( C ) suchthat δg = ∂g + g∂ = 1. Such a g is called a combinatorial propagator for C ([Fu]).For two choices g, g ′ of combinatorial propagators for C , g ′ − g is a δ -cycle. Thusthere exists h ∈ End b Λ ( C ) such that ∂h − h∂ = g ′ − g .3. Perturbation theory with holonomies in b Λ3.1.
Moduli space M Γ ( ~ξ ) of flow-graphs. Let f , f , . . . , f k : M → R be asequence of Morse functions and let ξ i be the gradient of f i . We consider a connectededge-oriented trivalent graph with its sets of vertices and edges labelled and with 2 k vertices and 3 k edges. By the labelling { , , . . . , k } → Edges(Γ) of Γ, we identifyedges with numbers. Choose some of the edges and split each chosen edge intotwo arcs. We attach elements of P ∗ ( f i ) on the two 1-valent vertices ( white-vertices )that appear after the splitting of the i -th edge. We call such obtained graph a ~C -graph ( ~C = ( C ∗ ( f ; b Λ) , . . . , C ∗ ( f k ; b Λ)), see Figure 1). A ~C -graph has two kinds of“edges”: a compact edge , which is connected, and a separated edge , which consistsof two arcs. We say that a separated edge obtained from a self-loop is closed . Wecall vertices that are not white vertices black vertices . If p i (resp. q i ) is the criticalpoint attached on the input (resp. output) white vertex of a separated edge i , wedefine the degree of i by deg( i ) = ind( p i ) − ind( q i ), where ind( · ) denotes the Morseindex. We define the degree of a compact edge i by deg( i ) = 1. We define thedegree of a ~C -graph by deg(Γ) = (deg(1) , deg(2) , . . . , deg(3 k )).We say that a continuous map I from a ~C -graph Γ to M is a flow-graph for thesequence ~ξ = ( ξ , ξ , . . . , ξ k ) if it satisfies the following conditions (see Figure 2). TADAYUKI WATANABE
Figure 2.
Flow-graphs for ~ξ = ( ξ , ξ , ξ )(1) Every critical point p i attached on the i -th edge is mapped by I to p i in M .(2) The restriction of I to each edge of Γ is a smooth embedding and at eachpoint x of the i -th edge that is not on a white vertex, the tangent vector of I at x (chosen along the edge orientation) is a positive multiple of ( − ξ i ) x .For a ~C -graph Γ, let M Γ ( ~ξ ) be the set of all flow-graphs for ~ξ from Γ to M . Byextracting black vertices, a natural map from M Γ ( ~ξ ) to the configuration space C k ( M ) of ordered tuples of 2 k points is defined. It follows from a property of thegradient that this map is injective. This induces a topology on the set M Γ ( ~ξ ). Lemma 3.1 (Fukaya [Fu, Wa1]) . If ~f = ( f , f , . . . , f k ) and µ are generic, thenfor a ~C -graph Γ with k black vertices and deg(Γ) = (1 , , . . . , , the space M Γ ( ~ξ ) is a compact 0-dimensional manifold. Moreover, this property can be assumed forall ~C -graphs with k black vertices simultaneously ∗ . The count of M Γ ( ~ξ ) . When the assumption of Lemma 3.1 is satisfied, we maydefine an orientation of M Γ ( ~ξ ) in a similar way as [Wa1]. Roughly, an orientation of M Γ ( ~ξ ) is defined as follows. The space M Γ ( ~ξ ) can be considered as the intersectionof several smooth manifold strata in M k each corresponds to the moduli space of anedge of Γ. We define an orientation of M Γ ( ~ξ ) by the coorientation V e ∈ Edges(Γ) v e of M Γ ( ~ξ ) in M k for some coorientations v e of the strata for e . If e is compact or non-closed separated, then e has two black vertices and v e is a vector in V ( T x M ⊕ T y M ),where x, y are the images from the black vertices of e . If e is closed separated, thenthe corresponding stratum is a flow-line γ pq of − ξ e between some pair p, q of criticalpoints with ind( p ) = ind( q ) −
1. In this case, the orientation of γ pq is defined by( − ind( q ) ε ( γ pq ) o , † where ε ( γ pq ) is the sign of γ pq in the definition of ∂ ( p ), and o isthe orientation of γ pq given by − ξ e .In [Wa1], the number M Γ ( ~ξ ) was defined as the sum of the signs determinedby the orientations. The following definition of M Γ ( ~ξ ) is different from that of[Wa1]. We count points of M Γ ( ~ξ ) with weights in Λ ⊗ k = Q [ t ± , t ± , . . . , t ± k ] as ∗ In [Wa1], we considered flow-graphs on punctured homology sphere. Nevertheless, the proofof the corresponding lemma is essentially the same. † This definition is consistent with that for non-closed separated edges. Namely, this ageeswith that induced on the intersection of the stratum for a non-closed separated edge with thediagonal in M × M . In the notation of [Wa1], the stratum for a separated edge e is cooriented by o ∗ M ( A q ( f e )) x ∧ o ∗ M ( D p ( f e )) y , whereas γ pq is cooriented by o ∗ M ( D p ( f e )) x ∧ o ∗ M ( A q ( f e )) x . On thediagonal, the two differ by ( − ind( q )(3 − ind( p )) = ( − ind( q ) . GENERALIZATION OF FUKAYA’S INVARIANT OF 3-MANIFOLDS II 5
Figure 3.
The relations AS, IHX, Orientation reversal, Linearityand Holonomy. p, q, r ∈ Λ (or p, q, r ∈ b Λ), α ∈ Q . The exponent ε i is 1 if the i -th edge is oriented toward v and otherwise − M Γ ( ~ξ ) = X I ∈ M Γ ( ~ξ ) ε ( I ) , ε ( I ) = ± t n t n · · · t n k k ,n i = (The intersection number of the i -th edge of I with Σ)We take the sign ± as the one determined by the orientation of I . The intersectionnumber of the i -th edge with Σ is determined by the orientations of the edge andof Σ and that of M , by ori( i -th edge) ∧ ori(Σ) = ori( M ).3.3. The generating series and its trace.
Let R be either Λ or b Λ. An R -colored ~C -graph is a pair of a ~C -graph Γ and a map φ : Edges(Γ) → R . We will write an R -colored ~C -graph as Γ( φ ) or Γ( φ (1) , φ (2) , . . . , φ (3 k )). We call φ an R -coloring ofΓ. We define an action of a monomial t n t n · · · t n k k on a ~C -graph as follows. t n t n · · · t n k k · Γ = Γ( t n , t n , . . . , t n k ) . The right hand side is a Λ-colored ~C -graph. By extending this by Q -linearity, theformal linear combination M Γ ( ~ξ ) · Γ of Λ-colored ~C -graphs is defined.Let A k (Λ) (resp. A k ( b Λ)) be the vector space over Q spanned by pairs (Γ , φ ),where Γ is an (unlabelled) edge-oriented trivalent graph with 2 k vertices and withvertex-orientation and φ is a Λ-coloring (resp. b Λ-coloring) of Γ, quotiented bythe relations AS, IHX, Orientation reversal, Linearity, Holonomy (Figure 3) andautomorphisms of oriented graphs ‡ .Now we shall define an element z k ( ~ξ ) ∈ A k ( b Λ). Let ~g = ( g (1) , g (2) , . . . , g (3 k ) ) bea sequence of combinatorial propagators for ~C = ( C ∗ ( f ; b Λ) , . . . , C ∗ ( f k ; b Λ)). Thenwe define(3.1) z k ( ~ξ ) = Tr ~g (cid:16)X Γ M Γ ( ~ξ ) · Γ (cid:17) . Here, the sum is taken over all ~C -graphs Γ with 2 k black vertices and deg(Γ) =(1 , , . . . , ~g is defined as follows. For simplicity, we assume that thelabels for the separated edges in a Λ-colored ~C -graph Γ( u ( t ) , u ( t ) , . . . , u k ( t )) is1 , , . . . , r . Let p i , q i be the critical points on the input and output of the i -th ‡ This definition is by Garoufalidis and Rozansky [GR]. The AS and the IHX relations are dueto Bar-Natan [BN].
TADAYUKI WATANABE edge of Γ, respectively and let g ( i ) q i p i ∈ b Λ be the coefficient of p i in g ( i ) ( q i ). ThenTr ~g (Γ( u ( t ) , u ( t ) , . . . , u k ( t ))) is the equivalence class in A k ( b Λ) of a b Λ-coloredgraph obtained by identifying each pair of the two white vertices of the separatededges in Γ( − g (1) q p u ( t ) , . . . , − g ( r ) q r p r u r ( t ) , u r +1 ( t ) , . . . , u k ( t )) . The definition of Tr can be generalized to graphs with other degrees in the samemanner.
Lemma 3.2. z k ( ~ξ ) does not depend on the choices of ~g and of the hypersurface Σ ⊂ M within the oriented bordism class.Proof. That z k ( ~ξ ) does not depend on the choice of ~g can be shown by the sameargument as in [Wa1, § ′ ⊂ M be another oriented 2-submanifold that is orientedbordant to Σ. Then by Morse theory, one may see that Σ ′ is obtained from Σ by afinite sequence of the following moves.(1) A homotopy in M .(2) An addition or a deletion of a 1-handle in a small ball B in M .We may assume that for each move of type (2), the small ball B is disjoint fromall the critical points of f i and the graphs in M Γ ( ~ξ ) for all Γ. Thus a move of type(2) does not change z k ( ~ξ ).The sum in (3.1) may change under a move of type (1) when a homotopy inter-sects a black vertex of a flow-graph or intersects a critical point of some f i . Whena homotopy intersects a black vertex v , a Λ-coloring for the three edges incident to v changes. The change of Λ-coloring is precisely the Holonomy relation. When asmall homotopy intersects a critical point p of some Morse function, say of f , theboundary operator of the twisted Morse complex ( C (1) ∗ , ∂ (1) ) for the first edge maychange. Let ∂ (1) be the resulting boundary operator. Let S p : C (1) ∗ → C (1) ∗ be thechain map of degree 0 defined for critical points x ∈ P (1) ∗ by S p ( x ) = (cid:26) t ± p if x = px otherwisewhere the sign ± p from above or below.Then we have ∂ (1) = S − p ◦ ∂ (1) ◦ S p and that g (1) = S − p ◦ g (1) ◦ S p : C (1) ∗ → C (1) ∗ +1 is a combinatorial propagator for ( C (1) ∗ , ∂ (1) ). There is an analogous left/rightaction of S ± p on a Λ-colored ~C -graph given as follows. For a Λ-colored ~C -graphΓ( φ ), Γ( φ ) ◦ S p (resp. S p ◦ Γ( φ )) is the Λ-colored ~C -graph obtained from Γ( φ ) byreplacing φ (1) with t ± φ (1) (resp. with t ∓ φ (1)) if p is the input (resp. the output)of the first edge of Γ and otherwise Γ( φ ) ◦ S p = Γ( φ ) (resp. S p ◦ Γ( φ ) = Γ( φ )).After the small homotopy that crosses p , the flow graph that was counted as Γ( φ )will be counted as S − p ◦ Γ( φ ) ◦ S p . Now we haveTr g (1) ,... ( S − p ◦ Γ( φ ) ◦ S p ) = Tr S p ◦ g (1) ◦ S − p ,... (Γ( φ )) = Tr g (1) ,... (Γ( φ )) . This completes the proof of the invariance under a move of type (1). (cid:3)
GENERALIZATION OF FUKAYA’S INVARIANT OF 3-MANIFOLDS II 7
The invariant e z k . For the independence of the choice of ~ξ , we shall define e z k by adding a correction term to z k ( ~ξ ). Though this could be done by thesame method as [Wa1], the following definition by Shimizu ([Sh1]) is nicer here.Take a compact oriented 4-manifold W with ∂W = M and with χ ( W ) = 0. Bythe condition χ ( W ) = 0, the outward normal vector field to M in T W | ∂W canbe extended to a nonsingular vector field ν W on W . Let T v W be the orthogonalcomplement of the span of ν W . Then T v W is a rank 3 subbundle of T W thatextends
T M . Take a sequence ~γ = ( γ , γ , . . . , γ k ) of generic sections of T v W sothat γ i is an extension of − ξ i . We define z anomaly2 k ( ~γ ) = X Γ M localΓ ( ~γ ) [Γ(1 , , . . . , ∈ A k ( b Λ) . The sum is taken over all ~C -graphs with 2 k vertices and with only compact edges.Here, M localΓ ( ~γ ) is the moduli space of affine graphs in the fibers of T v W whose i -thedge is a positive scalar multiple of γ i . The number M localΓ ( ~γ ) ∈ Z is the countof the signs of the affine graphs that are determined by transversal intersectionsof some codimension 2 chains in a configuration space bundle over W . See [Wa1]for detail. By the same argument as in [Sh1], it can be shown that z anomaly2 k ( ~γ ) − µ k sign W , where µ k ∈ A k ( b Λ) is the constant given in [Wa1], does not depend onthe choices of W , ν W and the extension ~γ of ~ξ . We define e z k ( ~ξ ) ∈ A k ( b Λ) by thefollowing formula. ˆ z k ( ~ξ ) = z k ( ~ξ ) − z anomaly2 k ( ~γ ) + µ k sign W, e z k ( ~ξ ) = X ε i = ± ˆ z k ( ε ξ , . . . , ε k ξ k ) . Theorem 3.3. e z k ( ~ξ ) is an invariant of the diffeomorphism type of M and of theoriented bordism class of Σ . § Proof of Theorem 3.3
Closedness of the 1-chain for a self-loop.
We will use the following lemmain the proof of Theorem 3.3. Let Γ be a ~C -graph whose i -th edge is a closedseparated edge. Let Γ ′ be the graph obtained from Γ by removing the i -th edge.Then there is a 1-chain O ( ξ i ) of M with untwisted b Λ-coefficients ¶ such that(4.1) M Γ ( ~ξ ) = M Γ ′ ( ξ , . . . , b ξ i , . . . , ξ k ) · pr − i O ( ξ i ) . Here, pr i : M k → M is the projection to the i -th factor, and the symbol · is theintersection between smooth manifold strata in M k , extended linearly to chains. § There are only two possibilities for the class of Σ that generates Ω ( M ) ∼ = Z , and the valuesof e z k for the two differ by turning every b Λ-coloring φ ( t ) on edge into φ ( t − ). Thus, they can beconsidered essentially the same. ¶ Notice that the holonomy along the closed separated edge does not depend on the positionof the black vertex on it.
TADAYUKI WATANABE
Let M p i q i ( ξ i ) denote the set of all flow-lines of − ξ i that flow from p i to q i . Then O ( ξ i ) can be written as follows.(4.2) O ( ξ i ) = X pi,qi ind( pi )=ind( qi )+1 X γ piqi ∈ M piqi ( ξ i ) ( − ind( q i ) g ( i ) q i p i ε ( γ p i q i )Hol( γ p i q i ) γ p i q i , where γ p i q i is the 1-chain obtained from the flow-line by compactification, andHol( γ p i q i ) = t n i , where n i is the intersection number of γ p i q i with Σ. Lemma 4.1. O ( ξ i ) + O ( − ξ i ) is a 1-cycle k .Proof. By (4.2), we have ∂O ( ξ i ) = X pi,qi ind( pi )=ind( qi )+1 X γ piqi ∈ M piqi ( ξ i ) ( − ind( q i ) g ( i ) q i p i ε ( γ p i q i )Hol( γ p i q i )( q i − p i ) . The coefficient of p i of index k in this formula is given by − X qi ind( qi )= k − ( − k − g ( i ) q i p i ∂ ( i ) p i q i + X ri ind( ri )= k +1 ( − k g ( i ) p i r i ∂ ( i ) r i p i = ( − k ( g ( i ) ◦ ∂ ( i ) + ∂ ( i ) ◦ g ( i ) ) p i p i = ( − k . Similarly, the coefficient of p i of index 3 − k in ∂O ( − ξ i ) is given by X qi ind( qi )=3 − ( k − ( − − k g ( i ) p i q i ∂ ( i ) q i p i − X ri ind( ri )=3 − ( k +1) ( − − ( k +1) g ( i ) r i p i ∂ ( i ) p i r i = − ( − k ( g ( i ) ∗ ◦ ∂ ( i ) ∗ + ∂ ( i ) ∗ ◦ g ( i ) ∗ ) p i p i = − ( − k , where g ( i ) ( t ) = g ( i ) ( t − ) etc., and g ( i ) ∗ etc. is given by the adjoint matrix. Thisproves ∂O ( ξ i ) + ∂O ( − ξ i ) = 0. (cid:3) Completing the proof of Theorem 3.3.
When the i -th edge of a ~C -graphΓ is a separated edge on which x, y ∈ P ( i ) are attached on the input/output re-spectively, we will write Γ = Γ( x, y ) i . This notation enables us to express thegraph Γ( x, y ) i with x, y replaced with x ′ , y ′ respectively, as Γ( x ′ , y ′ ) i . The notationΓ( ∅ , ∅ ) i will denote the graph obtained from Γ( x, y ) i by replacing the i -th edge witha compact edge.Since the proof is parallel to that of the main theorem of [Wa1], we only givean outline. We show that the value of e z k does not change if one Morse functionin the sequence ~f , say f , is replaced with another Morse function f ′ . As usualin Cerf theory ([Ce]), we use the fact that there is a smooth 1-parameter family { h s : M → R } s ∈ [0 , that restricts to f and f ′ on s = 0 , h s is Morse except for finitely many values of s and at the excluded valuesthe singularities of h s consist of birth-death singularities and Morse singularities.Moreover, there may be finitely many values of s at which the Morse complexfor e ξ s = grad e h s changes, namely, at which there is a flow-line of ξ s = grad h s between two Morse critical points of the same index j . Such a flow-line is called a j/j -intersection and corresponds to a handle-slide. k This lemma is implicit in [Sh2].
GENERALIZATION OF FUKAYA’S INVARIANT OF 3-MANIFOLDS II 9
Figure 4.
Degenerate flow-graphsLet J = [ s , s ] ⊂ [0 ,
1] be an interval that does not have birth-death parameter.By replacing f with the family { h s } , the moduli spaces M Γ ( ~ξ J ) and its naturalcompactifications M Γ ( ~ξ J ) for flow-graphs mapped along the fiber M in J × M aredefined. The moduli space M Γ ( ~ξ J ) is an oriented compact 1-dimensional manifoldimmersed in J × C k ( M ), where C k ( M ) is the differential geometric analogue ofthe Fulton–MacPherson compactification of the configuration space C k ( M ) ([AS,Ko1]). If M Γ ( ~ξ J ) does not have boundaries except the endpoints of J for every Γ,then it gives a cobordism between the moduli spaces on the endpoints of J and itfollows that the value of z k does not change between s and s . Here, the value of M Γ ( ~ξ s ), ~ξ s = ( ξ s , ξ , . . . , ξ k ), may change when a vertex of Γ intersects Σ, butthe difference is killed by the Tr as in Lemma 3.2 or by the Holonomy relation andthe trace is invariant.In general, M Γ ( ~ξ J ) may have boundaries on the interior of J . It follows fromresults in [Wa1, §
8] that the boundary of M Γ ( ~ξ J ) consists of degenerate flow-graphsas follows (see Figure 4).(1) A subgraph (or the whole) of Γ collapses into a point of M .(2) An edge of Γ, either compact or separated, splits in the middle by a criticalpoint.(3) The black vertex on a closed separated edge coincides with a critical point.Among the degenerations of type (1), if the whole of a graph collapses, then z k may change. However, ˆ z k does not change because the change is cancelled by thechange of the correction term, as shown in [Wa1, § §
5, 6] for detail about this paragraph.The degenerations of type (3) do not change e z k by (4.1) and Lemma 4.1.The degenerations of type (2) can be treated by the same argument as in [Wa1, § Z is replaced with Λ and thatthe coefficients of graphs belong to Λ ⊗ k . Namely, around a parameter s ∈ J ofthe inner boundary of M Γ ( ~ξ J ) where there are no j/j -intersections, the difference ˆ z k ( ~ξ s + ε ) − ˆ z k ( ~ξ s − ε ) is given by Tr ~g of the terms of degenerate flow-graphs oftype (2) and it is equal to Tr ~g of − k X i =1 X Γ ′ (˜ pi, ˜ qi ) i ind(˜ pi )=ind(˜ qi ) W Γ ′ (˜ p i ˜ q i ) i · (cid:16) X xi ∈ P ( i ) ∗ ind( xi )=ind(˜ pi )+1 ∂ ( i ) x i ˜ p i · Γ ′ ( x i , ˜ q i ) i + X yi ∈ P ( i ) ∗ ind( yi )=ind(˜ qi ) − ∂ ( i )˜ q i y i · Γ ′ (˜ p i , y i ) i + δ ˜ p i ˜ q i Γ ′ ( ∅ , ∅ ) i (cid:17) , where the second sum is taken over uncolored graphs of the form Γ ′ (˜ p i , ˜ q i ) i of degree( η , η , . . . , η k ) with η ℓ = 1 for ℓ = i and η i = 0, and W Γ ′ (˜ p i ˜ q i ) i = ( − ind(˜ p i ) M Γ ′ (˜ p i , ˜ q i ) i ( ~ξ J ) ∈ Λ ⊗ k . We consider ∂ ( i ) x i ˜ p i etc. as an element of Λ ⊗ k by identifying Λ with 1 ⊗ ( i − ⊗ Λ ⊗ ⊗ (3 k − i ) . For each fixed pair ˜ p i , ˜ q i ∈ P ( i ) ∗ with ind(˜ p i ) = ind(˜ q i ), we haveTr ~g (cid:16) W Γ ′ (˜ p i ˜ q i ) i · (cid:16) X xi ∈ P ( i ) ∗ ind( xi )=ind(˜ pi )+1 ∂ ( i ) x i ˜ p i · Γ ′ ( x i , ˜ q i ) i + X yi ∈ P ( i ) ∗ ind( yi )=ind(˜ qi ) − ∂ ( i )˜ q i y i · Γ ′ (˜ p i , y i ) i + δ ˜ p i ˜ q i Γ ′ ( ∅ , ∅ ) i (cid:17)(cid:17) = Tr ...,∂ ( i ) g ( i ) + g ( i ) ∂ ( i ) ,... (cid:16) W Γ ′ (˜ p i ˜ q i ) i · Γ ′ (˜ p i , ˜ q i ) i (cid:17) + Tr ~g (cid:16) W Γ ′ (˜ p i ˜ q i ) i · δ ˜ p i ˜ q i Γ ′ ( ∅ , ∅ ) i (cid:17) = Tr ..., id ,... (cid:16) W Γ ′ (˜ p i ˜ q i ) i · Γ ′ (˜ p i , ˜ q i ) i (cid:17) + Tr ~g (cid:16) W Γ ′ (˜ p i ˜ q i ) i · δ ˜ p i ˜ q i Γ ′ ( ∅ , ∅ ) i (cid:17) = 0 . Around a parameter s ∈ J of the inner boundary of M Γ ( ~ξ J ) at a j/j -intersection,the difference ˆ z k ( ~ξ s + ε ) − ˆ z k ( ~ξ s − ε ) is decomposed into two parts, as follows. Let g, g ′ be the combinatorial propagators for C (1) at s − ε and s + ε respectively andlet ~g = ( g, g (2) , . . . , g (3 k ) ) , ~g ′ = ( g ′ , g (2) , . . . , g (3 k ) ). Then we have X Γ Tr ~g ′ (cid:0) M Γ ( ~ξ s + ε ) · Γ (cid:1) − X Γ Tr ~g (cid:0) M Γ ( ~ξ s − ε ) · Γ (cid:1) = X Γ Tr ~g ′ (cid:16)(cid:0) M Γ ( ~ξ s − ε ) − M d ′′ Γ ( ~ξ J ) (cid:1) · Γ (cid:17) − X Γ Tr ~g (cid:0) M Γ ( ~ξ s − ε ) · Γ (cid:1) = X Γ Tr g ′ − g,... (cid:0) M Γ ( ~ξ s − ε ) · Γ (cid:1) − X Γ Tr ~g ′ (cid:0) M d ′′ Γ ( ~ξ J ) · Γ (cid:1) . Here, M d ′′ Γ ( ~ξ J ) is the sum of the counts of the degenerate flow-graphs includ-ing the j/j -intersection with appropriate signs. The first term in the last linecorresponds to the change of the combinatorial propagator and the other one cor-responds to the count of the degenerate flow-graphs including the j/j -intersection.The change of the combinatorial propagator can be described explicitly as fol-lows. The underlying b Λ-modules ~C do not change between s − ε and s + ε whilethe boundary operator ∂ (1) may change at s and the combinatorial propagator g changes accordingly. For an endomorphism 1 + h ∈ End b Λ ( C (1) ) corresponding toan elementary matrix, where h counts the j/j -intersection with holonomy, g ′ can GENERALIZATION OF FUKAYA’S INVARIANT OF 3-MANIFOLDS II 11 be given as g ′ = (1 + h ) ◦ g ◦ (1 − h ), which gives g ′ − g = hg ′ − g ′ h . The traceof a graph by hg ′ − g ′ h is cancelled by the part of the counts of the degenerateflow-graphs including the j/j -intersection. See [Wa1, §
10] for detail.When s crosses a parameter of a birth-death singularity, a separated edge will beglued together into a compact edge, or its reverse. The explicit form of the gluingis exactly the same as [Wa1, § (cid:3) Acknowledgments.
I would like to thank Tatsuro Shimizu for explaining to me his works. This workis supported by JSPS Grant-in-Aid for Scientific Research 26800041 and 26400089.
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