A note on the Θ-invariant of 3-manifolds

A note on the Θ-invariant of 3-manifolds arxiv:1903.04386v1 [math.gt] 11 Mar 2019 ALBERTO S. CATTANEO and TATSURO SHIMIZU Abstract In this note, we revisit the Θ-invariant as defined by R. Bott and the first author in [4]. The Θ-invariant is an invariant of rational homology 3-spheres with acyclic orthogonal local systems, which is a generalization of the 2-loop term of the Chern-Simons perturbation theory. The Θ- invariant can be defined when a cohomology group is vanishing. In this note, we give a slightly modified version of the Θ-invariant that we can define even if the cohomology group is not vanishing. 1 Introduction In 1998, R. Bott and the first author defined topological invariants of rational homology spheres with acyclic orthogonal local systems in [3], [4]. These invariant were inspired by the Chern-Simons perturbation theory developed by M. Kontsevich in [6], S. Axelrod and M. I. Singer in [2]. The Chern-Simons perturbation theory gives invariants of 3-manifolds with flat connections of the trivial G-bundle over the 3-manifold, where G is a semi-simple Lie group. The composition of adjoint representation of G and the holonomy representation of the flat connection gives an orthogonal local system. In [4], Bott and the first author constructed a real valued invariant, called Θ-invarant (In this note, we denote by Z Θ the corresponding term), which is a generalization of a 2-loop term of Chern-Simons perturbation theory. The vanishing of a cohomology group (denoted by H ( ;π 1 1 E π 1 2 E) in [4], H ( ;E ρ E ρ ) in this note) plays an important role in the construction of the Θ-invariant Z Θ. There are few gaps in the proof of this vanishing (Lemma 1.2 of [4]). In this note, we show that a linear combination of Z Θ and another term Z O O is, however, a topological invariant of closed 3-manifolds with orthogonal acyclic local systems, when the local system is given by using a holonomy representation of a flat connection. The term Z O O is also related to the 2-loop term of the Chern-Simons perturbation theory. We note that the second author proved that when G SU(2), Z Θ itself is an invariant of closed 3 manifolds with orthogonal local systems in [9]. The organization of this paper is as follows. In Section 2 we give a modified version of the Bott-Cattaneo Θ-invariant without proof. In Section 3 and University of Zurich, email: cattaneo@math.uzh.ch Research Institute for Mathematical Sciences, Kyoto University, email: shimizu@kurims.kyoto-u.ac.jp 1

Section 4 we prove a proposition and a theorem about well-definedness of the invariant stated in Section 2. Both the invariant defined in Section 2 of this note and the Θ-invariant depend on the choice of a framing of the 3-manifold. In Section 5 we introduce a framing correction. Orientation convention In this note, all manifolds are oriented. Boundaries are oriented by the outward normal first convention. Products of oriented manifolds are oriented by the order of the factors. The interval [0,1] R is oriented from 0 to 1. Acknowledgments A. S. C. acknowledges partial support of SNF Grant No. 200020 172498/1. This research was (partly) supported by the NCCR SwissMAP, funded by the Swiss National Science Foundation, and by the COST Action MP1405 QSPACE, supported by COST (European Cooperation in Science and Technology). T. S. expresses his appreciation to Professor Tadayuki Watanabe for his helpful comments and discussion on the Chern-Simons perturbation theory. This work was (partly) supported by JSPS KAKENHI Grant Number JP15K13437. 2 The invariant Let M be a closed oriented framed 3-manifold, namely a trivialization of the tangent bundle of M is fixed. We take a metric on M compatible with the framing. Let ρ : π 1 G be a representation of the fundamental group into a semi-simple Lie group G. We denote by ad : G g the adjoint representation of G, where g is the Lie algebra of G. Since G is semi-simple, the representation ad ρ is orthonormal with respect to the Killing form. A local system is a covariant functor from the fundamental groupoid of M to the category of finite dimensional vector spaces. Note that a representation of π 1 (M) gives a local system. We denote by E ρ the local system given by ad ρ. We assume that E ρ is acyclic, namely H (M;E ρ ) 0. In this note, we say that such a representation ρ is acyclic. 2.1 A compactification of a configuration space Let {(x,x) x M} M 2 be the diagonal. We identify with M by (x,x) x M. We orient by using this identification. We denote by ν the normal bundle of in M 2. We identify ν with the tangent bundle TM via the isomorphism defined by TM ν,(x,v) ((x,x),( v,v)) 2

where x M and v T x M. On the other hand, M is framed. Then TM is identified with M R 3. Thus ν is identified with M R 3. Let C 2 (M) Bl(M 2, ) be the compact 6-dimensional manifold with the boundary obtained by the real blowing up of M 2 along. We denote by the blow-down map. As manifolds, q : C 2 (M) M 2 C 2 (M) (M 2 \ ) Sν and q(sν ). Here Sν is the unit sphere bundle of ν with respect to the metric on M. C 2 (M) is a compactification of the configuration space M 2 \ of two distinct points. Obviously, C 2 (M) Sν. Sν is identified with S 2. We denote by p : C 2 (M) S 2 S 2 the projection. We use the same symbol q for the restriction map q C2(M) : C 2 (M)( S 2 ) of the blow-down map q. 2.2 The natural transformations c and Tr The killing form gives an isomorphism g g g g. Let 1 g g the element corresponding to the killing form in g g. By using an orthonormal basis e 1,...,e dimg g of g, 1 can be described as 1 dimg i1 e i e i. 1 g g is invariant under the diagonal action of π 1 (M). Thus we have a natural transformation c : R E ρ E ρ,1 1. Here R is the trivial local system, namely a local system corresponding to the 1-dimensional trivial representation of π 1 (M). We define a natural transformation as follows: for x,y,z g, Tr : E ρ E ρ E ρ R Tr(x y z) [x,y],z where, is the Killing form and [,] is the Lie bracket. Letπ 1,π 2 : M 2 M betheprojectionsdefinedbyπ 1 (x 1,x 2 ) x 1,π 2 (x 1,x 2 ) x 2. π 1 E ρ π 2 E ρ is a local system on M 2. We denote E ρ E ρ π 1 E ρ π 2 E ρ. We remark that E ρ E ρ E ρ E ρ. The pull-back F ρ q (E ρ E ρ ) is a local system on C 2 (M). Clearly, F ρ C2(M) q (E ρ E ρ ). 3

2.3 The involution T on C 2 (M) The involution T 0 : M 2 M 2 defined by T 0 (x 1,x 2 ) (x 2,x 1 ) induces an involution T : C 2 (M) C 2 (M). T 0,T induce homomorphisms T 0,T on the cohomology groups H (M 2,E ρ E ρ ), H (C 2 (M);F ρ ), H ( ;E ρ E ρ ) and on thespaceofdifferentialk formsω k (C 2 (M);F ρ ). WedenotebyH + (M2 ;E ρ E ρ ), H (M 2 ;E ρ E ρ ) the +1, 1eigenspaces ofthe homomorphism T 0 respectively. We use similar notations H + (C 2(M);F ρ ),H + (,E ρ E ρ ),Ω k + (C 2(M);F ρ )... in the same manner. Let T S 2 : S 2 S 2 be the involution defined as T S 2(x) x for any x S 2. Since the metric on M is compatible with the framing, we have p T C2(M) T S 2 p : C 2 (M) S 2. 2.4 The invariant Take a 2-form ω S 2 Ω 2 (S 2 ;R) on S 2 satisfying S 2 ω S 2 1 and T S 2 ω S 2 ω S 2. p ω S 2 is a closed 2-form on C 2 (M). Thus c (p ω S 2) p ω S 21 is a closed 2-form on C 2 (M) such that (T C2(M)) p ω S 21 p ω S 21. Thereforetheclosed2-formp ω S 21representsacohomologyclassinH 2 ( C 2(M);F ρ C2(M)): [p ω S 21] H 2 ( C 2 (M);F ρ C2(M)). Proposition 2.1. There exist 2 forms ω Ω 2 (C 2 (M);F ρ ) and ξ Ω 2 ( ;E ρ E ρ ) satisfying the following conditions: (1) ω C2(M) p ω S 21+q ξ, (2) T ω ω,(t 0 ) ξ ξ, namely ω Ω 2 (C 2(M);F ρ ),ξ Ω 2 ( ;E ρ E ρ ). Furthermore, the cohomology class [ξ] H 2 ( ;E ρ E ρ ) is independent of the choice of ξ. This proposition is proved in Section 3. Now, we have the following 2-forms: Then we obtain closed 6-forms q π 1ξ Ω 2 (C 2 (M);q (E 2 ρ R)), q π 2ξ Ω 2 (C 2 (M);q (R E 2 ρ )). ω 3 Ω 6 (C 2 (M);F 3 ρ ), (q π 1ξ)(q π 2ξ)ω Ω 6 (C 2 (M);F 3 ρ ). 4

Since Fρ 3 q (E 3 a natural transformation ρ Eρ 3 ), the natural transformation Tr : E 3 ρ R induces Tr 2 : F 3 ρ Therefore we get closed 6-forms Definition 2.2. Z Θ (ω) (R R )R. Tr 2 ω 3,Tr 2 ((q π 1 ξ)(q π 2 ξ)ω) Ω6 (C 2 (M);R). C 2(M) Tr 2 ω 3,Z O O (ω,ξ) C 2(M) Z 1 (M,ρ) Z Θ (ω) 3Z O O (ω,ξ). Tr 2 ((q π 1 ξ)(q π 2 ξ)ω), Theorem 2.3. Z 1 (M,ρ) is an invariant of M, ρ (independent of the choices of ω,ξ). Furthermore, Z 1 (M,ρ) is invariant under homotopy of the framing. This theorem is proved in Section 4. Remark 2.4. When we can take ξ 0, obviously Z O O (ω,ξ) 0 and then Z 1 (M,ρ) coincides with the Θ-invariant I (Θ,tr,tr) (M) of the framed 3-manifold M given in Theorem 2.5 in [4]. 3 Proof of Proposition2.1 In the following commutative diagram, the top horizontal line is a part of the long exact sequence of the pair (C 2 (M), C 2 (M)) and the bottom line is that of (M 2, ). Thanks to the excision theorem, the right vertical homomorphism q is an isomorphism. H ( C 2 2 (M);q (E ρ E ρ )) δ C 2 (M) H (C 3 2 (M), C 2 (M);F ρ ) (q C2 (M)) q H 2 ( ;E ρ E ρ ) δ M 2 H 3 (M 2, ;E ρ E ρ ) Since H 2 (M2 ;E ρ E ρ ) H 3 (M2 ;E ρ E ρ ) 0, the connected homomorphism δ M 2 on the bottom line is an isomorphism. Set Φ (δ M 2) 1 (q ) 1 δ C 2(M) : H2 ( C 2(M);q (E ρ E ρ )) H 2 ( ;E ρ E ρ ). We take a 2-form ξ Ω 2 ( ;E ρ E ρ ) such that Φ([p ω S 21]) [ξ] H 2 ( ;E ρ E ρ ). The above diagram implies that Φ(q [ξ]) [ξ]. Then Φ(p ω S 21 + q ξ) 0. Thus δ C 2(M) (p ω S 21 + q ξ) 0. Therefore there exists a closed 2-form ω Ω 2 (C 2(M);F ρ ) such that ω C2(M) p ω S 21+q ξ. Thesecondassertionisadirectconsequenceofthedefinition [ξ] Φ([p ω S 21]). 5

4 Proof of Theorem 2.3 The proof is reduced to the following two propositions: Proposition 4.1. Let ω,ω Ω 2 (C 2 (M);F ρ ) be closed 2-forms such that ω C2(M) ω C2(M) p ω S 21+q ξ. Then Z Θ (ω) Z Θ (ω ) and Z O O (ω,ξ) Z O O (ω,ξ) hold. Proposition 4.2. Let ω S 2,0,ω S 2,1 Ω 2 (S 2 ;R) be closed 2-forms satsfying S 2 ω S 2,0 S 2 ω S 2,1 1, T S 2 ω S 2,0 ω S 2,0 and T S 2 ω S 2,1 ω S 2,1. Let {p t : S 2 S 2 } t [0,1] be a homotopy such that p 0 p and p t T C2(M) T S 2 p t for t 0,1. Let ω 0,ω 1 Ω 2 (C 2(M);F ρ ) and ξ 0,ξ 1 Ω 2 ( ;E ρ E ρ ) be closed 2-forms satisfying ω 0 C2(M) p 0 ω S 2,01+q ξ 0,ω 1 C2(M) p 1 ω S 2,11+q ξ 1. Then Z Θ (ω 0 ) 3Z O O (ω 0,ξ 0 ) Z Θ (ω 1 ) 3Z O O (ω 1,ξ 1 ) holds. 4.1 Proof of Proposition4.1 Lemma 4.3. There exists a 1-form η Ω 1 (M 2 ;E ρ E ρ ) such that ω ω d(q η). Proof. In the following diagram, the top horizontal line is a part of the long exact sequence of the pair (C 2 (M), C 2 (M)) and the bottom line is that of (M 2, ). The left vertical homomorphism q is an isomorphism because of the excision theorem. H 2 (C 2 (M), C 2 (M);F ρ ) q H 2 (M2, ;E ρ E ρ ) H 2 (C 2 (M);F ρ ) q H 2 (M2 ;E ρ E ρ ) The closed 2-form ω ω gives a cohomology class in H 2 (C 2(M), C 2 (M);F ρ ) and then ((q ) 1 (ω ω )) M 2 gives a cohomology class in H 2 (M 2 ;E ρ E ρ ). Since H 2 (M 2 ;E ρ E ρ ) 0, there exists a 1-form η Ω 1 (M 2 ;E ρ E ρ ) such that dη ((q ) 1 (ω ω )) M 2. Thus we have d(q η) ω ω. 6

Thanks to Lemma 4.3 and Stokes s theorem, Z Θ (ω) Z Θ (ω ) Tr 2( (ω ω )(ω 2 +ωω +ω 2 ) ) 3 C 2(M) C 2(M) C 2(M) C 2(M) Tr 2( d(q η)(ω 2 +ωω +ω 2 ) ) Tr 2( (q η) C2(M)(ω 2 +ωω +ω 2 ) ) C2(M) Tr 2( (q η) C2(M)(p ω S 21+q ξ) 2) 6 Tr 2 (q (η )p ω S 21q ξ) S 2 6 Tr 2 (η ξ1). To simplify the notation, we set η η. Let l : E ρ E ρ E ρ be a natural transformation induced from the Lie bracket [,] : g g g. We have l(η) Ω 1 ( ;E ρ ), l(ξ) Ω 2 ( ;E ρ ). Let I : E ρ E ρ R be a natural transformation induced from the inner product of g. Then I(l(η)l(ξ)) is a 2-form in Ω 2 ( ;R). Lemma 4.4. Tr 2 (ηξ1) I(l(η)l(ξ)). Proof. Since T 0 id, Ω ( ;E E)) Ω ( ;(E E) ). Then we only need to check the claim on g 3 g 3. Let e 1,...,e dimg g be an orthonormal basis of g. Then {e i e j e j e i i < j} is a basis of (g g). It is enough to show the claim for this basis. ( Tr 2 (e i e j e j e i ) (e k e l e l e k ) ( ) e n e n ) n 2( [e i,e k ],[e j,e l ] [e i,e l ],[e j,e k ] ) 2( e i,[e k,[e j,e l ]] + e i,[e l,[e k,e j ]] ) 2(( e i,[e j,[e l,e k ]] e i,[e l,[e k,e j ]] )+ e i,[e l,[e k,e j ]] ) 2 e i,[e j,[e k,e l ]] 2 [e i,e j ],[e k,e l ] 1 2 2[e i,e j ],2[e k,e l ] 1 2 l(e i e j e j e i )l(e k e l e l e k ). Corollary 4.5. Tr 2 (ηξ1) 0. 7

Proof. Thanks to the above lemma, Tr 2 (ηξ1) I(l(η)l(ξ)). Since E ρ is acyclic, [l(ξ)] 0 H 2 ( ;E ρ ) 0. Thus there exists a 1-form ζ Ω 1 ( ;E ρ ) such that dζ l(ξ). Therefore I(l(η)l(ξ)) I(ηdζ) di(ηζ) 0. Thanks to the above lemma, we have Z Θ (ω) Z Θ (ω ) 0. Similarly, Z O O (ω,ξ) Z O O (ω,ξ) Tr 2 ((q π1 ξ)(q π2 ξ)(ω ω )) C 2(M) C 2(M) C 2(M) Tr 2 ((q π 1ξ)(q π 2ξ)dq η) Tr 2 (q ((π 1 ) ξ(π 2 ) ξη)). Since (π 1 ) ξ(π 2 ) ξη is a 5-form on the 3-dimensional manifold, the last term is vanishing. This completes the proof of Proposition 4.1. 4.2 Proof of Proposition 4.2 Since[ω S 2,0] [ω S 2,1] H 2 (S 2 ;R), thereexistsaclosed2-form ω S2 Ω 2 ([0,1] S 2 ;R) such that ω S 2 {t} S 2 ω S 2,t for t 0,1. Since [ξ 0 ] [ξ 1 ](Proposition 2.1), there exists a closed 1-form ξ Ω 1 ([0,1],π (E ρ E ρ )) such that ξ {0} ξ 0 and ξ {1} ξ 1. Here π : [0,1] is the projection. Let π C2(M) : [0,1] C 2 (M) C 2 (M) be the projection. Let q id [0,1] q : [0,1] C 2 (M) [0,1] M 2 and we also denote the restriction map q [0,1] C2(M) : [0,1] C 2 (M) [0,1] as q. By a similar argument as in Proposition 2.1, we can take a closed 2-form ω Ω 2 ([0,1] C 2 (M),π C 2(M) F ρ) 8

such that ω [0,1] C2(M) p ω S 21+ q ξ. Here p {p t } t : ([0,1] C 2 (M) )[0,1] S 2 S 2 is the homotopy between p 0 and p 1. Thanks to Proposition 4.1, both Z Θ (ω) and Z O O (ω,ξ) depend only on ω S 2 andξ. ThuswehaveZ Θ (ω 0 ) Z Θ ( ω {0} C2(M)),Z Θ (ω 1 ) Z Θ ( ω {1} C2(M)), Z O O (ω 0,ξ 0 ) Z O O ( ω {0} C2(M),ξ 0 )andz O O (ω 1,ξ 1 ) Z O O ( ω {1} C2(M),ξ 1 ). We note that, with our orientation convention, ([0,1] C 2 (M)) {1} C 2 (M) {0} C 2 (M) [0,1] C 2 (M). Therefore, by using Stokes theorem, 0 dtr 2 ω 3 [0,1] C 2(M) {1} C 2(M) Tr 2 ( ω 3 {1} C 2(M) ) Tr 2 ( ω 3 ) {0} C 2(M) Z Θ ( ω {1} C2(M)) Z Θ ( ω {0} C2(M)) Tr 2 ( ω 3 {0} C 2(M) ) Z Θ (ω 1 ) Z Θ (ω 0 ) Tr 2 (3 p ω S 21 q ξ2 ) Tr 2 ( p ω S 21+ q ξ) 3 We denote π i id [0,1] π i : [0,1] M 2 [0,1] M for i 1,2. We have, 0 dtr 2( ( q π ) 1 ξ)( q π 2 ξ) ω [0,1] C 2(M) Z O O (ω 1,ξ 1 ) Z O O (ω 0,ξ 0 ) Tr 2( ( q (( π 1 [0,1] ) ξ( π 2 [0,1] ) ξ) ω [0,1] C2(M)). Here, π 1 [0,1] π 2 [0,1] : [0,1] M. Thus( π 1 [0,1] ) ξ( π2 [0,1] ) ξ ξ 2 under the identification M. We have Z O O (ω 1,ξ 1 ) Z O O (ω 0,ξ 0 ) Tr 2 ( p ω S 21 q ξ2 ) Then we have Z Θ (ω 1 ) Z Θ (ω 0 ) 3(Z O O (ω 1,ξ 1 ) Z O O (ω 0,ξ 0 )). This completes the proof of Proposition 4.2. 9

5 A framing correction In this section, we introduce a correction term for framings to give an invariant of closed 3-manifolds with acyclic representations. Let M be a closed oriented 3-manifold (without framings). Recall that C 2 (M) is identified with the unit sphere bundle STM (see Section 2.1). Take a framing f : TM M R 3 of M. Then (M,f) is a framed 3-manifold. Let p : ( C 2 (M) )STM S 2 be the projection defined by the framing f. Let δ(f) Z be the signature defect (or Hirzebruch defect. For example, see [1], [5] for the details) of a framing f. For the convenience of the reader, we give a short review of the construction of δ(f) in the next section. Let ρ : π 1 (M) G be an acyclic representation as in Section 2.1. Theorem 5.1. is an topological invariant of M,ρ. Z 1 ((M,f),ρ) (dimg) 2 δ(f) 5.1 The signature defect δ(p) Let W be a compact 4-manifold such that W M and its Euler characteristic is zero. Then there exists an R 3 sub-bundle T v W of TW satisfying T v W M TM. Let ST v W W be the unit sphere bundle of T v W W. Thus ST v W is a 6-dimensional manifold with ST v W STM. We denote by F W ST v W the tangent bundle along the fiber of the S 2 bundle π : ST v W W. Take a closed 2-form α W Ω 2 (ST v W;R) such that α W STM p ω S 2 and [α W ] e(f W )/2 H 2 (ST v W;R), where e(f W ) is the Euler class of F W ST v W. Lemma 5.2. When W M, we have ST v W α3 W 3 4SignW. Here SignW is the signature of W. Proof. We give an outline of the proof. See Appendix of [8] or Proposition 2.45 of [7], for the details of the proof. Since W is closed, ST v W α3 W ( 1 ST v W 2 e(f W) ) 3. We denote by p1 (F W ) H 4 (ST v W;R) the 1st Pontrjagin class of the bundle F W. We remark that R F W π T v W and R T v W TW. Here R is the trivial R bundle over an appropriate manifold. Therefore, α 3 W 1 e(f W ) 3 ST v W 8 ST v W 1 e(f W )p 1 (F W ) 8 ST v W 1 e(f W )π p 1 (T v W) 8 ST v W 1 p 1 (TW) 4 W 3 4 SignW. 10

Thanks to the Novikov additivity for the signature, the following corollary holds. Corollary 5.3. The signature defect δ(f) defined to be δ(f) ST W α 3 W 3 4 SignW is independent of the choices of W and α W. 5.2 Proof of Theorem 5.1 Let f 0,f 1 : TM M R 3 be framings and let p 0,p 1 : C 2 (M) S 2 be the projections given by framings f 0,f 1 respectively. Since [p 0ω S 2] and [p 1ω S 2] are in H 2( S2 ;R) H 2 (S 2 ;R) R, [p 0 ω S 2] [p 1 ω S2]. Thus there exists a closed 2-form ω Ω 2 ([0,1] C 2 (M);R) such that ω {0} C2(M) p 0ω S 2, ω {1} C2(M) p 1ω S 2. We recall that ξ Ω 2 ( ;E ρ E ρ ) is a closed 2-form representing Φ([p ω S 21]) Φ c ([p ω S 2]) when we take a projection p : C 2 (M) S 2 given by a framing f. The homomorphism Φ c is independent from the choice of a framing. Then we can use same ξ Ω 2 ( ;E ρ E ρ ) for any framing. By a similar argument as in proof of Proposition2.1, we can take a closed 2-form ω Ω 2 ([0,1] C 2 (M);π C 2(M) F ρ) such that ω [0,1] C2(M) ω 1+Q ξ. Here, π C2(M) : [0,1] C 2 (M) C 2 (M) and Q : [0,1] C 2 (M) are the projections. We denote by Then, Thanks to Stokes theorem, 0 dtr 2 ( ω 3 ) [0,1] C 2(M) Z Θ (ω 1 ) Z Θ (ω 0 ) Z Θ (ω 1 ) Z Θ (ω 0 ) Z Θ (ω 1 ) Z Θ (ω 0 ) ω 0 ω {0} C2(M), ω 1 ω {1} C2(M). Z 1 ((M,f 0 ),ρ) Z Θ (ω 0 ) 3Z O O (ω 0,ξ), Z 1 ((M,f 1 ),ρ) Z Θ (ω 1 ) 3Z O O (ω 1,ξ). Z Θ (ω 1 ) Z Θ (ω 0 ) (dimg) 2 Tr 2 ( ω 3 ) Tr 2 ( ω 1 3 3 ) ω 3 Tr 2 (1 3 ) 11 3Tr 2 ( ω 2 1 2 Q ξ) 3Tr 2 ( ω 2 1 2 Q ξ) ω 3 3Tr 2 ( ω 1 2 2 Q ξ).

We denote π i : [0,1] M 2 M,(t,x 1,x 2 ) x i for i 1,2. We have, 0 dtr 2 (( q π 1 ξ)( q π 2 ξ) ω) [0,1] C 2(M) Z O O (ω 1,ξ) Z O O (ω 0,ξ) Tr 2( ( q ((π 1 [0,1] ) ξ(π 2 [0,1] ) ξ) ω 1 ) Z O O (ω 1,ξ) Z O O (ω 0,ξ) Tr 2( Q ξ 2 ω 1 ) 0 Thus we have Z 1 ((M,f 0 ),ρ) Z 1 ((M,f 1 ),ρ) (dimg) 2 ω 3 + 3Tr 2 ( ω 2 1 2 Q ξ). Lemma 5.4. Tr 2 ( ω 1 2 Q ξ) 0. Proof. Let T E : E ρ E ρ E ρ E ρ be the involution induced by g g g g,x y y x. Cleary, Tr 2 T 3 E Tr 2 : E 3 E 3 R. Since T E (1) 1 and TE (T 0 ) on Ω 1 ( ;E ρ E ρ ), we have Tr 2 ( ω 1 2 Q ξ) Tr 2( T 3 E ( ω 1 2 Q ξ) ) Tr 2 ( ω 1 2 Q ξ). Thus Tr 2 ( ω 1 2 Q ξ) 0. Lemma 5.5. δ(f 1 ) δ(f 0 ) Proof. We takeacompact 4-manifold W with W M and its Eulercharacteristic is zero. Take a collar neighborhood [0,1] M of M W in W such that {1} M W. Set W 0 W\([0,1] M). We can taket v W as T v W [0,1] M [0,1] TM. ThusST v W [0,1] M isidentified with[0,1] C 2 (M). Takeaclosed 2-form α W Ω 2 (ST v W;R) satisfying α W [0,1] STM ω and [α W ] 1 2 e(f W). Then we have ( δ(f 1 ) δ(f 0 ) α 3 W 3 ( ) ST v W 4 SignW (α W STv W 0 ) 3 3 ) ST v W 0 4 SignW 0 (α W [0,1] STM ) 3 [0,1] ST M ω 3. ω 3. 12

By the above two lemmas, Z 1 ((M,f 0 ),ρ) (dimg) 2 δ(f 0 ) Z 1 ((M,f 1 ),ρ) (dimg) 2 δ(f 1 ). Namely, Z 1 ((M,f),ρ) (dimg) 2 δ(f) is independent of the choice of a framing f. References [1] M. Atiyah. On framings of 3-manifolds. Topology, 29(1):1 7, 1990. [2] S. Axelrod and I. M. Singer. Chern-Simons perturbation theory. In Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics, Vol. 1, 2 (New York, 1991), pages 3 45. World Sci. Publ., River Edge, NJ, 1992. [3] R. Bott and A. S. Cattaneo. Integral invariants of 3-manifolds. J. Differential Geom., 48(1):91 133, 1998. [4] R. Bott and A. S. Cattaneo. Integral invariants of 3-manifolds. II. J. Differential Geom., 53(1):1 13, 1999. [5] R. Kirby and P. Melvin. Canonical framings for 3-manifolds. In Proceedings of 6th Gökova Geometry-Topology Conference, volume 23, pages 89 115, 1999. [6] M. Kontsevich. Feynman diagrams and low-dimensional topology. In First European Congress of Mathematics, Vol. II (Paris, 1992), volume 120 of Progr. Math., pages 97 121. Birkhäuser, Basel, 1994. [7] C. Lescop. On the Kontsevich-Kuperberg-Thurston construction of a configuration-space invariant for rational homology 3-spheres. arxiv:math/0411088v1, November 2004. [8] T. Shimizu. An invariant of rational homology 3-spheres via vector fields. Algebr. Geom. Topol., 16(6):3073 3101, 2016. [9] T. Shimizu. Morse homotopy for the su(2)-chern-simons perturbation theory. RIMS preprint 1857, 2016. 13