A note on the $\Theta$-invariant of 3-manifolds

In this note, we revisit the $\Theta$-invariant as defined by R. Bott and the first author. The $\Theta$-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 $\Theta$-invariant can be defined when a cohomology group is vanishing. In this note, we give a slightly modified version of the $\Theta$-invariant that we can define even if the cohomology group is not vanishing.


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 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.

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.

A compactification of a configuration space
We orient ∆ by using this identification. We denote by ν ∆ the normal bundle of ∆ in M 2 . We identify ν ∆ with the tangent bundle T M via the isomorphism defined by 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 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 dim g ∈ g of g, 1 can be described as 1 ∈ g ⊗ g is invariant under the diagonal action of π 1 (M ). Thus we have a natural transformation c : 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, where , is the Killing form and [, ] is the Lie bracket. Let π 1 , π 2 : M 2 → M be the projections defined by π 1 (x 1 ,

The involution T on C 2 (M)
The involution T 0 :

The invariant
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: This proposition is proved in Section 3. Now, we have the following 2-forms: ). Then we obtain closed 6-forms Therefore we get closed 6-forms This theorem is proved in Section 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.

Proof of Theorem 2.3
The proof is reduced to the following two propositions: Let ω S 2 ,0 , ω S 2 ,1 ∈ Ω 2 (S 2 ; R) be closed 2-forms satsfying 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.

Proof of Proposition 4.2
We note that, with our orientation convention, Therefore, by using Stokes' theorem, We denote π i = id Here, Then we have This completes the proof of Proposition 4.2.

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 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.
is an topological invariant of M, ρ.

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 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, Here R is the trivial R bundle over an appropriate manifold. Therefore, SignW.
Thanks to the Novikov additivity for the signature, the following corollary holds.
SignW is independent of the choices of W and α W .

Proof of Theorem 5.1
Let f 0 , f 1 : T M → 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 * 0 ω S 2 ] and [p * ) 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 Then, Thanks to Stokes' theorem, 3Tr ⊠2 ( ω 2 ∂ 1 ⊗2 Q * ξ).