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

TL;DR

This paper revisits the $ heta$-invariant of rational homology 3-spheres, providing a modified definition that applies even when certain cohomology groups do not vanish, extending its applicability.

Contribution

It introduces a modified $ heta$-invariant definition that works without the vanishing cohomology condition, broadening the invariant's scope.

Findings

Extended the $ heta$-invariant to non-vanishing cohomology cases

Provided a modified construction of the invariant

Maintained the invariant's relation to Chern-Simons theory

Abstract

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.

Equations

$$H^{\ast }(M;E_{\rho })=0.$$

$$\Delta \ni (x,x)\to x\in M.$$

$$TM\stackrel{\cong }{\to }{\nu }_{\Delta },(x,v)\mapsto ((x,x),(-v,v))$$

$$q:C_2(M)\to M^2$$

$$C_2(M)=(M^2\setminus \Delta )\cup S{\nu }_{\Delta }$$

$$p:\partial C_2(M)=\Delta \times S^2\to S^2$$

$$1=\sum _{i=1}^{\dim \mathfrak{g}}{e}_i\otimes e_i.$$

$$c:\underline{\mathbb{R}}\to E_{\rho }\otimes E_{\rho },1\mapsto 1.$$

$$\mathrm{Tr}:E_{\rho }\otimes E_{\rho }\otimes E_{\rho }\to \underline{\mathbb{R}}$$

$$\mathrm{Tr}(x\otimes y\otimes z)=⟨[x,y],z⟩$$

$$F_{\rho }=q^{\ast }(E_{\rho }⊠E_{\rho })$$

$$c_{\ast }(p^{\ast }{\omega }_{S^2})=p^{\ast }{\omega }_{S^2}1$$

$$[p^{\ast }{\omega }_{S^2}1]\in H_{-}^2(\partial C_2(M);F_{\rho }{|}_{\partial C_2(M)}).$$

$$q^{\ast }{\pi }_1^{\ast }\xi \in {\Omega }^2(C_2(M);q^{\ast }(E_{\rho }^{\otimes 2}⊠\underline{\mathbb{R}})),$$

$$q^{\ast }{\pi }_2^{\ast }\xi \in {\Omega }^2(C_2(M);q^{\ast }(\underline{\mathbb{R}}⊠E_{\rho }^{\otimes 2})).$$

$${\omega }^3\in {\Omega }^6(C_2(M);F_{\rho }^{\otimes 3}),$$

$$(q^{\ast }{\pi }_1^{\ast }\xi )(q^{\ast }{\pi }_2^{\ast }\xi )\omega \in {\Omega }^6(C_2(M);F_{\rho }^{\otimes 3}).$$

$${\mathrm{Tr}}^{⊠2}:F_{\rho }^{\otimes 3}\to (\underline{\mathbb{R}}⊠\underline{\mathbb{R}}=)\underline{\mathbb{R}}.$$

$${\mathrm{Tr}}^{⊠2}{\omega }^3,{\mathrm{Tr}}^{⊠2}((q^{\ast }{\pi }_1^{\ast }\xi )(q^{\ast }{\pi }_2^{\ast }\xi )\omega )\in {\Omega }^6(C_2(M);\mathbb{R}).$$

$$Z_{\Theta }(\omega )={\int }_{C_2(M)}{\mathrm{Tr}}^{⊠2}{\omega }^3,Z_{O-O}(\omega ,\xi )={\int }_{C_2(M)}{\mathrm{Tr}}^{⊠2}((q^{\ast }{\pi }_1^{\ast }\xi )(q^{\ast }{\pi }_2^{\ast }\xi )\omega ),$$

$$Z_1(M,\rho )=Z_{\Theta }(\omega )-3Z_{O-O}(\omega ,\xi ).$$

$$H_{-}^2(\partial C_2(M);q^{\ast }(E_{\rho }\otimes E_{\rho }))\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\Phi =({\delta }_{M^2}^{\ast }{)}^{-1}\circ (q^{\ast }{)}^{-1}\circ {\delta }_{C_2(M)}^{\ast }:H_{-}^2(\partial C_2(M);q^{\ast }(E_{\rho }\otimes E_{\rho }))\to H_{-}^2(\Delta ;E_{\rho }\otimes E_{\rho }).$$

$$\Phi ([p^{\ast }{\omega }_{S^2}1])=-[\xi ]\in H_{-}^2(\Delta ;E_{\rho }\otimes E_{\rho }).$$

$$\omega {|}_{\partial C_2(M)}=p^{\ast }{\omega }_{S^2}1+q^{\ast }\xi .$$

$$\omega {|}_{\partial C_2(M)}={\omega }^{\prime }{|}_{\partial C_2(M)}=p^{\ast }{\omega }_{S^2}1+q^{\ast }\xi .$$

$${\omega }_0{|}_{\partial C_2(M)}=p_0^{\ast }{\omega }_{S^2,0}1+q^{\ast }{\xi }_0,{\omega }_1{|}_{\partial C_2(M)}=p_1^{\ast }{\omega }_{S^2,1}1+q^{\ast }{\xi }_1.$$

$$H_{-}^2(C_2(M),\partial C_2(M);F_{\rho })\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$d\eta =((q^{\ast }{)}^{-1}(\omega -{\omega }^{\prime })){|}_{M^2}.$$

$$Z_{\Theta }(\omega )-Z_{\Theta }({\omega }^{\prime })$$

Full text

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 $\Theta$-invarant (In this note, we denote by $Z_{\Theta}$ 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 $H^{*}_{-}(\Delta;\pi_{1}^{-1}E\otimes\pi_{2}^{-1}E)$ in [4], $H^{*}_{-}(\Delta;E_{\rho}\boxtimes E_{\rho})$ in this note) plays an important role in the construction of the $\Theta$-invariant $Z_{\Theta}$. 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 $Z_{\Theta}$ 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 $G=SU(2)$, $Z_{\Theta}$ 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 $\Theta$-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 $\Theta$-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 $[0,1]\subset\mathbb{R}$ is oriented from [math] 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 $\rho:\pi_{1}\to G$ be a representation of the fundamental group into a semi-simple Lie group $G$. We denote by ${\rm ad}:G\to{\mathfrak{g}}$ the adjoint representation of $G$, where $\mathfrak{g}$ is the Lie algebra of $G$. Since $G$ is semi-simple, the representation ${\rm ad}\circ\rho$ 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 $\pi_{1}(M)$ gives a local system. We denote by $E_{\rho}$ the local system given by ${\rm ad}\circ\rho$. We assume that $E_{\rho}$ is acyclic, namely

[TABLE]

In this note, we say that such a representation $\rho$ is acyclic.

2.1 A compactification of a configuration space

Let $\Delta=\{(x,x)\mid x\in M\}\subset M^{2}$ be the diagonal. We identify $\Delta$ with $M$ by

[TABLE]

We orient $\Delta$ by using this identification. We denote by $\nu_{\Delta}$ the normal bundle of $\Delta$ in $M^{2}$. We identify $\nu_{\Delta}$ with the tangent bundle $TM$ via the isomorphism defined by

[TABLE]

where $x\in M$ and $v\in T_{x}M$. On the other hand, $M$ is framed. Then $TM$ is identified with $M\times\mathbb{R}^{3}$. Thus $\nu_{\Delta}$ is identified with $M\times\mathbb{R}^{3}$.

Let $C_{2}(M)=B\ell(M^{2},\Delta)$ be the compact 6-dimensional manifold with the boundary obtained by the real blowing up of $M^{2}$ along $\Delta$. We denote by

[TABLE]

the blow-down map. As manifolds,

[TABLE]

and $q(S\nu_{\Delta})=\Delta$. Here $S\nu_{\Delta}$ is the unit sphere bundle of $\nu_{\Delta}$ with respect to the metric on $M$. $C_{2}(M)$ is a compactification of the configuration space $M^{2}\setminus\Delta$ of two distinct points. Obviously, $\partial C_{2}(M)=S\nu_{\Delta}$.

$S\nu_{\Delta}$ is identified with $\Delta\times S^{2}$. We denote by

[TABLE]

the projection. We use the same symbol $q$ for the restriction map $q|_{\partial C_{2}(M)}:\partial C_{2}(M)(=\Delta\times S^{2})\to\Delta$ of the blow-down map $q$.

2.2 The natural transformations $c$ and ${\rm Tr}$

The killing form gives an isomorphism ${\mathfrak{g}}\otimes{\mathfrak{g}}\cong{\mathfrak{g}}^{*}\otimes{\mathfrak{g}}^{*}$. Let $\bf 1\in\mathfrak{g}\otimes\mathfrak{g}$ the element corresponding to the killing form in $\mathfrak{g}^{*}\otimes\mathfrak{g}^{*}$. By using an orthonormal basis $e_{1},\ldots,e_{\dim\mathfrak{g}}\in\mathfrak{g}$ of $\mathfrak{g}$, $\bf 1$ can be described as

[TABLE]

$\bf 1\in\mathfrak{g}\otimes\mathfrak{g}$ is invariant under the diagonal action of $\pi_{1}(M)$. Thus we have a natural transformation

[TABLE]

Here $\underline{\mathbb{R}}$ is the trivial local system, namely a local system corresponding to the 1-dimensional trivial representation of $\pi_{1}(M)$.

We define a natural transformation

[TABLE]

as follows: for $x,y,z\in\mathfrak{g}$,

[TABLE]

where $\langle,\rangle$ is the Killing form and $[,]$ is the Lie bracket.

Let $\pi_{1},\pi_{2}:M^{2}\to M$ be the projections defined by $\pi_{1}(x_{1},x_{2})=x_{1},\pi_{2}(x_{1},x_{2})=x_{2}$. $\pi_{1}^{*}E_{\rho}\otimes\pi_{2}^{*}E_{\rho}$ is a local system on $M^{2}$. We denote $E_{\rho}\boxtimes E_{\rho}=\pi_{1}^{*}E_{\rho}\otimes\pi_{2}^{*}E_{\rho}$. We remark that $E_{\rho}\boxtimes E_{\rho}|_{\Delta}=E_{\rho}\otimes E_{\rho}$. The pull-back

[TABLE]

is a local system on $C_{2}(M)$. Clearly, $F_{\rho}|_{\partial C_{2}(M)}=q^{*}(E_{\rho}\otimes E_{\rho})$.

2.3 The involution $T$ on $C_{2}(M)$

The involution $T_{0}:M^{2}\to M^{2}$ defined by $T_{0}(x_{1},x_{2})=(x_{2},x_{1})$ induces an involution $T:C_{2}(M)\to C_{2}(M)$. $T_{0},T$ induce homomorphisms $T_{0}^{*},T^{*}$ on the cohomology groups $H^{*}(M^{2},E_{\rho}\boxtimes E_{\rho})$, $H^{*}(C_{2}(M);F_{\rho})$, $H^{*}(\Delta;E_{\rho}\otimes E_{\rho})$ and on the space of differential $k$ forms $\Omega^{k}(C_{2}(M);F_{\rho})$. We denote by $H^{*}_{+}(M^{2};E_{\rho}\boxtimes E_{\rho})$, $H^{*}_{-}(M^{2};E_{\rho}\boxtimes E_{\rho})$ the $+1,-1$ eigenspaces of the homomorphism $T^{*}_{0}$ respectively. We use similar notations $H^{*}_{+}(C_{2}(M);F_{\rho}),H^{*}_{+}(\Delta,E_{\rho}\otimes E_{\rho}),\Omega^{k}_{+}(C_{2}(M);F_{\rho})...$ in the same manner.

Let $T_{S^{2}}:S^{2}\to S^{2}$ be the involution defined as $T_{S^{2}}(x)=-x$ for any $x\in S^{2}$. Since the metric on $M$ is compatible with the framing, we have $p\circ T|_{\partial C_{2}(M)}=T_{S^{2}}\circ p:\partial C_{2}(M)\to S^{2}$.

2.4 The invariant

Take a 2-form $\omega_{S^{2}}\in\Omega^{2}(S^{2};\mathbb{R})$ on $S^{2}$ satisfying $\int_{S^{2}}\omega_{S^{2}}=1$ and $T_{S^{2}}^{*}\omega_{S^{2}}=-\omega_{S^{2}}$. $p^{*}\omega_{S^{2}}$ is a closed 2-form on $\partial C_{2}(M)$. Thus

[TABLE]

is a closed 2-form on $\partial C_{2}(M)$ such that $(T|_{C_{2}(M)})^{*}p^{*}\omega_{S^{2}}{\bf 1}=-p^{*}\omega_{S^{2}}{\bf 1}$. Therefore the closed 2-form $p^{*}\omega_{S^{2}}{\bf 1}$ represents a cohomology class in $H^{2}_{-}(\partial C_{2}(M);F_{\rho}|_{\partial C_{2}(M)})$:

[TABLE]

Proposition 2.1**.**

There exist 2 forms $\omega\in\Omega^{2}(C_{2}(M);F_{\rho})$ and $\xi\in\Omega^{2}(\Delta;E_{\rho}\otimes E_{\rho})$ satisfying the following conditions:

• (1)

$\omega|_{\partial C_{2}(M)}=p^{*}\omega_{S^{2}}{\bf 1}+q^{*}\xi$,

• (2)

$T^{*}\omega=-\omega,(T_{0}|_{\Delta})^{*}\xi=-\xi$, namely $\omega\in\Omega^{2}_{-}(C_{2}(M);F_{\rho}),\xi\in\Omega^{2}_{-}(\Delta;E_{\rho}\otimes E_{\rho})$.

Furthermore, the cohomology class $[\xi]\in H^{2}_{-}(\Delta;E_{\rho}\otimes E_{\rho})$ is independent of the choice of $\xi$.

This proposition is proved in Section 3.

Now, we have the following 2-forms:

[TABLE]

[TABLE]

Then we obtain closed 6-forms

[TABLE]

[TABLE]

Since $F_{\rho}^{\otimes 3}=q^{*}(E_{\rho}^{\otimes 3}\boxtimes E_{\rho}^{\otimes 3})$, the natural transformation ${\rm Tr}:E_{\rho}^{\otimes 3}\to\underline{\mathbb{R}}$ induces a natural transformation

[TABLE]

Therefore we get closed 6-forms

[TABLE]

Definition 2.2**.**

[TABLE]

[TABLE]

Theorem 2.3**.**

$Z_{1}(M,\rho)$* is an invariant of $M$, $\rho$ (independent of the choices of $\omega,\xi$). Furthermore, $Z_{1}(M,\rho)$ is invariant under homotopy of the framing.*

This theorem is proved in Section 4.

Remark 2.4**.**

When we can take $\xi=0$, obviously $Z_{O-O}(\omega,\xi)=0$ and then $Z_{1}(M,\rho)$ coincides with the $\Theta$-invariant $I_{(\Theta,{\rm tr},{\rm tr})}(M)$ 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 $(C_{2}(M),\partial C_{2}(M))$ and the bottom line is that of $(M^{2},\Delta)$. Thanks to the excision theorem, the right vertical homomorphism $q^{*}$ is an isomorphism.

[TABLE]

Since $H^{2}_{-}(M^{2};E_{\rho}\otimes E_{\rho})=H^{3}_{-}(M^{2};E_{\rho}\otimes E_{\rho})=0$, the connected homomorphism $\delta^{*}_{M^{2}}$ on the bottom line is an isomorphism. Set

[TABLE]

We take a 2-form $\xi\in\Omega_{-}^{2}(\Delta;E_{\rho}\otimes E_{\rho})$ such that

[TABLE]

The above diagram implies that $\Phi(q^{*}[\xi])=[\xi]$. Then $\Phi(p^{*}\omega_{S^{2}}{\bf 1}+q^{*}\xi)=0$. Thus $\delta_{C_{2}(M)}^{*}(p^{*}\omega_{S^{2}}{\bf 1}+q^{*}\xi)=0$. Therefore there exists a closed 2-form $\omega\in\Omega^{2}_{-}(C_{2}(M);F_{\rho})$ such that

[TABLE]

The second assertion is a direct consequence of the definition $-[\xi]=\Phi([p^{*}\omega_{S^{2}}{\bf 1}])$.

4 Proof of Theorem 2.3

The proof is reduced to the following two propositions:

Proposition 4.1**.**

Let $\omega,\omega^{\prime}\in\Omega_{-}^{2}(C_{2}(M);F_{\rho})$ be closed 2-forms such that

[TABLE]

Then $Z_{\Theta}(\omega)=Z_{\Theta}(\omega^{\prime})$ and $Z_{O-O}(\omega,\xi)=Z_{O-O}(\omega^{\prime},\xi)$ hold.

Proposition 4.2**.**

Let $\omega_{S^{2},0},\omega_{S^{2},1}\in\Omega^{2}(S^{2};\mathbb{R})$ be closed 2-forms satsfying $\int_{S^{2}}\omega_{S^{2},0}=\int_{S^{2}}\omega_{S^{2},1}=1$, $T_{S^{2}}^{*}\omega_{S^{2},0}=-\omega_{S^{2},0}$ and $T_{S^{2}}^{*}\omega_{S^{2},1}=-\omega_{S^{2},1}$. Let $\{p_{t}:\Delta\times S^{2}\to S^{2}\}_{t\in[0,1]}$ be a homotopy such that $p_{0}=p$ and $p_{t}\circ T|_{\partial C_{2}(M)}=T_{S^{2}}\circ p_{t}$ for $t=0,1$. Let $\omega_{0},\omega_{1}\in\Omega_{-}^{2}(C_{2}(M);F_{\rho})$ and $\xi_{0},\xi_{1}\in\Omega^{2}_{-}(\Delta;E_{\rho}\otimes E_{\rho})$ be closed 2-forms satisfying

[TABLE]

Then $Z_{\Theta}(\omega_{0})-3Z_{O-O}(\omega_{0},\xi_{0})=Z_{\Theta}(\omega_{1})-3Z_{O-O}(\omega_{1},\xi_{1})$ holds.

4.1 Proof of Proposition4.1

Lemma 4.3**.**

There exists a 1-form $\eta\in\Omega_{-}^{1}(M^{2};E_{\rho}\boxtimes E_{\rho})$ such that $\omega-\omega^{\prime}=d(q^{*}\eta)$.

Proof.

In the following diagram, the top horizontal line is a part of the long exact sequence of the pair $(C_{2}(M),\partial C_{2}(M))$ and the bottom line is that of $(M^{2},\Delta)$. The left vertical homomorphism $q^{*}$ is an isomorphism because of the excision theorem.

[TABLE]

The closed 2-form $\omega-\omega^{\prime}$ gives a cohomology class in $H^{2}_{-}(C_{2}(M),\partial C_{2}(M);F_{\rho})$ and then $((q^{*})^{-1}(\omega-\omega^{\prime}))|_{M^{2}}$ gives a cohomology class in $H^{2}_{-}(M^{2};E_{\rho}\boxtimes E_{\rho})$. Since $H^{2}_{-}(M^{2};E_{\rho}\boxtimes E_{\rho})=0$, there exists a 1-form $\eta\in\Omega_{-}^{1}(M^{2};E_{\rho}\boxtimes E_{\rho})$ such that

[TABLE]

Thus we have $d(q^{*}\eta)=\omega-\omega^{\prime}$. ∎

Thanks to Lemma 4.3 and Stokes’s theorem,

[TABLE]

To simplify the notation, we set $\overline{\eta}=\eta|_{\Delta}$.

Let $l:E_{\rho}\otimes E_{\rho}\to E_{\rho}$ be a natural transformation induced from the Lie bracket $[,]:\mathfrak{g}\otimes\mathfrak{g}\to\mathfrak{g}$. We have $l(\overline{\eta})\in\Omega^{1}(\Delta;E_{\rho})$, $l(\xi)\in\Omega^{2}(\Delta;E_{\rho})$. Let $I:E_{\rho}\otimes E_{\rho}\to\underline{\mathbb{R}}$ be a natural transformation induced from the inner product of $\mathfrak{g}$. Then $I(l(\overline{\eta})l(\xi))$ is a 2-form in $\Omega^{2}(\Delta;\mathbb{R})$.

Lemma 4.4**.**

${\rm Tr}^{\boxtimes 2}(\overline{\eta}\xi{\bf 1})=I(l(\overline{\eta})l(\xi))$.

Proof.

Since $T_{0}|_{\Delta}={\rm id}$, $\Omega^{*}_{-}(\Delta;E\otimes E))=\Omega^{*}(\Delta;(E\otimes E)_{-})$. Then we only need to check the claim on $\mathfrak{g}^{\otimes 3}\otimes\mathfrak{g}^{\otimes 3}$. Let $e_{1},\ldots,e_{\dim\mathfrak{g}}\in\mathfrak{g}$ be an orthonormal basis of $\mathfrak{g}$. Then $\{e_{i}\otimes e_{j}-e_{j}\otimes e_{i}\mid i<j\}$ is a basis of $(\mathfrak{g}\otimes\mathfrak{g})^{-}$. It is enough to show the claim for this basis.

[TABLE]

∎

Corollary 4.5**.**

$\int_{\Delta}{\rm Tr}^{\boxtimes 2}(\overline{\eta}\xi{\bf 1})=0$.

Proof.

Thanks to the above lemma,

[TABLE]

Since $E_{\rho}$ is acyclic, $[l(\xi)]=0\in H^{2}(\Delta;E_{\rho})=0$. Thus there exists a 1-form $\zeta\in\Omega^{1}(\Delta;E_{\rho})$ such that $d\zeta=l(\xi)$. Therefore

[TABLE]

∎

Thanks to the above lemma, we have

[TABLE]

Similarly,

[TABLE]

Since $(\pi_{1}|_{\Delta})^{*}\xi(\pi_{2}|_{\Delta})^{*}\xi\overline{\eta}$ is a 5-form on the 3-dimensional manifold $\Delta$, the last term is vanishing. This completes the proof of Proposition 4.1.

4.2 Proof of Proposition 4.2

Since $[\omega_{S^{2},0}]=[\omega_{S^{2},1}]\in H^{2}(S^{2};\mathbb{R})$, there exists a closed 2-form $\widetilde{\omega_{S_{2}}}\in\Omega^{2}([0,1]\times S^{2};\mathbb{R})$ such that $\widetilde{\omega_{S^{2}}}|_{\{t\}\times S^{2}}=\omega_{S^{2},t}$ for $t=0,1$.

Since $[\xi_{0}]=[\xi_{1}]$(Proposition 2.1), there exists a closed 1-form

[TABLE]

such that $\widetilde{\xi}|_{\{0\}\times\Delta}=\xi_{0}$ and $\widetilde{\xi}|_{\{1\}\times\Delta}=\xi_{1}$. Here $\pi_{\Delta}:[0,1]\times\Delta\to\Delta$ is the projection.

Let $\pi_{C_{2}(M)}:[0,1]\times C_{2}(M)\to C_{2}(M)$ be the projection. Let $\widetilde{q}={\rm id}_{[0,1]}\times q:[0,1]\times C_{2}(M)\to[0,1]\times M^{2}$ and we also denote the restriction map $\widetilde{q}|_{[0,1]\times\partial C_{2}(M)}:[0,1]\times\partial C_{2}(M)\to[0,1]\times\Delta$ as $\widetilde{q}$. By a similar argument as in Proposition 2.1, we can take a closed 2-form

[TABLE]

such that

[TABLE]

Here $\widetilde{p}=\{p_{t}\}_{t}:([0,1]\times\partial C_{2}(M)=)[0,1]\times\Delta\times S^{2}\to S^{2}$ is the homotopy between $p_{0}$ and $p_{1}$.

Thanks to Proposition 4.1, both $Z_{\Theta}(\omega)$ and $Z_{O-O}(\omega,\xi)$ depend only on $\omega|_{\Delta\times S^{2}}$ and $\xi$. Thus we have $Z_{\Theta}(\omega_{0})=Z_{\Theta}(\widetilde{\omega}|_{\{0\}\times C_{2}(M)})$, $Z_{\Theta}(\omega_{1})=Z_{\Theta}(\widetilde{\omega}|_{\{1\}\times C_{2}(M)})$, $Z_{O-O}(\omega_{0},\xi_{0})=Z_{O-O}(\widetilde{\omega}|_{\{0\}\times C_{2}(M)},\xi_{0})$ and $Z_{O-O}(\omega_{1},\xi_{1})=Z_{O-O}(\widetilde{\omega}|_{\{1\}\times C_{2}(M)},\xi_{1})$. We note that, with our orientation convention,

[TABLE]

Therefore, by using Stokes’ theorem,

[TABLE]

We denote $\widetilde{\pi_{i}}={\rm id}_{[0,1]}\times\pi_{i}:[0,1]\times M^{2}\to[0,1]\times M$ for $i=1,2$. We have,

[TABLE]

Here, $\widetilde{\pi_{1}}|_{[0,1]\times\Delta}=\widetilde{\pi_{2}}|_{[0,1]\times\Delta}:[0,1]\times\Delta\to M$. Thus $(\widetilde{\pi_{1}}|_{[0,1]\times\Delta})^{*}\widetilde{\xi}(\widetilde{\pi_{2}}|_{[0,1]\times\Delta})^{*}\widetilde{\xi}=\widetilde{\xi}^{2}$ under the identification $\Delta=M$. We have

[TABLE]

Then we have

[TABLE]

This completes the proof of Proposition 4.2.

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 $\partial C_{2}(M)$ is identified with the unit sphere bundle $STM$ (see Section 2.1). Take a framing $f:TM\to M\times\mathbb{R}^{3}$ of $M$. Then $(M,f)$ is a framed 3-manifold. Let $p:(\partial C_{2}(M)=)STM\to S^{2}$ be the projection defined by the framing $f$. Let $\delta(f)\in\mathbb{Z}$ 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 $\delta(f)$ in the next section. Let $\rho:\pi_{1}(M)\to G$ be an acyclic representation as in Section 2.1.

Theorem 5.1**.**

[TABLE]

is an topological invariant of $M,\rho$.

5.1 The signature defect $\delta(p)$

Let $W$ be a compact 4-manifold such that $\partial W=M$ and its Euler characteristic is zero. Then there exists an $\mathbb{R}^{3}$ sub-bundle $T^{v}W$ of $TW$ satisfying $T^{v}W|_{M}=TM$. Let $ST^{v}W\to W$ be the unit sphere bundle of $T^{v}W\to W$. Thus $ST^{v}W$ is a 6-dimensional manifold with $\partial ST^{v}W=STM$. We denote by $F_{W}\to ST^{v}W$ the tangent bundle along the fiber of the $S^{2}$ bundle $\pi:ST^{v}W\to W$.

Take a closed 2-form $\alpha_{W}\in\Omega^{2}(ST^{v}W;\mathbb{R})$ such that $\alpha_{W}|_{STM}=p^{*}\omega_{S^{2}}$ and $[\alpha_{W}]=e(F_{W})/2\in H^{2}(ST^{v}W;\mathbb{R})$, where $e(F_{W})$ is the Euler class of $F_{W}\to ST^{v}W$.

Lemma 5.2**.**

When $\partial W=M=\emptyset$, we have $\int_{ST^{v}W}\alpha_{W}^{3}=\frac{3}{4}{\rm Sign}W$. Here ${\rm Sign}W$ 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, $\int_{ST^{v}W}\alpha_{W}^{3}=\int_{ST^{v}W}\left(\frac{1}{2}e(F_{W})\right)^{3}$. We denote by $p_{1}(F_{W})\in H^{4}(ST^{v}W;\mathbb{R})$ the 1st Pontrjagin class of the bundle $F_{W}$. We remark that $\underline{\mathbb{R}}\oplus F_{W}=\pi^{*}T^{v}W$ and $\underline{\mathbb{R}}\oplus T^{v}W=TW$. Here $\underline{\mathbb{R}}$ is the trivial $\mathbb{R}$ bundle over an appropriate manifold. Therefore,

[TABLE]

∎

Thanks to the Novikov additivity for the signature, the following corollary holds.

Corollary 5.3**.**

The signature defect $\delta(f)$ defined to be $\delta(f)=\int_{ST^{W}}\alpha_{W}^{3}-\frac{3}{4}{\rm Sign}W$ is independent of the choices of $W$ and $\alpha_{W}$.

5.2 Proof of Theorem 5.1

Let $f_{0},f_{1}:TM\to M\times\mathbb{R}^{3}$ be framings and let $p_{0},p_{1}:\partial C_{2}(M)\to S^{2}$ be the projections given by framings $f_{0},f_{1}$ respectively. Since $[p_{0}^{*}\omega_{S^{2}}]$ and $[p_{1}^{*}\omega_{S^{2}}]$ are in $H^{2}_{-}(\Delta\times S^{2};\mathbb{R})=H^{2}(S^{2};\mathbb{R})=\mathbb{R}$, $[p_{0}^{*}\omega_{S^{2}}]=[p_{1}^{*}\omega_{S^{2}}]$. Thus there exists a closed 2-form

[TABLE]

such that $\widetilde{\omega}_{\partial}|_{\{0\}\times\partial C_{2}(M)}=p_{0}^{*}\omega_{S^{2}},$ $\widetilde{\omega}_{\partial}|_{\{1\}\times\partial C_{2}(M)}=p_{1}^{*}\omega_{S^{2}}.$ We recall that $\xi\in\Omega^{2}_{-}(\Delta;E_{\rho}\otimes E_{\rho})$ is a closed 2-form representing $\Phi([p^{*}\omega_{S^{2}}{\bf 1}])=\Phi\circ c^{*}([p^{*}\omega_{S^{2}}])$ when we take a projection $p:\partial C_{2}(M)\to S^{2}$ given by a framing $f$. The homomorphism $\Phi\circ c^{*}$ is independent from the choice of a framing. Then we can use same $\xi\in\Omega^{2}_{-}(\Delta;E_{\rho}\otimes E_{\rho})$ for any framing.

By a similar argument as in proof of Proposition2.1, we can take a closed 2-form

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such that

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Here, $\pi_{C_{2}(M)}:[0,1]\times C_{2}(M)\to C_{2}(M)$ and $Q:[0,1]\times\partial C_{2}(M)\to\Delta$ are the projections. We denote by

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Then,

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Thanks to Stokes’ theorem,

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We denote $\overline{\pi}_{i}:[0,1]\times M^{2}\to M,(t,x_{1},x_{2})\mapsto x_{i}$ for $i=1,2$. We have,

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Thus we have

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Lemma 5.4**.**

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Proof.

Let $T_{E}:E_{\rho}\otimes E_{\rho}\to E_{\rho}\otimes E_{\rho}$ be the involution induced by $\mathfrak{g}\otimes\mathfrak{g}\to\mathfrak{g}\otimes\mathfrak{g},x\otimes y\mapsto y\otimes x$. Cleary, ${\rm Tr}^{\boxtimes 2}\circ T_{E}^{\otimes 3}={\rm Tr}^{\boxtimes 2}:E^{\otimes 3}\otimes E^{\otimes 3}\to\underline{\mathbb{R}}$. Since $T_{E}({\bf 1})={\bf 1}$ and $T_{E}^{*}=(T_{0}|_{\Delta})^{*}$ on $\Omega^{1}(\Delta;E_{\rho}\otimes E_{\rho})$, we have

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Thus ${\rm Tr}^{\boxtimes 2}(\widetilde{\omega}_{\partial}{\bf 1}^{\otimes 2}Q^{*}\xi)=0$. ∎

Lemma 5.5**.**

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Proof.

We take a compact 4-manifold $W$ with $\partial W=M$ and its Euler characteristic is zero. Take a collar neighborhood $[0,1]\times\partial M$ of $M=\partial W$ in $W$ such that $\{1\}\times M=\partial W$. Set $W_{0}=W\setminus([0,1]\times M)$. We can take $T^{v}W$ as $T^{v}W|_{[0,1]\times M}=[0,1]\times TM$. Thus $ST^{v}W|_{[0,1]\times M}$ is identified with $[0,1]\times\partial C_{2}(M)$. Take a closed 2-form $\alpha_{W}\in\Omega^{2}(ST^{v}W;\mathbb{R})$ satisfying $\alpha_{W}|_{[0,1]\times STM}=\widetilde{\omega}_{\partial}$ and $[\alpha_{W}]=\frac{1}{2}e(F_{W})$. Then we have

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∎

By the above two lemmas,

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Namely, $Z_{1}((M,f),\rho)-(\dim\mathfrak{g})^{2}\delta(f)$ is independent of the choice of a framing $f$.