2·1. Orientation conventions

First we mention some conventions to fix an orientation. An interval $[0,1]\subset {\mathbb{R}}$ is oriented from 0 to 1. Products of manifolds are oriented by the order of the factors. Boundaries of manifolds are oriented by the outward normal first convention. Let $A, B\subset X$ be oriented submanifolds of an oriented manifold X. We assume that A is transversal to B and $A\cap B\not =\emptyset$ . Let $\nu A$ be a normal bundle of A in X. We adopt the following conventions on orientations:

1. (O1) $T_xA\oplus \nu_xA=T_xX$ , for $x\in A$ .

2. (O2) $\nu_x(A\cap B)=\nu_x A\oplus \nu_x B$ .

3. ⇒ (O3) $T_x(A\cap B)\oplus \nu_xA\oplus \nu_xB=T_xX$ .

4. (O4) (The outward normal first convention) For an oriented manifold with a boundary W and $x\in \partial W$ ,

   \begin{align*}n_x\oplus T_x\partial W=T_xW.\end{align*}
   Here $n_x$ is a outward normal vector at $x\in\partial W$ .

5. (O5) Let $\pi\,:\,\widetilde X\to X$ be an F-bundle, where $\widetilde X,X$ and F are orientable manifolds. Let $U\subset X$ be an open set so that $\pi|_U$ is trivial. We choose orientations of $\widetilde X, X$ and F such that the condition $\pi^{-1}(U)=U\times F$ holds.

2·2. Local systems

Let M be a closed oriented 3-manifold.

Let $\Pi(M)$ be the “fundamental groupoid” of M; namely $\Pi(M)$ is a category such that the objects are the points of M and a morphism from $x\in M$ to $y\in M$ is a path from x to $y\in M$ up to homotopy relative to x, y. In particular, a morphism from a point $x\in M$ to x is an element of the fundamental group $\pi_1(M,x)$ . A “local system” over M is a covariant functor from $\Pi(M)$ to a category of vector spaces. We denote by $\gamma_*$ the isomorphism corresponding to a path $\gamma$ via the local system. Let E be a vector space over a field F. For a representation $\rho\,:\,\pi_1(M,x)\to \text{GL}(E)$ , there is a local system on M such that each object is isomorphic to E and $\gamma_*=\rho(\gamma)$ for any $\gamma\in \pi_1(M,x)$ . We call such a local system the local system corresponding to $\rho$ and we use the same symbol $\rho$ for the local system. We denote by $C_*(M;\,\rho)$ or $H_*(M;\,\rho)$ chain complexes or homology groups with the local coefficient $\rho$ respectively.

Let $V^*=\text{Hom}(V,F)$ be the dual vector space of V. For a representation $\rho\,:\,\pi_1(M)\to \text{GL}(V)$ , we denote by $\rho^*\,:\,\pi_1(M)\to \text{GL}(V^*)$ the dual representation of $\rho$ , namely $\rho^*$ is given by

\begin{align*}\rho^*(h)(\,f)=f\circ \rho(h^{-1})\end{align*}

for $h\in \pi_1(M)$ and $f\in V^*$ .

2·3. Chains with local coefficients

Let X be a manifold. Let E be a vector space and let $\rho$ be a local system corresponding to a representation $\rho\,:\,\pi_1(X)\to \text{GL}(E)$ . Let $c\,:\,(A,a)\to (X,c(a))$ be a continuous map, where A is a compact oriented manifold with a basepoint $a\in A$ . We denote by $E_{c(a)}^{c_*\pi_1(A,a)}$ the invariant part of $E_{c(a)}$ under the action of $c_*\pi_1(A,a)\lt\pi_1(X,c(a))$ via the representation $\rho$ . Take an element $e\in E_{c(a)}^{c_*\pi_1(A,a)}$ . Then c and $e\in E_{c(a)}$ give a $(\dim A)$ -dimensional chain. We denote this chain by

\begin{align*}\langle c\,:\,A\to X, a;\,e\rangle\in C_{\dim A}(X;\,\rho)\end{align*}

or shortly,

\begin{align*}\langle A, a;\,e\rangle=\langle c\,:\,A\to X, a;\,e\rangle.\end{align*}

For a chain $\alpha=\langle c,a;\,n\in{\mathbb{Z}}\rangle \in C_*(X;\,{\mathbb{Z}})$ and an element $e\in E_{c(a)}^{c_*\pi_1(A,a)}$ , we have a chain $\langle c,a;\,ne\rangle \in C_*(X;\,E)$ . To simplify the notation, we denote it by $\alpha\otimes e$ . Since any chain of $C_*(X;\,{\mathbb{Z}})$ is given as a weighted sum of singular simplexes, $\alpha\otimes e$ is defined for any $\alpha\in C_*(X;\,{\mathbb{Z}})$ .

Let F be a compact oriented manifold. Let $\pi\,:\,\widetilde X\to X$ be a F-bundle over X. A homomorphism $\rho\circ \pi_*\,:\,\pi_1(\widetilde{X})\stackrel{\pi_*}{\to}\pi_1(X)\to \text{GL}(E)$ gives a local system on $\widetilde X$ . Let $c=\langle A\to X,a;\,e\rangle\in C_k(X;\,E)$ be a k-chain. We denote by $c^*\widetilde X\to A$ the pull back of $\pi\,:\,\widetilde X\to X$ along c. Then we have a bundle map $\widetilde c\,:\,c^*\widetilde X\to \widetilde X$ induced by $c\,:\,A\to X$ .

By using $\widetilde c$ , the $(k+\dim F)$ -chain $\pi^!c\in C_{k+\dim F}(\widetilde X;\,\pi^*E)$ can be described as follows:

\begin{align*}\pi^!c=\langle \widetilde c\,:\,c^*\widetilde{X}\to \widetilde {X},v_a;\,e\rangle\in C_{k+\dim F}(\widetilde X;\,\rho\circ \pi_*).\end{align*}

Here $v_a\in \pi^{-1}(a)$ is any point.

2·4. The maximal free abelian representation $\rho_0$

Let us denote $H=H_1(M;\,{\mathbb{Z}})/\text{Tor}$ . We fix a basis

\begin{align*}\{t_1,\ldots,t_k\}\subset H.\end{align*}

We interpret H as a multiplicative group. In this paper, however, the operations of all homology groups including $H_1(M;\,{\mathbb{Z}})$ are summation with the exception of H:

\begin{align*}H_1(M;\,{\mathbb{Z}})/\text{Tor}=\{n_1t_1+\cdots+n_kt_k\mid n_i\in {\mathbb{Z}}\},\end{align*}

$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad H=\{t_1^{n_1}\cdots t_k^{n_k}\mid n_i\in {\mathbb{Z}}\}.$ Footnote ¹

For $h\in H$ , we denote by

\begin{align*}[h]_+\in H_1(M;\,{\mathbb{Z}})/\text{Tor}\end{align*}

the element of $H_1(M;\,{\mathbb{Z}})/\text{Tor}$ corresponding to h. For example, $[t_1^2t_2]_+=2t_1+t_2$ .

Let

\begin{align*}{\mathbb{R}} H=\left\{\sum_{n_1,\ldots,n_k}a_{n_1,\ldots,n_k}t_1^{n_1}\cdots t_k^{n_k}\mid \text{finitely many non-zero }a_{n_1,\ldots,n_k}\in {\mathbb{R}}\right\}\end{align*}

be the group ring of H over ${\mathbb{R}}$ and let

\begin{align*}Q(H)=\{f/g\mid f,g\in {\mathbb{R}} H, g\not=0\}\end{align*}

be the quotient field of ${\mathbb{R}} H$ . By using the basis, Q(H) can be identified with

\begin{align*}Q(H)={\mathbb{R}}(t_1,\ldots, t_k).\end{align*}

Let $\rho_0\,:\,{\mathbb{R}} H\to Q(H)$ be the inclusion given by $\rho_0(\,f)={f}/{1}$ . We also use the same symbol $\rho_0$ for the restriction of $\rho_0$ to H. Since Q(H) is a 1-dimensional vector space over Q(H), the representation $\rho_0\,:\,H\to \text{GL}(Q(H))=Q(H)^{\times}$ gives a representation of $\pi_1(M)$ . Then there is a local system corresponding to $\rho_0$ .

We assume that $\rho_0$ is acyclic. Namely, the corresponding local system is acyclic:

\begin{align*}H_*(M;\,\rho_0)=0.\end{align*}

2·5. Euler structures

As a general reference for Euler structures, for example, see [ Reference Turaev19 ]. Let $v,v'\in\Gamma TM$ be non-vanishing vector fields on M. We say that v and v’ are homologous, denoted by $v\sim v'$ , if $v|_{M\setminus \mathring{B}^3}$ is homotopic to $v'|_{M\setminus \mathring{B}^3}$ as non-vanishing vector fields on $M\setminus \mathring{B}^3$ for a ball $B^3\subset M$ . A smooth Euler structure, or shortly Euler structure, on M is a homology class of non-vanishing vectors field on M. We denote by $\text{Eul}(M)$ the set of all Euler structures on M:

\begin{align*}\text{Eul}(M)=\{\text{non-vanishing vector fields on} M\}/\sim.\end{align*}

Let $s\in \Gamma TM$ be a vector field on M which is transversal to the 0-section 0. Then $s^{-1}(0)$ is a 0-dimensional manifold consisting of finitely many singular points. The orientation $\pm1$ of each point $p\in s^{-1}(0)$ is defined as follows: The orientation of each fiber of the tangent bundle TM is induced from the orientation of M. Then TM is oriented. An orientation of the image of a section $s(M)\subset TM$ is induced from the orientation of M. Then an orientation of p is defined as the orientation of the intersection $0(M)\cap s(M)$ at p. Note that if $s=-{\textrm{grad}}\,f$ , the opposite of a gradient-like vector field of a Morse function f on M, the orientation of each singular point p coincides with $(-1)^{\dim M-\text{ind}(p)}=(-1)^{\text{ ind}(p)+1}$ . Here $\text{ind}(p)$ is the Morse index of p. Let $k=\sharp s^{-1}(0)/2$ .

From s and an Euler chain c, we construct an Euler structure on M. Let $N(c_1),\ldots, N(c_k)\subset M$ be disjoint neighbourhoods of $c_1([0,1]),\ldots,c_k([0,1])$ respectively such that each $N(c_i)$ is diffeomorphic to a Ball $B^3$ . In each $N(c_i)$ , there are two singular points. Since the orientation of the two singular points are opposite, $s_0|_{\partial N(c_i)}$ can be extended to $N(c_i)$ as a non-vanishing vector field. By replacing each $s|_{N(c_i)}$ with a non-vanishing vector field, we have a non-vanishing vector field on M. We denote by

\begin{align*}s/c\in \Gamma TM\end{align*}

such a non-vanishing vector field. There are several choices of $s/c$ . The Euler structure

\begin{align*}[s/c]\in \text{Eul}(M)\end{align*}

represented by a non-vanishing vector field $s/c$ is, however, uniquely determined from s and c.

For later sections, we give a more concrete construction of $s/c$ . Let $p_2\,:\,[0,1]\times M\to M$ be the projection. Then $p_2^*TM$ is an ${\mathbb{R}}^3$ -bundle over $[0,1]\times M$ . Let $\widetilde c_i$ be a 1-manifold of $[0,1]\times M$ defined by

\begin{align*}\widetilde c_i(t)=(t(1-t),c_i(t)), 0\le t\le 1.\end{align*}

Then $\widetilde c_i$ is a lift of $c_i$ along $p_2$ . Set $c=\sqcup_{i=1}^k\widetilde{c_i}$ .

Let $\widetilde{s'}=\{s'_t\}_{t\in [0,1]}$ be an alternative choice of $\widetilde{s}$ . By condition (iv), we have $s_1'=s_1$ on $M\setminus \sqcup_iN(c_i)$ . Then $s_1$ is homologous to $s_1'$ . Namely, the Euler structure $[s_1]\in \text{Eul}(M)$ does not depend on the choice of $\widetilde{s}=\{s_t\}_{t\in [0,1]}$ in lemma 2·2. Therefore we set

\begin{align*}[s/c]=[s_1](=[s_1'])\in \text{Eul}(M).\end{align*}

4·1 Preparation for local systems

Thanks to the Künneth theorem, $\rho^*\boxtimes \rho\,:\,\pi_1(M\times M)=\pi_1(M)\times \pi_1(M)\to \text{GL}(V^*\otimes V)$ gives an acyclic local system on a 6-dimensional manifold $M\times M$ . Here $\rho^*\boxtimes \rho$ is the exterior tensor product of $\rho$ and $\rho^*$ .

Let $\Delta=\{(x,x)\mid x\in M\}\subset M\times M$ be the diagonal. We orient $\Delta$ so that a diffeomorphism $M\ni x\mapsto (x,x)\in \Delta$ preserves the orientation. The restriction of $\rho^*\boxtimes \rho$ to $\Delta$ is $\rho^*\otimes \rho$ . The identity map $1\in V^*\otimes V=\text{Hom}(V,V)$ is invariant under the action of $\pi_1(M)$ via $\rho^*\otimes \rho$ . Then for any chains $c \in C_*(\Delta;\,{\mathbb{Z}})$ , we have a chain $c\otimes 1\in C_*(\Delta;\,\rho^*\otimes \rho)$ .

We now consider $\rho_0\,:\,\pi_1(M)\to Q(H)^{\times}$ . Since Q(H) is a 1-dimensional vector space, $\rho^*\otimes \rho$ is trivial and spanned by 1. Thus we have

\begin{eqnarray*}H_1(\Delta\,:\,\rho^*\otimes\rho)&=&H_1(\Delta;\,{\mathbb{Z}})\otimes_{{\mathbb{Z}}}Q(H)\\ &=&H_1(M;\,{\mathbb{Z}})\otimes_{{\mathbb{Z}}}Q(H)=H_1(M;\,{\mathbb{Z}})/\text{Tor}\otimes_{{\mathbb{Z}}}Q(H).\end{eqnarray*}

4·2. A twisted self-linking number of a link

The invariant $d'(M,\rho;\,e)$ is defined as a “twisted self-linking 1-dimensional homology class” of an embedded 3-manifold in a 6-manifold with local coefficients. Then we first review the usual linking number of a link.

For a 2-component oriented link $K_1\sqcup K_2\subset S^3$ , the linking number $\text{lk}_{S^3}(K_1,K_2)\in {\mathbb{Z}}$ is defined as follows: Take a 2-chain $\widetilde{K_1}\in C_2(S^3;\,{\mathbb{Z}})$ satisfying $\partial \widetilde{K_1}=K_1$ (we note that we can take a Seifert surface of $K_1$ as a $\widetilde{K_1}$ ). Then $\text{lk}_{S^3}(K_1,K_2)$ is defined by $\text{lk}_{S^3}(K_1,K_2)=[\widetilde K_1]\cap [K_2]\in H_0(K_2;\,{\mathbb{Z}})={\mathbb{Z}}$ . Let v be a normal vector field of $K_1$ in $S^3$ . The self-linking number

\begin{align*}\text{self.lk}_{S^3}(K_1,v)\in {\mathbb{Z}}\end{align*}

of a framed knot $(K_1,v)$ is defined as a linking number of $K_1$ and $v(K_1)$ . Here $v(K_1)$ is a copy of $K_1$ given by pushing $K_1$ along v.

4·3. A twisted self-linking homology class of an embedded 3-manifold

Let $Y_1,Y_2$ be closed 3-manifolds embedded in a closed 6-manifold X satisfying $Y_1\cap Y_2=\emptyset\subset X$ . Let $\tau\,:\,\pi_1(X)\to \text{GL}(E)$ be a representation, where E is a vector space. We assume that they satisfy the following conditions:

1. (i) $\tau$ is acyclic: $H_*(X;\,\tau)=0$ ;

2. (ii) There is a special element $1\in E$ such that 1 is invariant under the action of $\pi_1(M)$ .

Since $[Y_1]\otimes 1=0\in H_3(X;\,\tau)=0$ , there exists a 4-chain $\widetilde{Y_1}\otimes 1\in C_4(X;\,\tau)$ satisfying $\partial \widetilde{Y_1}=Y_1$ . The intersection of $[\widetilde{Y_1}]\in H_4(X,Y_1;\,\tau)$ and $[Y_2]\in H_3(Y_2;\,{\mathbb{Z}})$ gives a 1-dimensional homology class $[\widetilde{Y_1}]\cap[Y_2]=[\widetilde{Y_1}\cap Y_2]\in H_1(Y_2;\,\tau)$ .

Proof. The homology class $[\widetilde{Y_1}]\cap[Y_2]-[\widetilde{Y_1}']\cap[Y_2]=[\widetilde{Y_1}-\widetilde{Y_1}']\cap [Y_2]$ is in the image of the morphism

\begin{align*}\cap\,:\,H_4(X;\,\tau)\otimes H_3(Y_2;\,{\mathbb{Z}})\to H_1(Y_2;\,\tau).\end{align*}

Since $\tau$ is acyclic, $H_4(X;\,\tau)=0$ . Then $[\widetilde{Y_1}-\widetilde{Y_1}']\cap [Y_2]=0$ .

This lemma implies that a twisted linking homology class is well defined:

Definition 4·2.

\begin{align*}\text{lk}_{\tau}(Y_1,Y_2)=[\widetilde{Y_1}]\cap [Y_2]\in H_1(Y_2;\,\tau).\end{align*}

If there is a non-vanishing vector field v of the normal bundle of $Y_1$ in X, we can formulate a self-linking homology class of the framed 3-manifold $(Y_1,v)$ . Let $v(Y_1)\subset X$ be a parallel copy of $Y_1$ given by pushing $Y_1$ along v.

Definition 4·3. The twisted self-linking homology class of $(Y_1,v)$ is defined to be

\begin{align*}\text{self.lk}_{\tau}(Y_1,v)=\text{lk}_{\tau}(v(Y_1),Y_1).\end{align*}

More precisely, $v(Y_1)\subset X$ is obtained as follows: Let $\nu_{Y_1}$ be the normal bundle of $Y_1$ in X. Set $D\nu_{Y_1}=\{(x,v)\mid \|v\|\le 1\}$ . $D\nu_{Y_1}$ forms a $D^3$ -bundle over $Y_1$ . Let $S\nu_{Y_1}=\partial D\nu_{Y_1}$ be the unit sphere bundle. Let $N(Y_1)\subset X$ be a closed tubular neighbourhood of $Y_1$ . We identify $D\nu_{Y_1}$ with $N(Y_1)$ . We denote by $\pi\,:\, \partial N(Y_1)\to Y_1$ and $\pi\,:\, N(Y_1)\to Y_1$ the projections. We identify the local system $\tau|_{N(Y_1)}=\pi^*\tau$ with $\tau$ via $\pi^*$ . The normalisation ${v}/{\|v\|}$ of the non-vanishing vector field v is a section of the $S^2$ -bundle $\partial N(Y_1)\to Y_1$ . Then set

\begin{align*}v(Y_1)=\left\{\frac{v(x)}{\|v(x)\|}\middle| x\in Y_1\right\}\subset X.\end{align*}

Here $v(Y_1)$ is oriented by $v\,:\,Y_1\stackrel{\cong}{\to} v(Y_1)$ .

4·4. An invariant $d'(M,\rho;\,e)$

Take a non-vanishing vector field $v_e$ representing the Euler structure e. We identify the normal bundle $\nu_{\Delta}$ with the tangent bundle TM via the following isomorphism:

\begin{align*}\nu_{\Delta}\ni ((x,x),(-v,v))\mapsto (x,v)\in TM,\end{align*}

where $x\in M,v\in T_xM$ . Then $v_e$ gives a non-vanishing vector field of $\nu_{\Delta}$ . An invariant $d'(M,\rho;\,e)$ is defined to be a twisted self-linking homology class of $(\Delta,v_e)$ :

Definition 4·4.

\begin{eqnarray*}&&d'(M,\rho;\,e)=\text{self.lk}_{\rho^*\boxtimes \rho}(\Delta,v_e)=\text{lk}_{\rho^*\boxtimes \rho}(v_e(\Delta),\Delta)\\ &&\hskip50mm \in H_1(\Delta;\,\rho^*\otimes \rho)=H_1(M;\,\rho^*\otimes \rho).\end{eqnarray*}

Let

\begin{align*}\partial_*\,:\,H_4(M^2,M^2\setminus \mathring{N}(\Delta);\,\rho^*\boxtimes \rho)\to H_3(M^2\setminus \mathring{N}(\Delta);\,\rho^*\boxtimes\rho)\end{align*}

be the connecting homomorphism in the long exact sequence of homologies of the pair $(M^2,M^2\setminus \mathring{N}(\Delta))$ . Here $N(\Delta)$ is oriented as a part of $M^2$ . Since $H_4(M^2;\,\rho^*\boxtimes \rho)=0$ and $H_3(M^2;\,\rho^*\boxtimes \rho)=0$ , $\partial_*$ is an isomorphism. Let

\begin{align*}E\,:\,H_4(M^2,M^2\setminus \mathring{N}(\Delta),\rho^*\boxtimes\rho)\overset{\cong}{\to} H_4(N(\Delta),\partial N(\Delta);\,\rho^*\boxtimes \rho)\end{align*}

be an excision isomorphism. Let

\begin{align*}\psi \,:\,H_4(N(\Delta),\partial N(\Delta);\,\rho^*\boxtimes \rho)\overset{\cong}{\to} H_1(\Delta;\,\rho^*\otimes \rho)\end{align*}

be the Thom isomorphism. Here the sign convention for the Thom isomorphism $\psi$ is as follows: For a 4-chain $\alpha$ in $M^2$ representing a cycle in $C_4(N(\Delta),\partial N(\Delta);\,\rho^*\boxtimes \rho)$ , an equation $\psi(\alpha)=\alpha\cap \Delta$ holds.

Therefore we have an isomorphism

\begin{align*}\Phi=\psi\circ E\circ (\partial_*)^{-1}\,:\,H_3(M^2\setminus \mathring{N}(\Delta);\,\rho^*\boxtimes\rho)\stackrel{\cong}{\to} H_1(\Delta;\,\rho^*\otimes\rho).\end{align*}

By the definition of twisted self-linking homology class, $d'(M,\rho;\,e)$ can be written as follows:

Remark 4·6. An invariant $d(\rho_0)$ defined in [ Reference Shimizu18 ] can be computed from d^′ as follows:

\begin{align*}d(\rho_0)=d'(M,\rho_0;\,e_0).\end{align*}

Here $e_0$ is an Euler structure associated to a parallelisation of M.

5·1. An operator D

In this section, we introduce an operator $D\,:\,Q(H)^{\times}\to\left(H_1(M;\,{\mathbb{Z}})/\text{Tor}\right)\otimes_{{\mathbb{Z}}} Q(H)$ . Recall that for $h\in H$ , $[h]_+\in H_1(M;\,{\mathbb{Z}})/\text{Tor}$ is the element corresponding to h. We first define D on ${\mathbb{R}} H\setminus 0$ .

Definition 5·1. For $f=\sum_{h\in H}a_hh\in {\mathbb{R}} H\setminus 0 (a_h\in {\mathbb{R}})$ , we set

\begin{align*}\displaystyle{D(\,f)=D\left(\sum_{h\in H}a_hh\right)=\sum_h[h]_+\otimes \frac{a_hh}{f}}.\end{align*}

Example 5·2. Let $H=\langle t_1,t_2\rangle$ .

\begin{eqnarray*}&&D(1+t_1^3+t_1^2t_2^6)\\ &=&[1]_+\otimes \frac{1}{1+t_1^3+t_1^2t_2^6}+[t_1^3]_+\otimes\frac{t_1^3}{1+t_1^3+t_1^2t_2^6}+[t_1^2t_2^6]_+\otimes\frac{t_1^2t_2^6}{1+t_1^3+t_1^2t_2^6}\\ &=&0\otimes \frac{1}{1+t_1^3+t_1^2t_2^6}+3[t_1]_+\otimes\frac{t_1^3}{1+t_1^3+t_1^2t_2^6}+(2[t_1]_++6[t_2]_+)\otimes\frac{t_1^2t_2^6}{1+t_1^3+t_1^2t_2^6}\\ &=&[t_1]_+\otimes \frac{3t_1^3+2t_1^2t_2^6}{1+t_1^3+t_1^2t_2^6}+[t_2]_+\otimes \frac{6t_1^2t_2^6}{1+t_1^3+t_1^2t_2^6}.\end{eqnarray*}

Proof. (i) Let $f=\sum_ha_hh, g=\sum_hb_hh$ . We have

\begin{eqnarray*}&&\left(1\otimes \frac{f}{f+g}\right)D(\,f)+\left(1\otimes \frac{g}{f+g}\right)D(g)\\ &=& \sum_h[h]_+\otimes \left(\frac{f}{f+g}\cdot\frac{a_hh}{f}+\frac{g}{f+g}\cdot\frac{b_hh}{g}\right)\\ &=& \sum_h[h]_+\otimes \left(\frac{a_hh}{f+g}+\frac{b_hh}{f+g}\right)\\ &=&D(f+g).\end{eqnarray*}

(ii) Let $f=\sum_ha_hh, g=\sum_{h'}b_{h'}h'$ . Then $fg=\sum_{h,h'}a_hb_{h'}hh'$ . We have

\begin{eqnarray*}D(fg)&=& \sum_{h,h'}[hh']_+\otimes \frac{a_hb_{h'}hh'}{fg}\\ &=& \sum_{h,h'}([h]_++[h']_+)\otimes \frac{a_hb_{h'}hh'}{fg}\\ &=& \sum_{h,h'}[h]_+\otimes \frac{a_hb_{h'}hh'}{fg}+\sum_{h,h'}[h']_+\otimes \frac{a_hb_{h'}hh'}{fg}\\ &=& \sum_{h}\left([h]_+\otimes \sum_{h'}\frac{a_hb_{h'}hh'}{fg}\right)+\sum_{h'}\left([h']_+\otimes \sum_{h}\frac{a_hb_{h'}hh'}{fg}\right)\\ &=& \sum_{h}\left([h]_+\otimes \frac{a_hh}{f}\frac{\sum_{h'}b_{h'}h'}{g}\right)+\sum_{h'}\left([h']_+\otimes \frac{b_{h'}h'}{g}\frac{\sum_ha_hh}{f}\right)\\ &=& \sum_{h}\left([h]_+\otimes \frac{a_hh}{f}\right)+\sum_{h'}\left([h']_+\otimes \frac{b_{h'}h'}{g}\right)\\ &=& D(\,f)+D(g).\end{eqnarray*}

As a direct consequence of the above lemma, we have the following corollary.

Corollary 5·4. Let $f_1,g_1,f_2,g_2\in {\mathbb{R}} H\setminus 0$ . If $g_1/f_1=g_2/f_2\in Q(H)$ then

\begin{align*}D(g_1)-D(f_1)=D(g_2)-D(f_2).\end{align*}

Proof. It is enough to show that for any $h\in {\mathbb{R}} H\setminus 0$ , $D(g_1h)-D(f_1h)=D(g_1)-D(h_1)$ . Thanks to lemma 5·3 (ii), we have

\begin{eqnarray*}D(g_1h)-D(f_1h)= D(g_1)+D(h)-\left(D(f_1)+D(h)\right)= D(g_1)-D(f_1).\end{eqnarray*}

The corollary guarantees that D can be extended to $Q(H)^{\times}$ as follows:

Let $x_1,\ldots,x_m$ be arbitrary elements of H. We next give an explicit formula of D(f) for any rational function $f(x_1,\ldots, x_m)\in Q(H)^{\times}$ in the language of $x_1,\ldots,x_m$ :

Proposition 5·6. For any rational function $f(x_1,\ldots,x_m)\in Q(H)^{\times}$ of $x_1,\ldots,x_m$ ,

\begin{align*}D(\,f)=\sum_{i=1}^m\left([x_i]_+\otimes \frac{x_i}{f}\frac{\partial f}{\partial x_i}\right).\end{align*}

In particular, for any monomial $ax_1^{n_1}\cdots x_m^{n_m}\in {\mathbb{R}} H$ , where $a\in{\mathbb{R}}\setminus 0$ and $n_1,\ldots,n_m\in{\mathbb{Z}}$ , we have

\begin{align*}D(ax_1^{n_1}\cdots x_m^{n_m})=(n_1[x_1]_++\cdots+n_m[x_m]_+)\otimes 1.\end{align*}

Any rational function of $x_1,\ldots,x_m$ can be written as a quotient of polynomials of $x_1,\ldots,x_m$ , and any polynomial of $x_1,\ldots,x_m$ can be written as a linear combination of products of $x_1,\ldots,x_m$ . Thus we can compute D ^′ completely by the properties (i), (ii), (iii) and (iv). Then $D'=D$ for any rational functions of $x_1,\ldots,x_m$ .

Example 5·7. Let $t_1,t_2\in H$ and $f=1+t_1^3+t_1^2t_2^6$ .

\begin{eqnarray*}D(\,f)&=&[t_1]_+\otimes \frac{t_1}{f}\frac{\partial}{\partial t_1}(1+t_1^3+t_1^2t_2^6)+[t_2]_+\otimes \frac{t_2}{f}\frac{\partial}{\partial t_2}(1+t_1^3+t_1^2t_2^6)\\ &=&[t_1]_+\otimes \frac{3t_1^3+2t_1^2t_2^6}{f}+[t_2]_+\otimes \frac{6t_1^2t_2^6}{f}.\end{eqnarray*}

Let $T=t_1^2\in H$ . Then $f=1+t_1^3+t_1^2t_2^6$ can be represented as $f(t_1,t_2,T)=1+t_1T+t_2^6T$ . Then

\begin{eqnarray*}&&D(1+t_1T+t_2^6T)\\ &=&[t_1]_+\otimes \frac{t_1}{f}\frac{\partial}{\partial t_1}(1+t_1T+t_2^6T)+[t_2]_+\otimes \frac{t_2}{f}\frac{\partial}{\partial t_2}(1+t_1T+t_2^6)\\ &&+[T]_+\otimes \frac{T}{f}\frac{\partial}{\partial T}(1+t_1T+t_2^6T)\\ &=&[t_1]_+\otimes \frac{t_1T}{f}+[t_2]_+\otimes \frac{6t_2^6T}{f}+[T]_+\otimes \frac{t_1T+t_2^6T}{f}\\ &=&[t_1]_+\otimes \frac{t_1^3}{f}+[t_2]_+\otimes \frac{6t_1^3t_2^6}{f}+2[t_1]_+\otimes \frac{t_1^3+t_1^2t_2^6}{f}\\ &=&[t_1]_+\otimes \frac{3t_1^3+2t_1^2t_2^6}{f}+[t_2]_+\otimes \frac{6t_1^3t_2^6}{f}.\end{eqnarray*}

5·2. Main theorem

Recall that for the maximal free abelian representation $\rho_0$ ,

\begin{align*}\text{Tor}(M,\rho_0;\,e)\in Q(H)^{\times}/{\pm},\ \text{and}\ d'(M,\rho_0;\,e)\in (H_1(M;\,{\mathbb{Z}})/\text{Tor})\otimes_{{\mathbb{Z}}}Q(H).\end{align*}

The following theorem is the main theorem of this paper.

\begin{align*}d'(M,\rho_0;\,e)=D(\text{Tor}(M,\rho_0;\,e)).\end{align*}

Example 5·8. Let $M=S^1\times S^2$ . In this case $H=H_1(M;\,{\mathbb{Z}})=\langle t\rangle$ . Here $t\in H$ is represented by $S^1\times \{\text{pt}\}$ for a point $\text{pt}\in S^2$ . It is known that, for a suitable Euler structure e, the Reidemeister–Turaev torsion is given as

\begin{align*}\text{Tor}(S^1\times S^2,\rho_0;\,e)=\frac{1}{(t-1)^2}\in Q(H).\end{align*}

Thus

\begin{align*}d'(M,\rho_0;\,e)=D\left(\frac{1}{(t-1)^2}\right)=[t]_+\otimes\frac{-2t}{t-1}.\end{align*}

6·1. A Morse function and a combinatorial propagator

Recall that $d'(M,\rho;\,e)$ is defined to be a twisted self-linking homology class of $\Delta\subset M\times M$ . Thus $d'(M,\rho;\,e)$ can be computed from a 4-chain $\Sigma^0_f$ of $C_4(M^2;\,\rho_0^*\boxtimes \rho_0)$ bounded by $v_{e}(\Delta)\otimes 1$ . We give $\Sigma_0^{\kern0.5pt f}$ explicitly by using a Morse function. The description is related to the construction in [ Reference Shimizu17 , Reference Shimizu18 ]. It is inspired by [ Reference Fukaya6 , Reference Watanabe22 ].

Take a Morse function $f\,:\,M\to {\mathbb{R}}$ and a gradient-like vector field ${\textrm{grad}}\, f$ of f satisfying the Morse–Smale condition. We denote by $\text{Crit}_i(\,f)$ the set of critical points of index i and set $\text{Crit}(\,f)=\sqcup_i\text{Crit}_i(\,f)$ .

Let $(\Phi_t^{\kern0.5pt f}\,:\,M\stackrel{\cong}{\to} M)_{t\in{\mathbb{R}}}$ be the one-parameter family of diffeomorphisms associated to $-{\textrm{grad}}\, f$ . For $p\in \text{Crit}(\,f),$ let $\mathcal{A}_p$ and $\mathcal{D}_p$ be the ascending manifold of p and the descending manifold of p respectively:

\begin{align*}\mathcal A_p=\{ x\in M\mid \lim_{t\to\infty}\Phi_t^{\kern0.5pt f}(x)=p\},\end{align*}

\begin{align*}\mathcal D_p=\{ x\in M\mid \lim_{t\to-\infty}\Phi_t^{\kern0.5pt f}(x)=p\}.\end{align*}

We orient $\mathcal A_p$ and $\mathcal D_p$ so that the condition $T_p\mathcal A_p\oplus T_p\mathcal D_p=T_pM$ is satisfied. Let $\mathcal M(p,q)$ be the set of all trajectories from $p\in\text{Crit}_i(\,f)$ to $q\in\text{Crit}_{i-1}(\,f)$ :

\begin{align*}\mathcal M(p,q)=\left\{\gamma\,:\,{\mathbb{R}}\to M\mid d\gamma/dt=-{\textrm{grad}}\, f,\lim_{t\to-\infty}\gamma(t)=p,\lim_{t\to\infty}\gamma(t)=q\right\}/{\mathbb{R}}.\end{align*}

Here $s\in {\mathbb{R}}$ acts as $s\cdot\gamma(t)=\gamma(t+s)$ . A trajectory $\gamma$ gives an isomorphism $\gamma_*\,:\,V_p\stackrel{\cong}{\to} V_q$ . Let $\gamma_{\downarrow}$ be an oriented 1-manifold given by the closure of $\gamma({\mathbb{R}})$ with the downward orientation (the direction from p to q). Let $\gamma_{\cap}$ be an oriented 1-manifold given as a closure of $\gamma({\mathbb{R}})$ oriented as the intersection $\mathcal A_q\cap\mathcal D_p$ in M.

By using the orientation of $\gamma_{\cap}$ , we assign a sign ${\varepsilon}(\gamma)\in\{+1,-1\}$ for each $\gamma\in\mathcal M(p,q)$ such that the equality

\begin{align*}\gamma_{\downarrow}={\varepsilon}(\gamma)\gamma_{\cap}\end{align*}

holds. We denote by $C_*^{\kern1.5pt f}=(C_i^{\kern1.5pt f}(M;\,\rho),\partial^{\kern0.5pt f}_i\,:\,C_i^{\kern1.5pt f}(M;\,\rho)\to C_{i-1}^{\kern1.5pt f}(M;\,\rho))_i$ the Morse–Smale complex of f, namely $C_i^{\kern0.5pt f}(M;\,\rho)=\oplus_{p\in\text{Crit}_i(\,f)}V_p$ and

\begin{align*}\partial_i^{\kern0.5pt f}(x)=\sum_{q\in\text{Crit}_{\text{i}-1}(\,f)}(-1)^i\sum_{\gamma\in \mathcal M(p,q)}{\varepsilon}(\gamma)\gamma_*(x)\end{align*}

for any $x\in V_p$ and $p\in \text{Crit}_i(\,f)$ .

Take homomorphisms $g^{\kern0.5pt f}=\{g_i^{\kern1pt f}\,:\,C_{i-1}^{\kern1.5pt f}(M;\,\rho)\to C_i^{\kern1.5pt f}(M;\,\rho)\}_{i=1,2,3,4}$ satisfying

\begin{align*}\partial_{i+1}^{\kern0.5pt f}\circ g^{\kern0.5pt f}_{i+1}+g_i^{\kern0.5pt f}\circ \partial^{\kern0.5pt f}_i=\text{id}_{C_i^{\kern1.5pt f}(M;\,\rho)}\end{align*}

as in the definition of $\text{Tor}(M;\,\rho)$ in section 3. This $g^{\kern0.5pt f}$ is called a combinatorial propagator in [ Reference Fukaya6 , Reference Watanabe22 ].

For $p\in \text{Crit}_i(\,f)$ and $q\in \text{Crit}_{i-1}(\,f)$ , we denote

\begin{align*}\partial_{q,p}=\pi_{V_q}\circ \partial_i^{\kern0.5pt f}|_{V_p}\,:\,V_p\to V_q,\end{align*}

where $\pi_{V_q}\,:\,\oplus_{q'\in \text{Crit}_{i-1}(\,f)}V_{q'}\to V_q$ is the projection. Similarly, a combinatorial propagator $g^{\kern0.5pt f}$ gives a homomorphism

\begin{align*}g_{p,q}=\pi_{V_p}\circ g_i^{\kern0.5pt f}|_{V_q}\,:\,V_q\to V_p,\end{align*}

where $\pi_{V_p}\,:\,\oplus_{r\in \text{Crit}_i(\,f)}V_r\to V_p$ is the projection.

6·2. Preparation for manifolds

Recall that $D\nu_{\Delta}=\{(x,v)\mid \|v\|\le 1\}$ is a disk bundle of the normal bundle $\nu_{\Delta}$ of $\Delta$ in $M\times M$ . $S\nu_{\Delta}=\partial D\nu_{\Delta}$ is the unit sphere bundle of $\Delta$ . We denote by $B\ell(M^2,\Delta)$ the real blowing up of $M^2$ along $\Delta$ , namely

\begin{align*}&B\ell(M^2,\Delta)\notag\\ &\quad =(M^2\setminus \Delta)\cup_{D\nu_{\Delta}\setminus \Delta}\left\{((x,tu),u)\in \nu_{\Delta}\times S\nu_{\Delta}\mid x\in \Delta, u\in (S\nu_{\Delta})_x,t\in [0,\infty) \right\}\notag\\ &\quad \cong (M^2\setminus\Delta)\cup S\nu_{\Delta}.\end{align*}

We denote by $\pi\,:\,B\ell(M^2,\Delta)\to M^2$ the blow-down map. Namely, $\pi|_{M^2\setminus\Delta}=\text{id}\,:\,M^2\setminus\Delta\to M^2\setminus \Delta\subset M^2$ and $\pi|_{S\nu_{\Delta}}\,:\,S\nu_{\Delta}\to \Delta\subset M^2$ are the projections. By construction,

\begin{align*}\text{int} B\ell(M^2,\Delta)=M^2\setminus\Delta,\end{align*}

\begin{align*}\partial B\ell(M^2,\Delta)=\pi^{-1}(\Delta)=S\nu_{\Delta}.\end{align*}

Recall that $\nu_{\Delta}$ is identified with the tangent bundle TM by the bundle isomorphism given by

\begin{align*}\nu_{\Delta}\ni ((x,x),(-v,v))\mapsto (x,v)\in TM,\end{align*}

where $x\in M, v\in T_xM$ . Under the identification, we have $D\nu_{\Delta}=DTM$ and $S\nu_{\Delta}=STM$ . We orient $\partial B\ell(M^2,\Delta)$ as a boundary of $B\ell(M^2,\Delta)$ . Then $S\nu_{\Delta}$ and STM are oriented by the above identification.

Take an open tubular neighbourhood $N(\Delta)$ of $\Delta$ in $M^2$ and identify $N(\Delta)$ with $\nu_{\Delta}$ . We can take a diffeomorphism between $B\ell(M^2,\Delta)$ and $M^2\setminus \mathring{D}\nu_{\Delta}$ by “shrinking” a collar of $\partial B\ell(M^2,\Delta)$ . More precisely, we take the inclusion map

\begin{align*}\iota\,:\,B\ell(M^2,\Delta)\hookrightarrow M^2\end{align*}

as follows: Let $D_2\nu_{\Delta}=\{(x,v)\in \nu_{\Delta}\mid \|v\|\le 2\}\subset M^2$ . Take a $C^{\infty}$ function $h\,:\,[0,\infty)\to[1,\infty)$ satisfying

\begin{align*}\frac{dh}{dt}(t)\ge 0\ \forall\ t\ \text{and}\ h(t)=\begin{cases}t& (t\ge 2)\\ 1 &(t=0)\end{cases}.\end{align*}

The inclusion $\iota\,:\,B\ell(M^2,\Delta)\hookrightarrow M^2$ is given as the canonical extension of the map $\iota_0\,:\,(M^2\setminus\Delta=)\text{int}B\ell(M^2,\Delta)\to B\ell(M^2,\Delta)$ defined by

\begin{eqnarray*}\iota_0|_{B\ell(M^2)\setminus D_2\nu_{\Delta}}&=&\text{id},\\\iota_0|_{D_2\nu_{\Delta}\setminus \Delta}(x,v)&=& \left(x,\frac{h(\|v\|)}{\|v\|}v\right)\text{for}\ (x,v)\in D_2\nu_{\Delta}\setminus \Delta.\end{eqnarray*}

Obviously, $\iota(B\ell(M^2,\Delta))=M^2\setminus \mathring{N}(\Delta)$ . Thus we can identify $B\ell(M^2,\Delta)$ with $M^2\setminus \mathring{N}(\Delta)$ via $\iota$ . Then $\iota(B\ell(M^2,\Delta))\subset M^2$ .

6·3. A Morse theoretic description of $d'(M,\rho,e)$

By using the Morse function f and the gradient-like vector field ${\textrm{grad}}\, f$ , we construct a 4-chain

\begin{align*}\Sigma^0_f\in C_4(M\times M;\,\rho^*\boxtimes \rho)\end{align*}

such that $\partial \Sigma_f^0=v_e(\Delta)\otimes 1$ (Definition 6·5 and Proposition 6·6).

Let $M_{\to}(\,f)$ be the set of all pairs of two points in M connected by a trajectory:

\begin{align*}M_{\to}(\,f)=\{(x,\Phi_t^{\kern0.5pt f}(x))\mid x\in M\setminus \text{Crit}(\,f), t\gt0\}\subset M^2\setminus {\Delta}.\end{align*}

We orient $M_{\to}(\,f)$ so that an inclusion $(M\setminus \text{Crit}(\,f))\times (0,1]\ni(x,t)\mapsto (x,\Phi_t^{\kern0.5pt f}(x))\in M_{\to}(\,f)$ preserves the orientation. There is a compactification $\overline{M}_{\to}(\,f)\subset B\ell(M^2,\Delta)(\subset M^2)$ which consists of all broken trajectories (see [ Reference Shimizu17 , lemma 5·13] or [ Reference Lescop15 , lemma 4·3], [ Reference Watanabe22 , proposition 3·14]). $\overline{M}_{\to}(\,f)$ with the local coefficient given by $1\in \text{Hom}(V,V)$ near $\Delta$ gives a 4-chain in $C_4(B\ell(M^2,\Delta);\,\rho^*\boxtimes \rho)$ and in $C_4(M^2;\,\rho^*\boxtimes \rho)$ :

\begin{align*}\overline{M}_{\to}(\,f)=\langle \overline{M}_{\to}(\,f),\text{pt};\,1\rangle=\overline{M}_{\to}(\,f)\otimes 1,\end{align*}

where pt is any point of $\partial B\ell(M^2,\Delta)$ .

Similarly, we have compactifications $\overline{\mathcal D_p}$ of $\mathcal D_p$ and $\overline{\mathcal A_q}$ of $\mathcal A_q$ by adding all the broken trajectories (see [ Reference Shimizu17 , lemma 5·14] or [ Reference Watanabe22 , proposition 3·14]). For $p\in\text{Crit}_i(\,f)$ and $q\in\text{Crit}_{i-1}(\,f)$ , the combinatorial propagator $g^{\kern0.5pt f}$ gives a homomorphism $1\otimes g_{p,q}^{\kern0.5pt f}\,:\,V_q^*\otimes V_q\to V_q^*\otimes V_p$ . Thus we have a 4-chain

\begin{align*}\langle \overline{\mathcal A_q}\times \overline{\mathcal D_p},(q,p);\, (1\otimes g_{p,q}^{\kern0.5pt f})1\rangle\in C_4(M^2;\,\rho^*\boxtimes \rho).\end{align*}

Under the identification $V_{q}^*\otimes V_p=\text{Hom}(V_q,V_p)$ , the coefficient $(1\otimes g_{p,q}^{\kern0.5pt f})1\in V_q^*\otimes V_p$ can be written as $g_{p,q}^{\kern0.5pt f}\in \text{ Hom}(V_q,V_p)$ :

\begin{align*}\langle \overline{\mathcal A_q}\times \overline{\mathcal D_p},(q,p);\, (1\otimes g_{p,q}^{\kern0.5pt f})1\rangle=\langle \overline{\mathcal A_q}\times \overline{\mathcal D_p},(q,p);\, g_{p,q}^{\kern0.5pt f}\rangle.\end{align*}

To simplify the notation, we use the notation on the right hand side rather than the left one. Let

\begin{align*}\langle (\overline{\mathcal A_q}\times \overline{\mathcal D_p})^{B\ell},(q,p);\, g_{p,q}^{\kern0.5pt f}\rangle\in C_4(B\ell(M^2,\Delta);\,\rho^*\boxtimes \rho)\end{align*}

be the 4-chain given from the closure of $\overline{\mathcal A_q}\times \overline{\mathcal D_p}\setminus\Delta$ in $B\ell(M^2,\Delta)$ . See [ Reference Watanabe22 , proposition 3·14] for more details.

Set

\begin{eqnarray*}\mathcal M_0(\,f)=\langle \overline{M}_{\to}(\,f),\text{pt};\,1\rangle+\sum_{p,q;\,\text{ind}(p)=\text{ind}(q)+1}\langle (\overline{\mathcal {A}_q}\times \overline{\mathcal{D}_p})^{B\ell},(q,p);\,g^{\kern0.5pt f}_{p,q}\rangle&&\\\in C_4(B\ell(M^2,\Delta);\,\rho^*\boxtimes \rho).&&\end{eqnarray*}

The 4-chain $\overline{M}_{\to}(\,f)$ has

\begin{eqnarray*}\left\{\frac{-{\textrm{grad}}_x{\kern1pt f}}{\|{\textrm{grad}}_x{\kern1pt f}\|}\middle| x\in M\setminus \text{Crit}(\,f)\right\}\otimes 1&&\in C_3(\partial B\ell(M^2,\Delta);\,\rho^*\otimes \rho)\\ &&= C_3(STM;\,\rho^*\otimes \rho)\end{eqnarray*}

as a part of the boundary $\partial \overline{M}_{\to}(\,f)$ .

Due to our orientation conventions, we have

\begin{align*}(\overline{\mathcal{A}_q}\times \overline{\mathcal{D}_p})\cap\Delta=(-1)^{(3-\dim \mathcal A_q)\dim \mathcal D_p}\overline{\mathcal{A}_q}\cap \overline{\mathcal{D}_p}=\overline{\mathcal{A}_q}\cap \overline{\mathcal{D}_p}\end{align*}

and

\begin{align*}\overline{\mathcal{A}_q}\cap \overline{\mathcal{D}_p}=\sum_{\gamma\in\mathcal M(p,q)}\gamma_{\cap}.\end{align*}

Thus $\langle(\overline{\mathcal{A}_q}\times \overline{\mathcal{D}_p})^{B\ell},(q,p);\,g_{p,q}^{\kern0.5pt f}\rangle$ has

\begin{eqnarray*}\sum_{\gamma\in\mathcal M(p,q)}\pi^{!}\langle \gamma_{\cap},p;\,g_{p,q}^{\kern0.5pt f}\circ\gamma_*\rangle=\sum_{\gamma\in\mathcal M(p,q)}\pi^{-1}(\gamma_{\cap})\otimes (g_{p,q}^{\kern0.5pt f}\circ\gamma_*)\in C_3(S\nu_{\Delta};\,\rho^*\otimes \rho)\end{eqnarray*}

as a part of the boundary. Here $\pi\,:\,S\nu_{\Delta}=STM\to M$ is the projection.

By a similar argument as in [ Reference Shimizu17 , lemma 5·16] (or [ Reference Shimizu18 , proposition 6·2]), we can check that the other boundary strata of $\partial \overline{M}_{\to}(\,f)$ and $\sum_{p,q}\partial (\mathcal{A}_q\times \mathcal{D}_p)^{B\ell}$ with local coefficients cancel: $\sum_{p}\langle \overline{\mathcal{A}}_p\times\overline{\mathcal{D}}_p,(p,p);\,1\rangle$ which appears with a plus sign in the boundary of $\sum_{p,q}(\mathcal{A}_q\times \mathcal{D}_p)^{B\ell}$ and with a minus sign in the boundary of $\overline{M}_{\to}(\,f)$ . Then we have

Lemma 6·1.

\begin{align*}&\partial\mathcal M_0(\,f)\notag\\ &\def\negativespace{}\def\negativespace{}\def\negativespace{}\quad=\overline{\left\{ \left(x,\frac{-{\textrm{grad}}_x{f}}{\|{\textrm{grad}}_x(\,f)\|}\right)\middle| x\in M\setminus \text{Crit}(\,f) \right\}}\otimes 1\\&\quad+\sum_{\substack{p,q;\\ \text{ind}(p)=\text{ind}(q)+1}}\sum_{\gamma\in\mathcal M(p,q)}\pi^!\langle\gamma_{\cap},p;\,g_{p,q}^{\kern0.5pt f}\circ\gamma_*\rangle \notag\\ &\def\negativespace{}\def\negativespace{}\def\negativespace{}\quad=\overline{\left\{ \left(x,\frac{-{\textrm{grad}}_x{f}}{\|{\textrm{grad}}_x(\,f)\|}\right)\mid x\in M\setminus \text{Crit}(\,f) \right\}} \def\negativespace{}\otimes 1\\&\quad+\sum_{\substack{p,q;\\ \text{ind}(p)=\text{ind}(q)+1}}\sum_{\gamma\in\mathcal M(p,q)}\pi^{-1}(\gamma_{\cap})\otimes (g_{p,q}^{\kern0.5pt f}\circ\gamma_*).\end{align*}

Set

\begin{eqnarray*}\Gamma&=&\sum_{p,q;\,\text{ind}(p)=\text{ind}(q)+1}\sum_{\gamma\in\mathcal M(p,q)}\langle\gamma_{\cap},p;\,g_{p,q}^{\kern0.5pt f}\circ\gamma_*\rangle\\ &=& \sum_{p,q;\,\text{ind}(p)=\text{ind}(q)+1}\sum_{\gamma\in\mathcal M(p,q)}\gamma_{\cap} \otimes (g_{p,q}^{\kern0.5pt f}\circ\gamma_*)\in C_1(M;\,\rho^*\otimes \rho).\end{eqnarray*}

Let

\begin{align*}c^0(\,f)=\sum_{p\in\text{Crit}(\,f)}(-1)^{\text{ind}(p)}p\end{align*}

be the 0-chain given by critical points. The 1-chain $\Gamma$ has the following property:

Lemma 6·2.

\begin{align*}\partial \Gamma=-c^0(\,f)\otimes 1\in C_0(M;\,\rho^*\otimes \rho).\end{align*}

Proof. Let $c_p$ be the coefficient of $p\in \text{Crit}(\,f)$ in $\partial \Gamma$ . We show that $c_p=(-1)^{\text{ ind}(p)+1}$ . If a trajectory $\gamma$ is from a critical point $r\in \text{Crit}_{\text{ind}(p)+1}(\,f)$ to p and $\gamma_{\cap}$ is oriented from p to r, then the coefficient of p in $\partial \gamma_{\cap}$ is $-1$ and ${\varepsilon}(\gamma)=-1$ . If the orientation of $\gamma_{\cap}$ is opposite, namely from r to p, then the coefficient of p in $\partial \gamma_{\cap}$ is $+1$ and ${\varepsilon}(\gamma)=+1$ . If a trajectory $\gamma_{\cap}$ connecting p and $q\in \text{Crit}_{\text{ind}(p)-1}(\,f)$ is oriented from p to q, then the coefficient of p in $\partial \gamma_{\cap}$ is $-1$ and ${\varepsilon}(\gamma)=+1$ . If the orientation of $\gamma_{\cap}$ is opposite, namely from q to p, then the coefficient of p in $\partial\gamma_{\cap}$ is $+1$ and ${\varepsilon}(\gamma)=-1$ . Therefore $c_p$ is computed as follows:

\begin{eqnarray*}c_p&=&\sum_{r\in\text{Crit}_{\text{ind}(p)+1}(\,f)}\sum_{\gamma\in \mathcal M(r,p)}{\varepsilon}(\gamma)g_{r,p}^{\kern0.5pt f}\circ\gamma_*+\sum_{q\in\text{Crit}_{\text{ind}(p)-1}(\,f)}\sum_{\gamma\in\mathcal M(p,q)}-{\varepsilon}(\gamma)g_{p,q}^{\kern0.5pt f}\circ\gamma_*\\ &=& \sum_{r\in\text{Crit}_{\text{ind}(p)+1}(\,f)}(-1)^{\text{ind}(p)+1}g_{r,p}^{\kern0.5pt f}\circ\partial_{p,r}^{\kern0.5pt f}+\sum_{q\in\text{Crit}_{\text{ind}(p)-1}(\,f)}(-1)^{\text{ind}(p)+1}g_{p,q}^{\kern0.5pt f}\circ\partial_{q,p}^{\kern0.5pt f}\\ &=&(-1)^{\text{ind}(p)+1}\pi_{V_p}\circ \left(\partial^{\kern0.5pt f}\circ g^{\kern0.5pt f}+g^{\kern0.5pt f}\circ\partial^{\kern0.5pt f}\right)|_{V_p}(1)\\ &=&(-1)^{\text{ind}(p)+1}.\end{eqnarray*}

Here $\pi_{V_p}\,:\,\sum_{p'\in \text{Crit}_{\text{ind}(p)}}V_{p'}\to V_p$ is the projection.

We prepare two more chains.

Take an Euler chain $c_f$ for $-{\textrm{grad}}\, f$ (see section 2·5 for more details on Euler chains). Take a section $\widetilde s\in\Gamma p_2^*TM$ of $p_2^*TM$ on $[0,1]\times M$ so that $\widetilde s|_{0\times M}=-{\textrm{grad}}\, f$ as in lemma 2·2.

Then $\widetilde s^{-1}(0)$ is a 1-manifold of $[0,1]\times M$ such that $\partial \widetilde s^{-1}(0)=c^0(\,f)$ .

The restriction of $\widetilde s|_{0\times M}=-{\textrm{grad}}\, f$ to the 4-manifold

\begin{align*}\widetilde{S}=\overline{\left\{\frac{\widetilde{s}(x)}{\|\widetilde{s}(x)\|}\middle|x\in [0,1]\times M\setminus \widetilde{s}^{-1}(0)\right\}}\subset [0,1]\times STM\end{align*}

gives a 4-chain in $STM=\partial B\ell(M^2,\Delta)$ :

\begin{align*}\left((p_2)_*\widetilde{S}\right)\otimes 1\in C_4(\partial B\ell(M^2,\Delta);\,\rho^*\otimes \rho).\end{align*}

Since $\widetilde{s}|_{\{0\}\times M}=-{\textrm{grad}}\, f$ , $\widetilde{s}|_{\{1\}\times M}=-{\textrm{grad}}\, f/c_f$ and $p_2(\widetilde{s}^{-1}(0))=c_f$ , we have the following lemma.

Lemma 6·3.

\begin{align*}&\partial \left(((p_2)_*\widetilde{S})\otimes 1\right)\\ & \quad = \pi^{-1}(c_f)\otimes 1\\ & \qquad -\overline{\left\{\frac{-{\textrm{grad}}_x{\kern1.5pt f}}{\|{\textrm{grad}}_x{\kern1.5pt f}\|}\middle| x\in M\setminus \text{Crit}(\,f)\right\}}\otimes 1+\left\{\frac{-({\textrm{grad}}\, f/c_f)_x}{\|({\textrm{grad}}\, f/c_f)_x\|}\middle| x\in M\right\}\otimes 1.\end{align*}

The disk bundle $D\nu_{\Delta}=DTM$ forms a $D^3$ -bundle over M. We orient $D\nu_{\Delta}$ such that $\partial D\nu_{\Delta}=-S\nu_{\Delta}=-\partial B\ell(M^2,\Delta)$ . Then $B\ell(M^2,\Delta)\cup D\nu_{\Delta}=M^2$ . Let $\widetilde{\pi}\,:\,D\nu_{\Delta}=DTM\to M$ be the projection. Thanks to lemma 6·2,

\begin{align*}(c_f\otimes 1)+\Gamma\end{align*}

is a 1-cycle of $C_1(M;\,\rho^*\otimes \rho)$ . Thus we have a 4-chain

\begin{align*}\widetilde{\pi}^{!}((c_f\otimes 1)+\Gamma)\in C_4(D\nu_{\Delta};\,\rho^*\otimes \rho).\end{align*}

By construction, the boundary of this chain is written as follows:

Lemma 6·4.

\begin{eqnarray*}\partial(\widetilde{\pi}^{!}((c_f\otimes 1)+\Gamma))&=&-\pi^{!}((c_f\otimes 1)+\Gamma)\\ &=&-((\pi^{!}c_f)\otimes 1)-\pi^!\Gamma.\end{eqnarray*}

Here $\pi\,:\,STM(=\partial D\nu_{\Delta})\to M$ is the projection.

Definition 6·5.

\begin{align*}\Sigma_0^{\kern0.5pt f}=\iota_*\mathcal M_0(\,f)+((p_2)_*\widetilde{S})\otimes 1+\widetilde{\pi}^{-1}((c_f\otimes 1)+\Gamma)\in C_4(M\times M;\,\rho^*\boxtimes \rho).\end{align*}

Here $\iota\,:\,B\ell(M^2,\Delta)\to M^2$ is an inclusion defined in the last part of section 6·2.

The following proposition is a direct consequence of Lemmas 6·1, 6·3, and 6·4.

Proposition 6·6.

\begin{align*}\partial \Sigma_0^{\kern0.5pt f}=\left\{\frac{-({\textrm{grad}}\, f/c_f)_x}{\|({\textrm{grad}}\, f/c_f)_x\|}\middle| x\in M\right\}\otimes 1=v_e(\Delta)\otimes 1.\end{align*}

$d'(M,\rho;\,e)$ is defined as $d'(M,\rho;\,e)=\text{lk}_{\rho^*\boxtimes\rho}(v_e(\Delta)\otimes 1,\Delta)=[\Sigma_0^{\kern0.5pt f}\cap \Delta]$ . Thus we have the following formula.

Proposition 6·7.

\begin{eqnarray*}d'(M,\rho;\,e)&=&[(c_f\otimes 1)+\Gamma]\\ &=&[(c_f\otimes1)+\sum_{p,q;\,\text{ind}(p)=\text{ind}(q)+1}\sum_{\gamma\in \mathcal M(p,q)}\gamma_{\cap}\otimes (g_{p,q}^{\kern0.5pt f}\circ\gamma_*)]\\ &&\in H_1(M;\,\rho^*\otimes\rho).\end{eqnarray*}

Proof. By the definition of $d'(M,\rho;\,e)$ , we have

\begin{eqnarray*}d'(M,\rho;\,e)&=&[\Sigma_0^{\kern0.5pt f}\cap \Delta]\\ &=&[\left(\mathcal M_0(\,f)+((p_2)_*\widetilde S)\otimes 1+\widetilde{\pi}^{!}((c_f\otimes 1)+\Gamma)\right)\cap\Delta].\end{eqnarray*}

Since $\iota_*\mathcal M_0(\,f)$ and $(p_2)_*\widetilde S$ are in $B\ell(M^2,\Delta)$ , these are far from $\Delta$ . Thus we have

\begin{align*}d'(M,\rho;\,e)=[\Sigma_0^{\kern0.5pt f}\cap \Delta]=[\widetilde{\pi}^{!}((c_f\otimes 1)+\Gamma)\cap \Delta]=[(c_f\otimes 1)+\Gamma].\end{align*}

Let $B_{c_f}$ be a 3-ball in M containing all neighbourhoods $N(c_1),\ldots$ of intervals $c_1,\ldots$ consisting $c_f$ . Then any trajectory $\gamma_{\downarrow}$ can be transformed into a 1-cycle called $\gamma{}'_{\downarrow}$ by adding a 1-chain supported in $B_{c_f}$ . Similarly, a 1-cycle $\gamma_{\cap}'$ is given from $\gamma_{\cap}$ . By using these cycles, we have the following slightly different description of $d'(M,\rho;\,e)$ :

Proposition 6·8.

\begin{align*}d'(M,\rho;\,e)=\sum_{p,q;\,\text{ind}(p)=\text{ind}(q)+1}\sum_{\gamma\in\mathcal M(p,q)}[\gamma_{\cap}']\otimes (g_{p,q}^{\kern0.5pt f}\circ\gamma_*)\in H_1(M;\,\rho^*\otimes \rho).\end{align*}

Here $[\gamma_{\cap}']$ is a 1-cycle of $H_1(M;\,{\mathbb{Z}})$ represented by $\gamma_{\cap}'$ .

6·4. Behaviours of d ^′ and Tor under replacing an Euler structure

Let $c_f'$ be an Euler chain for f. Let $e'=[-{\textrm{grad}}\, f/c_f']\in \text{Eul}(M)$ . Since $\partial c_f'=\partial c_f=c^0(\,f)$ , the difference $c_f'-c_f$ is a cycle in $C_1(M;\,{\mathbb{Z}})$ and then it represents a homology class $[c_f'-c_f]$ in $H_1(M;\,{\mathbb{Z}})$ .

(ii) Let $c_1,\ldots,c_k$ be intervals consisting $c_f$ , namely $c_f=\sqcup_{i=1}^kc_i$ . Here $k=\sharp \text{ Crit}(\,f)/2$ . Similarly let $c_f'=\sqcup_{i=1}^kc_i'$ . Take an arbitrary isomorphism $V_x\stackrel{\cong}{\to}V$ for each $x\in \text{Crit}(\,f)$ . Then each $(c_i)_*$ or $(c_i)'_*$ can be considered as an element of $\text{Hom}(V,V)=V$ . The isomorphism $(c_f)_*\,:\,C_{odd}\to C_{even}$ is identified with an isomorphism from $V^{\oplus k}$ to $V^{\oplus k}$ . Thus $\det(c_f)_*$ is given by $\det(c_f)_*=\pm \det(c_1)_*\ldots \det(c_k)_*$ . We have

\begin{eqnarray*}\text{Tor}(M,\rho;\,e')\text{Tor}(M,\rho;\,e)^{-1}&=&\det ((c_f')_*\circ (\partial_*^{\kern0.5pt f}+g^{\kern0.5pt f}))\det ((c_f)_*\circ \partial_*^{\kern0.5pt f}+g^{\kern0.5pt f})^{-1}\\ &=& (\det (c_f')_*)(\det (c_f)_*)^{-1} \\ &=&\det(\Pi_{i=1}^k(c_i')_*\Pi_{i=1}^k(c_i)_*^{-1})\\ &=&\det\rho([c_f'-c_f])\in F^{\times}/{\pm}.\end{eqnarray*}

7·1. An explicit description of the boundary operators $\partial_*^{\kern0.5pt f}$ and a propagator $g^{\kern0.5pt f}$

From now we consider $\rho_0\,:\,\pi_1(M)\to Q(H)$ .

Thanks to lemma 6·9, it is enough to show the theorem for a special Euler structure. Then we choose a special Morse function and an Euler chain for it to simplify the computation. We take a Morse function such that $\sharp \text{Crit}_3(\,f)=\sharp\text{Crit}_0(\,f)=1$ . Let

\begin{eqnarray*}&&\text{Crit}_3(\,f)=\{\text{NP}\},\\ &&\text{Crit}_2(\,f)=\{p_1,\ldots, p_n\},\\ &&\text{Crit}_1(\,f)=\{q_1,\ldots, q_n\},\\ &&\text{Crit}_0(\,f)= \{\text{SP}\}.\end{eqnarray*}

We denote by

\begin{eqnarray*}&&C_{\text{NP}}=C_3^{\kern0.5pt f}(M;\,\rho_0)=Q(H)_{NP},\\ &&C_{\bf p}=C_2^{\kern0.5pt f}(M;\,\rho_0)=\oplus_{i=1}^nQ(H)_{p_i},\\ &&C_{\bf q}=C_1^{\kern0.5pt f}(M;\,\rho_0)=\oplus_{i=1}^nQ(H)_{q_i},\\ &&C_{\text{SP}}=C_0^{\kern0.5pt f}(M;\,\rho_0)=Q(H)_{SP}.\end{eqnarray*}

Let $c_f$ be an Euler chain for f and let $e=[-{\textrm{grad}}\, f/c_f]$ . Take a 3-ball $B_{c_f}$ in M containing all neighbourhoods $N(c_1),\ldots,N(c_{n+1})$ of intervals $c_1,\ldots,c_{n+1}$ consisting the Euler chain $c_f$ . We identify $Q(H)_{\text{NP}}$ with Q(H). Then $Q(H)_p$ , $p\in \text{Crit}(\,f)$ , are identified with $Q(H)_{\text{NP}}=Q(H)$ via isomorphisms $\phi_*$ associated to paths $\phi$ inside $B_{c_f}$ .

Two bases ${b}_e$ and ${b}_f$ of $C^{\kern1pt f}_*(M;\,\rho_0)$

To simplify the computations, we first introduce two bases $\text{b}_e$ and $\text{b}_f$ of $C_*^{\kern0.5pt f}(M,\rho_0)$ .

For each $p\in \text{Crit}(\,f)$ , we denote by $p\in Q(H)_p$ the element corresponding to the generator $1\in Q(H)$ under the identification $Q(H)_p=Q(H)$ . Therefore, we now have a basis ${\bf b}_e$ of $C_*^{\kern0.5pt f}(M,\rho_0)$ :

\begin{align*}{\bf b}_e=\{\text{NP},p_1,\ldots,p_n,q_1,\ldots,q_n,\text{SP}\}.\end{align*}

We next introduce the basis ${\bf b}_f$ . Let $\pi_{Q(H)_{p_1}}\,:\,C_{\bf p}\to Q(H)_{p_1}$ be the projection. Without loss of generality, we assume that $\pi_{Q(H)_{p_1}}\circ\partial_3^{\kern0.5pt f}(\text{NP})\not=0$ . We take a basis of $C_{\bf p}$ as

\begin{align*}\partial_3^{\kern0.5pt f}(\text{NP}),p_2,\ldots, p_n.\end{align*}

Since $\dim_{Q(H)}\text{Im}\partial_2^{\kern0.5pt f}=n-1\not= n=\dim_{Q(H)}C_{\bf q}$ , at least one of $q_1,\ldots,q_n$ does not belong to $\text{Im} \partial_2^{\kern0.5pt f}\subset C_{\bf q}$ . We may assume that $q_1\not\in \text{Im}\partial_2^{\kern0.5pt f}$ . Then

\begin{align*}q_1,\partial_2^{\kern0.5pt f}(p_2),\ldots, \partial_2^{\kern0.5pt f}(p_n)\end{align*}

is a basis of $C_{\bf q}$ . Since $H_0(M;\,\rho_0)=0$ , then $\partial_1^{\kern0.5pt f}(q_1)\not=0$ . We take $\partial_1^{\kern0.5pt f}(q_1)$ as a basis of $C_{\text{SP}}$ . Now we have a basis ${\bf b}_f$ :

\begin{align*}{\bf b}_f=\{\text{NP}, \partial_3^{\kern0.5pt f}(\text{NP}),p_2,\ldots, p_n,q_1,\partial_2^{\kern0.5pt f}(p_2),\ldots, \partial_2^{\kern0.5pt f}(p_n),\partial_1^{\kern0.5pt f}(q_1)\}.\end{align*}

Let $(\partial_1^{ij})_{ij}, (\partial_2^{ij})_{ij}$ and $(\partial_3^{ij})_{ij}$ be representation matrices of $\partial_1^{\kern0.5pt f},\partial_2^{\kern0.5pt f}$ and $\partial_3^{\kern0.5pt f}$ respectively under the basis ${\bf b}_e$ . Namely,

\begin{eqnarray*}\partial_3^{\kern0.5pt f}(\text{NP})&=&(p_1,\ldots,p_n)\left(\begin{array}{c}\partial^{11}_3\\\vdots\\\partial^{n1}_3\end{array}\right),\\\partial_2^{\kern0.5pt f}(p_1,\ldots,p_n)&=&(q_1,\ldots,q_n)\left(\begin{array}{ccc}\partial^{11}_2& \cdots& \partial_2^{1n}\\\vdots &\ddots &\vdots\\\partial^{n1}_2&\cdots &\partial_2^{nn}\end{array}\right),\\\partial_1^{\kern0.5pt f}(q_1,\ldots,q_n)&=&(\text{SP})(\partial_1^{11}~\cdots~\partial_1^{1n}).\end{eqnarray*}

A propagator $g^{\kern0.5pt f}$

We give a combinatorial propagator $g^{\kern0.5pt f}$ explicitly written by using the basis ${\bf b}_f$ as follows:

1. (i) $g^{\kern0.5pt f}_3(\partial_3^{\kern0.5pt f}(\text{NP}))=\text{NP}$ , $g^{\kern0.5pt f}_3(p_2)=0,\ldots,g^{\kern0.5pt f}_3(p_n)=0$ ;

2. (ii) $g^{\kern0.5pt f}_2(q_1)=0$ , $g^{\kern0.5pt f}_2(\partial^{\kern0.5pt f}_2(p_2))=p_2,\ldots,g^{\kern0.5pt f}_2(\partial^{\kern0.5pt f}_2(p_n))=p_n$ ;

3. (iii) $g^{\kern0.5pt f}_1(\partial^{\kern0.5pt f}_1(q_1))=q_1$ .

Obviously, these homomorphisms $g^{\kern0.5pt f}=\{g_i\}_i$ satisfies $\partial_{i+1}^{\kern0.5pt f}\circ g_{i+1}^{\kern0.5pt f}+g_i^{\kern0.5pt f}\circ \partial_i^{\kern0.5pt f}=\text{id}$ for any i, namely $g^{\kern0.5pt f}$ is a combinatorial propagator.

An explicit description of $g^{\kern0.5pt f}$ with respect to the basis ${\bf b}_e$ .

1. (i) A description of $g^{\kern0.5pt f}_3$

   We have $\partial^{\kern0.5pt f}_3(\text{NP})=\sum_{i=1}^n\partial_3^{i1}p_i$ . Since $g_3^{\kern0.5pt f}(p_i)=0$ for $i\ge 2$ , we have $\text{NP}=g_3^{\kern0.5pt f}(\partial_3^{\kern0.5pt f}(\text{NP}))=g_3^{\kern0.5pt f}(\partial_3^{11}p_1)=\partial_3^{11}g_3^{\kern0.5pt f}(p_1)$ . Then $g_3^{\kern0.5pt f}(p_1)=(\partial_3^{11})^{-1}(NP)$ . We have

   \begin{align*}g^{\kern0.5pt f}_{\text{NP},p_1}=(\partial_3^{11})^{-1}, g^{\kern0.5pt f}_{\text{NP},p_2}=0,\ldots, g^{\kern0.5pt f}_{\text{NP},p_n}=0.\end{align*}

2. (ii) A description of $g^{\kern0.5pt f}_2$

   Since $g_2^{\kern0.5pt f}(q_1)=0$ , it suffices to determine the restriction $g_2^{\kern0.5pt f}|_{\langle q_2,\ldots,q_n\rangle}$ of $g_2^{\kern0.5pt f}$ to the linear span $\langle q_2,\ldots,q_n\rangle$ . We know that the image of $g_2^{\kern0.5pt f}$ is the linear span $\langle p_2,\ldots,p_n\rangle$ and that $g_2^{\kern0.5pt f}|_{\langle q_2,\ldots,q_n\rangle}$ is the inverse of the map $\Pi\circ \partial_2^{\kern0.5pt f}|_{\langle p_2,\ldots,p_n\rangle}$ , where $\Pi\,:\,C_{\bf q}\to \langle q_2,\ldots,q_n\rangle$ is the projection with respect to the basis ${\bf b}_e$ . The map $\Pi\circ \partial_2^{\kern0.5pt f}|_{\langle p_2,\ldots,p_n\rangle}$ is represented as

   \begin{eqnarray*}\Pi\circ \partial_2^{\kern0.5pt f}|_{\langle p_2,\ldots,p_n\rangle}(p_2,\ldots,p_n)=(q_2,\ldots,q_n)A_2.\end{eqnarray*}
   Here the matrix $A_2$ is defined by
   \begin{align*}A_2=\left(\begin{array}{ccc}\partial_2^{22}&\cdots&\partial_2^{2n}\\ \vdots&\ddots&\vdots\\\partial_2^{n2}&\cdots&\partial_2^{nn}\end{array}\right).\end{align*}
   Let $(A_2^{ij})$ be the (i, j)-cofactor of $A_2$ , namely
   \begin{align*}\left(\frac{1}{\det A_2}A_2^{ji}\right)_{i,j}=A_2^{-1}.\end{align*}

   Therefore we have

   \begin{align*}g^{\kern0.5pt f}_{p_j,q_i}=\begin{cases}\frac{1}{\det A_2}A_2^{ji}&(i,j\ge 2),\\0& i=1\text{or}\, j=1.\end{cases}\end{align*}

3. (iii) A description of $g^{\kern0.5pt f}_1$

   Since $\partial_1^{\kern0.5pt f}(q_1)=\partial_1^{11}\text{SP}$ , we have

   \begin{align*}g_1^{\kern0.5pt f}(\text{SP})=(\partial_1^{11})^{-1} g_1^{\kern0.5pt f}(\partial_1^{11}\text{SP})=(\partial_1^{11})^{-1}g_1^{\kern0.5pt f}(\partial_1^{\kern0.5pt f}(q_1))=(\partial_1^{11})^{-1}q_1.\end{align*}

   Then we have

   \begin{align*}g^{\kern0.5pt f}_{q_1,\text{SP}}=(\partial_1^{11})^{-1}, g^{\kern0.5pt f}_{q_2,\text{SP}}=0,\ldots, g^{\kern0.5pt f}_{q_n,\text{SP}}=0.\end{align*}

7·2. The computation of $D(\text{Tor}(M,\rho_0;\,e))$

Under the identification $Q(H)_p=Q(H)$ for $p\in\text{Crit}(\,f)$ by using $B_{c_f}$ , each $c_i$ consisting $c_f$ gives $(c_i)_*=1$ . Then $\text{Tor}(M,\rho_0;\,e)=\det(\partial^{\kern0.5pt f}+g^{\kern0.5pt f})$ .

The isomorphism

\begin{align*}\partial^{\kern0.5pt f}+g^{\kern0.5pt f}=g_3^{\kern0.5pt f}+\partial_2^{\kern0.5pt f}+g_1^{\kern0.5pt f}\,:\,C_{even}^{\kern0.5pt f}\to C_{odd}^{\kern0.5pt f}\end{align*}

is represented by the following matrix with respect to the basis ${\bf b}_e$ :

\begin{eqnarray*}(g_3^{\kern0.5pt f}+\partial_2^{\kern0.5pt f}+g_1^{\kern0.5pt f})(p_1,p_2,\ldots,p_n,\text{SP})=(\text{NP}, q_1, q_2,\ldots,q_n)T.\end{eqnarray*}

Here

\begin{eqnarray*}T=\left(\begin{array}{ccccc}(\partial_3^{11})^{-1}&0&\cdots&0&0\\[2pt]\partial_2^{11}&\partial_2^{12}&\cdots&\partial_2^{1n}&(\partial_1^{11})^{-1}\\[3pt]\partial_2^{21}&\partial_2^{22}&\cdots&\partial_2^{2n}&0\\\vdots&\vdots&\ddots&\vdots&\vdots\\\partial_2^{n1}&\partial_2^{n2}&\cdots&\partial_2^{nn}&0\end{array}\right).\end{eqnarray*}

Therefore, we have

\begin{eqnarray*}\text{Tor}(M,\rho_0;\,e)=\det T=(-1)^{n-1}(\partial_3^{11})^{-1}(\det A_2)(\partial_1^{11})^{-1}.\end{eqnarray*}

We next compute $D(\text{Tor}(M,\rho_0;\,e))$ . To simplify the notations, we use same symbol $\gamma{}'_{\downarrow}$ for an element of H represented by a cycle $\gamma{}'_{\downarrow}$ . In other word, $\gamma{}'_{\downarrow}=\gamma_{*}(1)\in H.$ Recall that $[\gamma{}'_{\downarrow}]_+$ is an element of $H_1(M;\,{\mathbb{Z}})/\text{Tor}$ corresponding to $\gamma{}'_{\downarrow}\in H$ .

Since the boundary homomorphisms $\partial_i^{\kern0.5pt f}$ are written by a rational function of $\gamma{}'_{\downarrow}$ for all $\gamma$ , the Reidemeister–Turaev torsion $\text{Tor}(M,\rho_0;\,e)$ is also written by a rational function of all $\gamma{}'_{\downarrow}$ . Then thanks to Proposition 5·6, $D(\text{Tor}(M,\rho_0;\,e))$ is given by

\begin{eqnarray*}D(\text{Tor}(M,\rho_0;\,e))&=&\sum_{\gamma}\left([\gamma{}'_{\downarrow}]_+\otimes \frac{\gamma{}'_{\downarrow}}{\text{Tor}(M,\rho_0;\,e)}\frac{\partial}{\partial \gamma{}'_{\downarrow}}\text{Tor}(M,\rho_0;\,e)\right)\\ &=&\sum_{\gamma}\left([\gamma{}'_{\downarrow}]_+\otimes \frac{\gamma{}'_{\downarrow}}{\text{Tor}(M,\rho_0;\,e)}\frac{\partial}{\partial \gamma{}'_{\downarrow}}(-1)^{n-1}(\partial_3^{11})^{-1}(\det A_2)(\partial_1^{11})^{-1}\right)\\ &=&\sum_{\gamma}\left([\gamma{}'_{\downarrow}]_+\otimes A(\gamma)\right),\end{eqnarray*}

where $\displaystyle A(\gamma)=-\frac{\gamma{}'_{\downarrow}}{\partial_3^{11}}\frac{\partial}{\partial \gamma{}'_{\downarrow}}\partial_3^{11}+\frac{\gamma{}'_{\downarrow}}{\det A_2}\frac{\partial}{\partial \gamma{}'_{\downarrow}}\det A_2-\frac{\gamma{}'_{\downarrow}}{\partial_1^{11}}\frac{\partial}{\partial \gamma{}'_{\downarrow}}\partial_1^{11}$ .

We compute $A(\gamma)$ for each trajectory $\gamma$ . We denote by $s(\gamma)=p,t(\gamma)=q$ for a trajectory $\gamma$ from p to q. We break the computation into cases depending on $s(\gamma)$ and $t(\gamma)$ .

Case 1: $s(\gamma)=\text{NP},t(\gamma)=p_i$

In this case the term $\gamma'_{\downarrow}$ only appears in $\partial_3^{i1}$ . We note that $\partial_3^{i1}=\sum_{\gamma\in \mathcal M(\text{NP},p_i)}-{\varepsilon}(\gamma)\gamma{}'_{\downarrow}$ . Thus

\begin{eqnarray*}A(\gamma)&=&-\frac{\gamma{}'_{\downarrow}}{\partial_3^{11}}\frac{\partial}{\partial \gamma{}'_{\downarrow}}\partial_3^{11}=\begin{cases}{\varepsilon}(\gamma)\frac{\gamma'}{\partial_3^{11}}&(i=1),\\0&(i\not=1).\end{cases}\end{eqnarray*}

Case 2-1. $s(\gamma)=\text{p}_i,t(\gamma)=q_j$ , $i=1$ or $j=1$

In this case there are no $\gamma{}'_{\downarrow}$ in $(\partial_3^{11})^{-1}(\det A_2)(\partial_1^{11})^{-1}$ . Then we have

\begin{align*}A(\gamma)=0.\end{align*}

Case 2-2. $s(\gamma)=\text{p}_i,t(\gamma)=q_j$ , $i,j\ge 2$

\begin{eqnarray*}A(\gamma)&=&\frac{\gamma{}'_{\downarrow}}{\det A_2}\frac{\partial}{\partial \gamma{}'_{\downarrow}}\det A_2=\frac{\gamma{}'_{\downarrow}}{\det A_2}\frac{\partial}{\partial \gamma_{\downarrow'}}\left(\sum_{a=2}^n \partial_2^{ai}A_2^{ai}\right)=\frac{\gamma{}'_{\downarrow}}{\det A_2}{\varepsilon}(\gamma)A_2^{ji}.\end{eqnarray*}

Case 3. $s(\gamma)=q_j,t(\gamma)=\text{SP}$

\begin{eqnarray*}A(\gamma)&=&-\frac{\gamma{}'_{\downarrow}}{\partial_1^{11}}\frac{\partial}{\partial \gamma{}'_{\downarrow}}\partial_1^{11}=\begin{cases}{\varepsilon}(\gamma)\frac{\gamma{}'_{\downarrow}}{\partial_1^{11}}&(j=1),\\0&(j\not=1).\end{cases}\end{eqnarray*}

Therefore, $D(\text{Tor}(M,\rho_0;\,e))$ is given as follows:

\begin{eqnarray*}&&D(\text{Tor}(M,\rho_0;\,e))\\ &=&\sum_{\gamma\in\mathcal M(\text{NP},p_1)}[\gamma{}'_{\downarrow}]_+\otimes \frac{{\varepsilon}(\gamma)\gamma{}'_{\downarrow}}{\partial_3^{11}}+\sum_{i,j\ge 2}\sum_{\gamma\in\mathcal M(p_i,q_j)}[\gamma{}'_{\downarrow}]_+\otimes \frac{{\varepsilon}(\gamma)\gamma{}'_{\downarrow}}{\det A_2}A^{ji}_2\\ &&+\sum_{\gamma\in\mathcal M(q_1,\text{SP})}[\gamma{}'_{\downarrow}]_+\otimes \frac{{\varepsilon}(\gamma)\gamma{}'_{\downarrow}}{\partial_1^{11}}.\end{eqnarray*}

We get the following lemma:

Lemma 7·2.

\begin{eqnarray*}&&D(\text{Tor}(M,\rho_0;\,e))\\ &=&\sum_{\gamma\in\mathcal M(\text{NP},p_1)}[\gamma_{\cap}']_+\otimes \frac{\gamma{}'_{\downarrow}}{\partial_3^{11}}+\sum_{i,j\ge 2}\sum_{\gamma\in\mathcal M(p_i,q_j)}[\gamma_{\cap}']_+\otimes \frac{\gamma'_{\downarrow}}{\det A_2}A^{ji}_2\\ &&+\sum_{\gamma\in\mathcal M(q_1,\text{SP})}[\gamma_{\cap}']_+\otimes \frac{\gamma{}'_{\downarrow}}{\partial_1^{11}}.\end{eqnarray*}

7·3. The computation of $d'(M,\rho_0;\,e)$ by using $g^{\kern0.5pt f}$

By using the explicit description of $g^{\kern0.5pt f}$ with respect to the basis ${\bf b}_e$ and the formula given in Proposition 6·8, we have the following description of $d'(M,\rho_0;\,e)$ :

Lemma 7·3.

\begin{eqnarray*}&&d'(M,\rho_0;\,e)\\ &=&\sum_{\gamma\in\mathcal M(\text{NP},p_1)}[\gamma_{\cap}']_+\otimes\gamma{}'_{\downarrow}(\partial_3^{11})^{-1}+\sum_{i,j\ge 2}\sum_{\gamma\in\mathcal M(p_i,q_j)}[\gamma_{\cap}']_+\otimes \frac{\gamma{}'_{\downarrow}}{\det A_2}A_2^{ji}\\ &&+\sum_{\gamma\in\mathcal M(q_1,\text{SP})}[\gamma_{\cap}']_+\otimes\gamma{}'_{\downarrow}(\partial_1^{11})^{-1}.\end{eqnarray*}

7·4. A proof of Theorem

As a direct consequence of lemma 7·2 and lemma 7·3, we have

\begin{align*}D(\text{Tor}(M,\rho_0;\,e))=d'(M,\rho_0;\,e).\end{align*}