2.1. Twisted intersection forms and signatures

Let $M$ be a connected, compact, oriented manifold of dimension $2k$ with a representation $\alpha \colon \pi _1(M) \to U(n)$ for some $n\in \mathbb {N}$. Let $\pi :=\pi _1(M)$. Let $\widetilde {M}$ be the universal cover of $M$, so that $\pi$ acts on the left on $C_*(\widetilde {M})$ by deck transformations, and on $\mathbb {C}^{n}$ through the representation $\alpha$ (we write $\mathbb {C}^{n}_\alpha$ for the left $\mathbb {Z}[\pi ]$-module coming from this action). We consider the twisted homology groups

\begin{align*} & H_i(M;\mathbb{C}^{n}_\alpha):= H_i((\mathbb{C}^{n}_\alpha)^{t} \otimes_{\mathbb{Z}[\pi]} C_*(\widetilde{M})),\\ & H^{i}(M;\mathbb{C}^{n}_{\xi^{\alpha}}):= H^{i}({\operatorname{Hom}}_{\mathbb{Z}[\pi]} (C_*(\widetilde{M}),\mathbb{C}^{n}_\alpha)), \end{align*}

where the $t$ denotes the fact that we are turning the left $\mathbb {Z}[\pi ]$-module structure of $\mathbb {C}^{n}$ into a right one by means of the involution $g\to g^{-1}$ in $\pi$. Relative homology and cohomology groups are defined in a similar manner, as well as twisted homology groups with different $\mathbb {Z}[\pi ]$-modules (see e.g [Reference Friedl, Nagel, Orson and Powell18, Definition A.2]).

Remark 2.1 If $M=M_1\sqcup \cdots \sqcup M_N$ is a disjoint union of connected manifolds $M_j$, we will make the abuse of notation of writing $\alpha \colon \pi _1(M) \to U(n)$ to denote a collection of representations $\alpha _j\colon \pi _1(M_j)\to U(n)$. In this case, we define then

\[ H_i(M;\mathbb{C}^{n}_\alpha):= \bigoplus_{j=1}^{N} H_i(M_j;\mathbb{C}^{n}_\alpha). \]

We recall the following facts.

Fact 1 Given a subset $N\subseteq M$, there is a well-defined cup product [Reference Friedl, Nagel, Orson and Powell18, Lemma A.11]

\[{\cup}\colon H^{p}(M,N;\mathbb{C}^{n}_\alpha)\times H^{q}(M,N;\mathbb{C}^{n}_{\overline{\alpha}})\to H^{p+q}(M,N;\mathbb{C}^{n}_\alpha\otimes_{\mathbb{Z}} \mathbb{C}^{n}_{\overline{\alpha}}), \]

where $\overline {\alpha }$ denotes the complex-conjugate representation of $\alpha$.

Fact 2 As the representation $\alpha$ is Hermitian, the standard Hermitian product of $\mathbb {C}^{n}$ gives rise to a well-defined map of $\mathbb {Z}[\pi ]$-modules $\mathbb {C}^{n}_\alpha \otimes _\mathbb {Z}\mathbb {C}^{n}_{\overline {\alpha }}\to \mathbb {C}$ (where $\mathbb {C}$ is the $\mathbb {Z}[\pi ]$-module associated with the trivial one-dimensional representation). This induces a group homomorphism

\[ H^{p+q}(M,N;\mathbb{C}^{n}_\alpha\otimes_{\mathbb{Z}} \mathbb{C}^{n}_{\overline{\alpha}})\to H^{p+q}(M,N;\mathbb{C}). \]

Composing the above map with the cup product of Fact 1, and observing that $H^{q}(M,\,N;\mathbb {C}^{n}_{\overline {\alpha }})$ coincides with the complex-conjugate vector space $\overline {H^{q}(M,\,N;\mathbb {C}^{n}_\alpha )}$, we get a bilinear map

\[ \cup_\mathbb{C}\colon H^{p}(M,N;\mathbb{C}^{n}_\alpha)\times\overline{H^{q}(M,N;\mathbb{C}^{n}_\alpha)}\to H^{p+q}(M,N;\mathbb{C}). \]

Fact 3 There are twisted Poincaré duality isomorphisms [Reference Friedl, Nagel, Orson and Powell18, Theorem A.15]

\[ \text{PD}\colon H^{p}(M,\partial M;\mathbb{C}^{n}_\alpha) \xrightarrow{\sim}H_{2k-p}(M;\mathbb{C}^{n}_\alpha). \]

Fact 4 Since the representation $\alpha$ is unitary, the evaluation map in twisted homology gives rise to an isomorphism [Reference Conway and Nagel11, Lemma 2.7]

\[ {\operatorname{ev}}_\mathbb{C}\colon H^{q}(M;\mathbb{C}^{n}_\alpha)\xrightarrow{\sim}{\operatorname{Hom}}_\mathbb{C} (\overline{H_q(M;\mathbb{C}^{n}_\alpha)},\mathbb{C}).\]

Using the first three facts, we define the twisted intersection form as the sesquilinear form

\[ I^{\alpha}_M \colon H_k(M;\mathbb{C}^{n}_\alpha)\times H_{k}(M;\mathbb{C}^{n}_\alpha)\to \mathbb{C}, \]

given by

\[ I^{\alpha}_M(a,b)=\langle \text{PD}^{{-}1}(a)\cup_\mathbb{C} \overline{\text{PD}^{{-}1}(b)}, [M]\rangle \]

(compare e.g. with [Reference Friedl, Nagel, Orson and Powell18, Section 13]). As for the ordinary intersection pairing, the properties of the cup product make the form $I_M^{\alpha }$ Hermitian if $k$ is even and skew-Hermitian if $k$ is odd. This leads to the following definition.

Definition 2.2 Let $M$ be a compact oriented manifold of dimension $2k$ with a representation $\alpha \colon \pi _1(M)\to U(n)$. The signature of $M$ twisted by $\alpha$ is the integer

\[ \sigma_\alpha(M):= \begin{cases}{\operatorname{sign}} (I_M^{\alpha}) , \text{ if }k\text{ is even},\\ {\operatorname{sign}}(i\,I_M^{\alpha}), \text{ if }k \text{ is odd}, \end{cases} \]

where $\operatorname { {sign}}$ denotes the signature of a Hermitian form.

Observe that the form $I^{\alpha }_M$ can be defined alternatively using the sequence of maps

\[ H_k(M;\mathbb{C}^{n}_\alpha)\xrightarrow{j_*}H_k(M,\partial M;\mathbb{C}^{n}_\alpha) \xrightarrow{\text{PD}^{{-}1}} H^{k}(M;\mathbb{C}^{n}_\alpha)\xrightarrow{{\operatorname{ev}}_\mathbb{C}}{\operatorname{Hom}} (\overline{H_k(M;\mathbb{C}^{n}_\alpha)},\mathbb{C}) \]

(see e.g. [Reference Conway, Nagel and Toffoli12, Definition 2.7]), where $\operatorname { {ev}}_\mathbb {C}$ is the map discussed in Fact 4. From this point of view, we see that the radical of $I_M^{\alpha }$ coincides with the kernel of $j_*$, as the two other maps in the composition are isomorphisms. As a consequence, $I_M^{\alpha }$ is non-degenerate whenever $M$ is closed.

2.2. Basics on the rho invariant

Let $N$ be a closed, oriented, Riemannian manifold of dimension $2k+1$, with a representation $\alpha \colon \pi _1(N)\to U(n)$. Let $E_\alpha \to N$ be the associated flat vector bundle, and consider the subspace

\[ \Omega^{{\operatorname{ev}}}(N, E_\alpha):=\bigoplus_{q=0}^{k} \Omega^{2q}(N, E_\alpha) \]

of twisted differential forms of even degree (see [Reference Atiyah, Patodi and Singer2, Section 2] for details). Let $D^{\alpha }_N$ be the twisted odd signature operator, i.e. the first-order differential operator on $\Omega ^{\operatorname { {ev}}}(N,\, E_\alpha )$ defined by

\[ D^{\alpha}_N \,\phi := ({-}1)^{q+1}i^{k}({\star} d - d \star)\phi, \quad \text{ for } \phi\in \Omega^{2q}(N, E_\alpha). \]

The operator $D^{\alpha }_N$ can be extended to a self-adjoint elliptic operator with discrete spectrum and, by the results of Atiyah, Patodi, and Singer [Reference Atiyah, Patodi and Singer3, Theorem 4.5], its eta function

\[ \eta(s)=\sum_{\substack{ \lambda\in {\operatorname{Spec}} (D^{\alpha}_N)\\ \lambda\ne 0} } \ {\operatorname{sgn}} \lambda\, \lvert \lambda \rvert ^{{-}s} \]

has a meromorphic extension which is holomorphic at $s=0$, leading to the eta invariant

\[ \eta_\alpha(N):=\eta(0)\in \mathbb{R}. \]

We say that a compact Riemannian manifold $M$ has metric of product form near the boundary, if there exists a neighbourhood of $\partial M$ that is isometric to $(-\varepsilon,\, 0] \times \partial M$ with the product metric. The main result about the eta invariant of the twisted signature operator is the following [Reference Atiyah, Patodi and Singer2, Theorem 2.2].

Theorem 2.3 Atiyah–Patodi–Singer

Let $M$ be an even-dimensional compact, oriented manifold with $\partial M=N$, equipped with Riemannian metric of product form near $N$, and let $\alpha \colon \pi _1(M)\to U(n)$ be a representation. Then

\[ \sigma_\alpha(M)= n\int_M \!L(p) -\eta_\alpha(N) , \]

where $L(p)$ is the Hirzebruch $L$-polynomial in the Pontryagin forms of $M$.

Note that both summands on the right-hand term depend on the Riemannian metric on $N$. We shall not dwell upon the geometrical significance of the integral of the $L$-polynomial, as it is going to get simplified soon.

For the eta invariant associated with the untwisted odd signature operator $D_N$ on $\Omega ^{\operatorname { {ev}}}(N,\, \mathbb {C})$, we use the notation $\eta (N):=\eta (D_N)$. Following Atiyah, Patodi, and Singer [Reference Atiyah, Patodi and Singer2], we are now going to define the rho invariant.

Definition 2.5 Let $N$ be a closed, oriented manifold of odd dimension, and let $\alpha \colon \pi _1(N)\to U(n)$ be a representation. The Atiyah–Patodi–Singer rho invariant of $N$ associated with $\alpha$ is the real number

\[ \rho_\alpha(N):=\eta_\alpha(N)-n\,\eta(N), \]

where the eta invariants are computed for an arbitrary Riemannian metric on $N$.

We shall see in a moment that the difference $\eta _\alpha (N)-n\,\eta (N)$ is independent of the Riemannian metric, so that $\rho _\alpha (N)$ is well defined. Moreover, $\rho _\tau (N)$ is $0$ for trivial representations $\tau$, and it satisfies

(2.1)\begin{equation} \rho_\alpha({-}N)={-}\rho_\alpha(N). \end{equation}

The main theorem about the rho invariant is the following.

We state one more well-known result that will turn useful later on.

2.3. Rho invariants and Casson–Gordon invariants

We will now review the definition of an invariant of Casson and Gordon and relate it to the Atiyah–Patodi–Singer rho invariant. Let $N$ be a closed, oriented three manifolds, and let $\alpha \colon H_1(N;\mathbb {Z})\to U(1)$ be a representation. Assume that the image of $\alpha$ is finite. Using a bordism argument, Casson and Gordon observe that there exists a compact, oriented four-manifold $W$ such that the boundary of $W$ is the disjoint union of $r$ copies of $N$ for some $r\in \mathbb {N}$ (we will write $\partial W =r N$) with a representation $\alpha '\colon H_1(W;\mathbb {Z})\to U(1)$ that restrict to $\alpha$ on each boundary component [Reference Casson and Gordon8, p. 183]. They define then an invariant as

(2.2)\begin{equation} \sigma(N,\alpha):={\displaystyle\frac{1}{r}}(\sigma_{\alpha'}(W)-\sigma(W)). \end{equation}

By additivity of the signature and again some bordism theory, they show that the invariant $\sigma (N,\,\alpha )$ is independent of the choice of $W$ and of the extension $\alpha '$ [Reference Casson and Gordon8, p. 183–184] (see also [Reference Conway, Nagel and Toffoli12, Corollary 2.11] for a more detailed version of their proof). Using the Atiyah–Patodi–Singer index theorem, their invariant can be immediately reinterpreted as an Atiyah–Patodi–Singer rho invariant. We state this explicitly for further reference, albeit it is surely known to the experts.

Proposition 2.8 Let $N$ be a closed, oriented three-manifold, and let $\alpha \colon H_1(N;\mathbb {Z})\to U(1)$ be a representation with finite image. Then, we have

\[ \sigma(N,\alpha)={-} \rho_\alpha(N). \]

Proof. Let $W$ be a compact, oriented four-manifold with $\partial W = rN$, with a representation $\alpha '\colon H_1(W;\mathbb {Z})\to U(1)$ that restricts to $\alpha$ as discussed above, so that $\sigma (N,\,\alpha )$ is described by (2.2). Using Theorem 2.6, on the other hand, we have

\[ r\rho_{\alpha}(N)=\rho_{\alpha}(rN)=\sigma(W)-\sigma_{\alpha'}(W). \]

Comparing this with (2.2), we obtain the desired statement.

2.4. The rho invariant of lens spaces

Given coprime integers $p$ and $q$, the three-dimensional lens space $L(p,\,q)$ can be built as the union of two solid tori $Y_1,\,Y_2$ along any orientation-reversing diffeomorphism $f\colon \partial Y_2\to \partial Y_1$ such that

\[ f_*(\mu_2)={-} q \mu_1 + p\lambda_1, \]

where $\mu _1,\, \mu _2$ are the meridians, respectively, of $Y_1,\, Y_2$, and $\lambda _1$ is a longitude of $Y_1$. This construction is well defined for both positive and negative values of $p$ and $q$. For positive $p$ it coincides, up to some explicit orientation-preserving diffeomorphism, with the classical definition of $L(p,\,q)$ as a quotient of $S^{3}$ (compare with [Reference Friedl15, Lemma 91.3]). In general, there are orientation-preserving diffeomorphisms

\[ L({-}p,q)\cong L(p,-q) \cong{-} L(p,q). \]

In particular, for both positive and negative $p$, we have an identification of $\pi _1(L(p,\,q))$ with $\mathbb {Z}/p$, and under this identification, the element $[1]\in \mathbb {Z}/p$ corresponds to the generator of $\pi _1(Y_1)$.

Rho invariants of three-dimensional lens spaces can be computed explicitly. As every representation $\alpha \colon \mathbb {Z}/p\to U(n)$ can be written as a direct sum of one-dimensional representations, we shall focus on one-dimensional representations. Moreover, we shall exclude the case $p=0$, as $L(0,\,1)$ is diffeomorphic to $S^{2}\times S^{1}$ and its rho invariant is $0$ for any choice of $\alpha$. We observe in this case that the representations $\alpha \colon \mathbb {Z}/p\to U(1)$ are in a natural bijection with the set of $\left |{p}\right |$^th roots of unity: to each such root $\omega$, we associate the representation $\alpha _\omega$ sending $1$ to $\omega$.

Notation 2.9 Given a $\left |{p}\right |$^th root of unity $\omega$, we write

\[ \rho(L(p,q),\omega):= \rho_{\alpha_\omega}(L(p,q)). \]

Formulas for the rho invariants of lens spaces were given since the original paper of Atiyah, Patodi and Singer [Reference Atiyah, Patodi and Singer2, Proposition 2.12]. Introducing the periodic sawtooth function $\left ( \! \left ({\cdot }\right ) \! \right )\colon \mathbb {R}\to (-\frac {1}{2},\, \frac {1}{2})$ defined by

\[ \left( \! \left({x}\right) \! \right):= \begin{cases} x- \lfloor x\rfloor - {\displaystyle\frac{1}{2}}, & \text{ if }x\in \mathbb{R}\setminus \mathbb{Z},\\ 0, & \text{ if }x\in \mathbb{Z}.\end{cases}, \]

we give the following description.

Proposition 2.10 Let $p,\,q$ be two coprime integers with $p\ne 0,$ and let $\zeta =e^{2\pi i /p}$. Then, for $k\in \{0,\,1,\,\dotsc,\, \left |{p}\right |-1\},$ we have

\[ \rho(L(p,q),\zeta^{kq})={-}4\sum_{j=1}^{k-1} \Big( \!\! \Big({{\displaystyle\frac{qj}{p}}}\Big)\!\! \Big) - 2\Big( \!\! \Big({{\displaystyle\frac{qk}{p}}}\Big)\!\! \Big). \]

Proof. We first suppose $p,\,q>0$. Set $z:=e^{2\pi i n/p}$, with $n=\gcd (p,\,k)$, and set $r:=k/n$. Then, we have $\zeta ^{kq}=z^{rq}$. The representation sending $1$ to $\omega ^{q}$ has as its image the set of $m$^th roots of unity, with $m$ such that $p=mn$. In this setting, Casson and Gordon [Reference Casson and Gordon8, pp.187–188] proved that

(2.3)\begin{equation} \sigma(L(p,q),\alpha_\omega)= 4\left({\operatorname{area}}\Delta\left(nr,{\displaystyle\frac{rq}{m}}\right)- \displaystyle{\operatorname{int}}\Delta\left(nr,{\displaystyle\frac{rq}{m}}\right)\right) , \end{equation}

where $\Delta (x,\,y)$ is the triangle with vertices $(0,\,0)$, $(x,\,0)$ and $(x,\,y)$ and the number $\operatorname { {int}}(\Delta (x,\,y))$ is given by counting:

• • $+1$ for every point of $\mathbb {Z}^{2}$ that lies in the interior of $\Delta (x,\,y)$;

• • $+1/2$ for every point of $\mathbb {Z}^{2}$ that lies in the interior of its edges;

• • $+1/4$ for every point of $\mathbb {Z}^{2}\setminus \{(0,\,0)\}$ that coincides with one of the vertices.

Using Proposition 2.8 and the definition of $r$, we rewrite (2.3) as

(2.4)\begin{equation} \rho(L(p,q),\zeta^{kq})=4(\displaystyle{\operatorname{int}}\Delta(k,{\displaystyle\frac{kq}{p}})-{\operatorname{area}}\Delta(k,{\displaystyle\frac{kq}{p}})). \end{equation}

We will now express the right-hand term of (2.4) in a more explicit way. First of all, it is clear that $4\operatorname { {area}}\Delta (k,\,{\displaystyle \frac {kq}{p}})= {\displaystyle \frac {2q}{p}}k^{2}.$ Moreover, we can count the lattice points inside the triangle by following vertical lines $\{(x,\,y)\,\vert \,x=j\}$, for $j=1,\,\ldots,\,k$, and then summing over $j$. We obtain

\begin{align*} 4{\operatorname{int}}\Delta\Big(k,{\displaystyle\frac{kq}{p}}\Big)& = 4\sum_{j=1}^{k-1}\Big({\displaystyle\frac{1}{2}}+ \Big\lfloor{{\displaystyle\frac{jq}{p}}}\Big\rfloor\Big) +4\Big({\displaystyle\frac{1}{4}}+{\displaystyle\frac{1}{2}} \Big\lfloor{{\displaystyle\frac{kq}{p}}}\Big\rfloor\Big)\\ & =2k -1 +4\sum_{j=1}^{k-1} \Big\lfloor{{\displaystyle\frac{jq}{p}}}\Big\rfloor +2 \Big\lfloor{{\displaystyle\frac{kq}{p}}}\Big\rfloor. \end{align*}

Taking the difference, we obtain thus

(2.5)\begin{equation} \rho(L(p,q),\zeta^{kq})={-}{\displaystyle\frac{2q}{p}}k^{2} +2k -1 +4\displaystyle\sum_{j=1}^{k-1} \Big\lfloor{ {\displaystyle\frac{jq}{p}}}\Big\rfloor +2 \Big\lfloor{{\displaystyle\frac{kq}{p}}}\Big\rfloor. \end{equation}

Expanding the expression in the statement, it is now immediate to see that it coincides with (2.5). As the sawtooth function $\left ( \! \left ({\cdot }\right ) \! \right )$ is odd, we see that both sides of the identity change sign when either $p$ or $q$ is changed of sign. As a consequence, the result keeps holding for non-positive choices of $p$ and $q$.

Corollary 2.11 Let $n$ be any integer, and let $\omega \in U(1)$ be an $\left |{n}\right |$^th root of unity. Then, we have

\[ \rho(L(n,1),\omega)= 2n\theta(1-\theta) -{\operatorname{sgn}} (n), \]

where $\theta \in [0,\,1)$ is such that $\omega =e^{2\pi i\theta }$.

Proof. For $n=0$, we have $\rho (L(n,\,1),\,\omega )=0$ for all $\omega \in U(1)$ because $L(0,\,1)=S^{2}\times S^{1}$, so that the result is trivially satisfied in this case. For $n>0$, there has to be a $k\in \{0,\,1,\, \cdots n-1\}$ such that $\theta =k/n$, with $k\in \{0,\,1,\, \cdots n-1\}$. From (2.5), we easily see that

\[ \rho(L(n,1),\omega)={-}{\displaystyle\frac{2k^{2}}{n}} +2k -1 = 2n\theta(1-\theta) -1, \]

and the desired formula is satisfied in this case. For negative $n$, we obtain now from the last equation that

\[ \rho(L(n,1),\omega)={-}\rho(L({-}n,1),\omega)=2n\theta(1-\theta) +1, \]

which leads to the general formula of the statement.

3.1. Complex symplectic spaces and the Maslov triple index

A complex symplectic space is a pair $(H,\,\omega )$ such that $H$ is a finite-dimensional complex vector space, and $\omega \colon H\times H\to \mathbb {C}$ is a non-degenerate skew-Hermitian form, called the symplectic form. We shall often omit $\omega$ from the notation and simply call $H$ a complex symplectic space. We recall that a subspace $L\subseteq H$ is Lagrangian if it coincides with its orthogonal complement with respect to the symplectic form $\omega$. Let $\mathcal {L} ag(H)$ denote the set of all Lagrangian subspaces of $H$. We are now going to define a function

\[ \tau\colon \mathcal{L} ag (H) \times \mathcal{L} ag (H) \times \mathcal{L} ag (H) \to \mathbb{Z}. \]

Given three Lagrangian subspaces $L_1,\, L_2,\, L_3\in \mathcal {L} ag(H)$, it is immediate to verify that the sesquilinear form

\begin{align*} \psi_{L_1L_2L_3}\colon(L_1+L_2)\cap L_3 \times (L_1+L_2)\cap L_3 & \to \mathbb{C} \\ (a_1+a_2,b_1+b_2) & \mapsto \omega(a_1,b_2) \end{align*}

(with $a_1,\,b_1\in L_1$ and $a_2,\,b_2\in L_2$) is well defined and Hermitian. In particular, we can give the following definition.

Definition 3.1 The Maslov triple index of $(L_1,\,L_2,\,L_3)$ is the integer

\[ \tau(L_1, L_2, L_3):={\operatorname{sign}} \psi_{L_1L_2L_3}. \]

The Maslov triple index satisfies several elementary properties, among which are the following two.

Example 3.3 Suppose that $(H,\,\omega )$ is a complex symplectic space of dimension $2$. Let $(\mu,\,\lambda )$ be an ordered basis in which $\omega$ is represented by the matrix $\left (\begin {smallmatrix} 0 & -1\\1 & 0\end {smallmatrix}\right )$, i.e. such that

\[ \omega(\mu,\mu)=\omega(\lambda,\lambda)=0, \quad \omega(\mu,\lambda)={-}1 \]

we shall call such pair a symplectic basis. Then, it is easy to verify that a one-dimensional subspace is Lagrangian if and only if it is the span of some vector $a\mu +b\lambda$ with $a,\,b\in \mathbb {R}$. We set in this case the notation

\[ \tau(v_1,v_2,v_3):= \tau ({\operatorname{Span}}_\mathbb{C}\{v_1\}, {\operatorname{Span}}_\mathbb{C}\{v_2\}, {\operatorname{Span}}_\mathbb{C}\{v_3\}). \]

Using the definition of the Maslov triple index, we easily compute that

\[ \tau (\mu,\lambda, a\mu+b\lambda)={-} {\operatorname{sgn}}(ab). \]

3.2. Complex symplectic spaces and twisted homology

In the applications, complex symplectic spaces will arise from the following setting. Let $\Sigma$ be a $2k$-dimensional closed, oriented manifold, and let $\alpha \colon \pi _1(\Sigma )\to U(n)$ be a representation. As we have seen, the twisted intersection form on $H:=H_{k}(\Sigma ;\mathbb {C}^{n}_\alpha )$ is Hermitian if $k$ is even, and it is skew-Hermitian if $k$ is odd. Moreover, it is non-degenerate because $\Sigma$ is closed. We can thus always consider the non-degenerate, skew-Hermitian form

\[ \omega:= \begin{cases} I_\Sigma^{\alpha}, \text{ if }k \text{ is odd},\\ i\, I_\Sigma^{\alpha} , \text{ if }k\text{ is even}, \end{cases} \]

which makes the pair $(H,\,\omega )$ a complex symplectic space. We introduce the following notation.

Notation 3.4 Given a compact, connected $(2k+1)$-dimensional manifold $X$ with a representation $\alpha \colon \pi _1(X)\to U(n)$, we set

\[ V_X^{\alpha}=\ker(H_{k}(\partial X;\mathbb {C}^{n}_{\alpha}) \to H_{k}(X;\mathbb {C}^{n}_{\alpha})). \]

A well-known argument based on Poincaré duality shows that $V_X^{\alpha }$ is a Lagrangian subspace of $H_k(\partial X;\mathbb {C}^{n}_\alpha )$ with respect to $\omega$ (compare with [Reference Degtyarev, Florens and Lecuona14, Lemma 2.14] and [Reference Toffoli28, Proposition 1.4.6]). We can thus give the following definition.

3.3. Wall's non-additivity of the signature

We shall now review a result of Wall. We start with some notation that will be useful throughout the paper.

Notation 3.6 Given two topological spaces $X,\,Y$ with a common subspace $A$, we set

\[ X\cup_A Y:= (X\sqcup Y) \,/ \! \sim, \]

where $\sim$ is the relation that identifies every element in $A\subseteq X$ with its copy in $A\subseteq Y$. We say that $X\cup _A Y$ is obtained by gluing $X$ and $Y$ along $A$. If $X$ and $Y$ are manifolds with $\partial X= \partial Y$, we also write $X\cup _\partial Y$ to denote the gluing along their common boundary.

Suppose that $M$ is a compact, oriented manifold of even dimension that is split as

\[ M=M_1\cup_{X_0} M_2 \]

along a properly embedded submanifold $X_0$ of codimension $1$, which is allowed to have boundary $\Sigma$. Let $X_1:=\partial M_1 \setminus \operatorname { {int}}(X_0)$ and $X_2:=\partial M_2 \setminus \operatorname { {int}}(X_0)$. As unoriented manifolds, we have then $\partial X_0=\partial X_1 =\partial X_2 = \Sigma$ and

(3.1)\begin{equation} \partial M_1= X_1\cup_\Sigma X_0, \quad \partial M_2= X_0\cup_\Sigma X_2, \quad \partial M= X_1\cup_\Sigma X_2 . \end{equation}

We pick on $X_1$ the orientation coming from being a codimension $0$ submanifold of $\partial M_1$, and we give $\Sigma$ the orientation coming from being the boundary of $X_1$. Suppose now that $\alpha \colon \pi _1(M)\to U(n)$ is a representation. In our setting, $\Sigma$ is (up to orientation) the common boundary of $X_0$, $X_1$ and $X_2$, so that the canonical Lagrangians $V_{X_0}^{\alpha }$, $V_{X_1}^{\alpha }$ and $V_{X_2}^{\alpha }$ all live in the same space $H_{k}(\Sigma ;\mathbb {C}^{n}_\alpha )$. In particular, it makes sense to compute their Maslov triple index. In fact, the following result holds.

Theorem 3.7 Wall's non-additivity

Let $M$ be a closed, oriented, even-dimensional manifold, and let $\alpha \colon \pi _1(M)\to U(n)$ be a representation. Then, if $M$ decomposes as $M=M_1\cup _{X_0} M_2$ as above, we have

\[ \sigma_\alpha( M) = \sigma_\alpha (M_1) + \sigma_\alpha (M_2) - \tau(V_{X_0}^{\alpha},V^{\alpha}_{X_1},V^{\alpha}_{X_2}). \]

3.4. The cut-and-paste formula for the rho invariant

Suppose to have a closed, oriented $(2k+1)$-dimensional manifold that is split by a codimension-one closed manifold $\Sigma$, yielding a decomposition $X_1\cup _\Sigma X_2$. Let $X_0$ be a compact, oriented manifold with $\partial X_0=-\Sigma$. Then, we can replace $X_1\cup _\Sigma X_2$ with the disjoint union of $X_1\cup _\Sigma X_0$ and $-X_0\cup _\Sigma X_2$. We will call this manipulation cut-and-paste. Schematically, we have

\[ X_1\cup_\Sigma X_2 \quad \leadsto \ X_1\cup_\Sigma X_0 \ \sqcup \ -X_0\cup_\Sigma X_2, \]

and pictorially we can represent the operation as in the next figure.

Suppose now that $\alpha \colon \pi _1(X_1\cup _\Sigma X_2)\to U(n)$ is a representation. In particular, $\alpha$ is defined on $\pi _1(X_1)$ and on $\pi _1(\Sigma )$. In order to have rho invariants of $X_1 \cup _\Sigma X_0$ and $-X_0\cup _\Sigma X_2$ to be compared to $\rho _\alpha (X_1\cup _\Sigma X_2)$, we need to extend $\alpha$ to the fundamental groups of these manifolds. This is possible if and only if we can construct an extension of $\alpha$ from $\pi _1(\Sigma )$ to $\pi _1(X_0)$: then, using Seifert–Van Kampen's theorem, this will be patched with $\alpha \colon \pi _1(X_1)\to U(n)$ to produce representations $\pi _1(X_1 \cup _\Sigma X_0)\to U(n)$ and $\pi _1(-X_0\cup _\Sigma X_2)\to U(n)$. For simplicity, we will use the same notation $\alpha$ for all of these representations. Then, we want to compute the correction term $C$ in the formula

(3.2)\begin{equation} \rho_\alpha(X_1\cup_\Sigma X_2) = \rho_\alpha(X_1 \cup_\Sigma X_0) + \rho_\alpha({-}X_0\cup_\Sigma X_2) + C. \end{equation}

Now, if $X_1\cup _\Sigma X_0$ and $-X_0\cup _\Sigma X_2$ bound manifolds $W_1$ and $W_2$ such that the representation extends, then Wall's theorem, together with the Atiyah–Patodi–Singer signature theorem, tells us how to compute the correction term. Namely, in that case, we have

(3.3)\begin{equation} C=\tau(V^{\alpha}_{X_0},V_{X_1}^{\alpha},V^{\alpha}_{X_2}) - n\, \tau(V_{X_0},V_{X_1},V_{X_2}). \end{equation}

The content of the main result of this section is that the correction term $C$ of (3.2) is always given by (3.3), no matter whether the manifolds $W_1$ and $W_2$ exist. This should be compared with an analogous result for the untwisted eta invariant [Reference Bunke5, 2.5].

Theorem 3.9 Let $X_1,$ $X_2$ and $X_0$ be compact, oriented manifolds of dimension $2k+1$ with $\partial X_1=-\partial X_0=-\partial X_2=:\Sigma,$ and let $\alpha \colon \pi _1(X_1\cup _\partial X_2)\to U(n)$ be a representation that extends to $\pi _1(X_0)$. Then, for every choice of such an extension, we have

\[ \rho_\alpha(X_1\cup_\Sigma X_2) = \rho_\alpha(X_1 \cup_\Sigma X_0) + \rho_\alpha({-}X_0\cup_\Sigma X_2)+ C, \]

where

\[ C=\tau(V^{\alpha}_{X_0},V_{X_1}^{\alpha},V^{\alpha}_{X_2}) - n\, \tau(V_{X_0},V_{X_1},V_{X_2}), \]

with the first Maslov triple index performed on $H_{k}(\partial X_1;\mathbb {C}^{n}_\alpha ),$ and the second on $H_{k}(\partial X_1;\mathbb {C})$.

Proof. Consider the oriented manifolds

\[ M_1:=[0,1]\times (X_1 \cup_\Sigma X_0),\quad M_2:=[0,1]\times ({-}X_0\cup_\Sigma X_2). \]

We glue then $M_1$ with $M_2$ along $\{1\}\times X_0$, obtaining a topological oriented manifold $M$ to which $\alpha$ extends.

The boundary of $M$ can be described topologically as

(3.4)\begin{equation} \partial M=(-(X_1 \cup_\Sigma X_0)\sqcup{-}({-}X_0\cup_\Sigma X_2) ) \sqcup (X_1\cup_\Sigma X_2 ), \end{equation}

and we can equip $M$ with a smooth structure such that (3.4) is satisfied in the smooth sense [Reference tom Dieck29, 15.10.3]. Thanks to Theorem 2.6 (ii), we have then

(3.5)\begin{equation} \rho_\alpha(\partial M)=n\,\sigma(M)-\sigma_\alpha(M). \end{equation}

By (3.4), the left-hand term is given by

(3.6)\begin{equation} \rho_\alpha(\partial M)={-}\rho_\alpha(X_1 \cup_\Sigma X_0) -\rho_\alpha({-}X_0\cup_\Sigma X_2) +\rho_\alpha (X_1\cup_\Sigma X_2 ). \end{equation}

By Wall's non-additivity (Theorem 3.7), we can compute the twisted and untwisted signature of $M$ as

\begin{align*} \sigma(M)& =\sigma(M_1)+\sigma(M_2) -\tau(V_{X_0},V_{X_1},V_{X_2}),\\ \sigma_\alpha(M)& =\sigma_\alpha(M_1)+\sigma_\alpha(M_2) -\tau(V^{\alpha}_{X_0},V_{X_1}^{\alpha},V^{\alpha}_{X_2}). \end{align*}

The manifolds $M_1$ and $M_2$ are of the form $[0,\,1]\times X$, with $X$ a closed manifold. In particular, they admit an orientation-reversing self-diffeomorphism defined by

\begin{align*} [0,1]\times X & \to [0,1]\times X\\ (t,x)& \mapsto (1-t, x). \end{align*}

As orientation-reversing diffeomorphisms have the effect of changing sign to the signature, this shows that the ordinary signatures of $M_1$ and $M_2$ vanish. Since the above diffeomorphism is trivial on the fundamental group, the same argument can be applied to the twisted signatures, which therefore also vanish. The computation of $\sigma (M)$ and $\sigma _\alpha (M)$ is hence reduced to

(3.7)\begin{equation} \sigma(M)={-}\tau(V_{X_0},V_{X_1},V_{X_2}),\quad \sigma_\alpha(M)={-}\tau(V^{\alpha}_{X_0},V_{X_1}^{\alpha},V^{\alpha}_{X_2}). \end{equation}

Substituting (3.6) and (3.7) into (3.5), we get the desired formula.

4.1. Links and framings

Let $L=K_1\cup \cdots \cup K_k$ be an oriented link in $S^{3}$ (from now on, just a link). By removing from $S^{3}$ the interior of a closed tubular neighbourhood $N(L)$, we get its link exterior

\[ X_L:=S^{3} \setminus {\operatorname{int}} (N(L)). \]

The link exterior $X_L$ is a compact, oriented three-manifold, whose boundary is a union of tori: to each link component $K_i\subseteq L$, there corresponds a boundary component $T_i=-\partial (N(K_i))$ (this is the orientation coming from being part of the boundary of $X_L$, and it is the one we shall always consider). The link component $K_i$ determines the following two elements:

• • the meridian of $K_i$ is the only element $\mu _i\in H_1(T_i;\mathbb {Z})$ whose image in $H_1(N(K_i);\mathbb {Z})$ is $0$ and such that $\operatorname { {lk}} (\mu _i,\, K_i)=1$;

• • the standard longitude of $K_i$ is the only element $\lambda _i^{s}\in H_1(T_i;\mathbb {Z})$ whose image in $H_1(N(K_i);\mathbb {Z})$ is homologous to $K_i$ and such that $\operatorname { {lk}} (\lambda _i^{s},\, K_i)=0$.

For a knot $K$, we shall often just use the notation $T$, $\mu$ and $\lambda$ for the boundary torus, the meridian and the standard longitude. Observe that the algebraic intersection of these two elements is given by

(4.1)\begin{equation} \mu_i\cdot\lambda_i^{s}={-}1. \end{equation}

We shall often consider the images of the above elements into homology with rational or complex coefficients without changing their names.

Definition 4.3 A rational framing on a link $L=K_1\cup \cdots \cup K_k$ is a $k$-tuple of rational numbers $r=(r_1,\,\dotsc,\, r_k)$. The pair $(L,\,r)$ is called a rationally framed link. If all $r_i$'s are integers, we say that $r$ is an integer framing. The framed longitudes of a rationally framed link $(L,\,r)$ are the elements

\[ \lambda_i:=\lambda_i^{s} + r_i \mu_i\in H_1(T_i;\mathbb{Q}). \]

Without changing notation for them, we shall now consider the images of the meridians and framed longitudes in the homology of the link complement $X_L$. It is an elementary well-known fact that $H_1(X_L;\mathbb {Z})$ is a free $\mathbb {Z}$-module generated by the meridians, and that each standard longitude satisfies

(4.2)\begin{equation} \lambda_i^{s} = \displaystyle\sum_{j\ne i}{\operatorname{lk}} (K_i,K_j)\, \mu_j \in H_1(X_L;\mathbb{Z}). \end{equation}

Next, we define the following matrix associated with a rationally framed link.

Definition 4.4 The framed linking matrix of a framed $k$-component link $(L,\,r)$ is the symmetric matrix $\Lambda _r=(\Lambda _{ij})_{i,j}\in \mathbb {Q}^{k\times k}$ defined by

\[ \Lambda_{ij}= \begin{cases} {\operatorname{lk}} (K_i, K_j), \quad & \text{if }i\ne j,\\ r_i, \quad & \text{if } i=j. \end{cases} \]

Example 4.5 The Seifert framing on a link $L=K_1\cup \cdots \cup K_k$ is the integer framing $(f_1,\, \cdots,\, f_k)$ defined by

\[ f_i:={-}\sum_{j\ne i} {\operatorname{lk}}(K_i,K_j). \]

In particular, the coefficients of its framed linking matrix satisfy

\[ \Lambda_{ii}={-}\sum_{j\ne i}\Lambda_{ij}. \]

The framed longitudes $\lambda _i=\lambda _i^{s} + f_i\mu _i$ associated with the Seifert framing correspond to the intersections of a Seifert surface with the boundary tori $T_i$.

From (4.2), together with the definition of the framed longitudes and of the framed linking matrix, it follows now immediately that the framed longitudes in the rational homology of $X_L$ are equal to

(4.3)\begin{equation} \lambda_i = \displaystyle\sum_{j=1}^{k}\Lambda_{ij}\, \mu_j \in H_1(X_L;\mathbb{Q}). \end{equation}

For the computations involving the Maslov triple index, a good understanding of the complex symplectic space $H_1(\partial X_L;\mathbb {C})$ is required. The next result summarizes some easy facts that we will need later.

Proof. $(i)$ is an immediate consequence of the definition of the framed meridians together with (4.1). $(ii)$ is an immediate consequence of (4.3). The fact that $\mathcal {M}$ and $\mathcal {L}_r$ are Lagrangians is obvious from $(i)$. In order to prove $(iii)$, we compute the Maslov triple index using the definition. As $\mathcal {M}$ and $\mathcal {L}_r$ are transverse, we have $(\mathcal {M} + \mathcal {L}_r)\cap V_{X_L} = V_{X_L}$, and every generator $v_i$ can be written in a unique, obvious way as the sum of an element in $\mathcal {M}$ and one in $\mathcal {L}_r$. By Definition 3.1, then, $\tau (\mathcal {M},\, \mathcal {L}_r,\, V_{X_L})$ is the signature of the Hermitian form $\psi \colon V_{X_L}\times V_{X_L}\to \mathbb {C}$ defined on the basis elements of $V_{X_L}$ by

\[ \psi (v_i, v_j)= \left(- \sum_{s=1}^{k}\Lambda_{is} \mu_s \right)\cdot v_j={-}\sum_{s=1}^{k} \Lambda_{is} (\mu_s\cdot \lambda_j) =\Lambda_{ij}. \]

It follows that $\tau (\mathcal {M},\, \mathcal {L}_r,\, V_{X_L})= \operatorname { {sign}} \Lambda _r$, as it was desired.

Given a rationally framed link $(L,\,r)$, we can consider the closed manifold $S_L(r)$ obtained by the Dehn surgery along the framing. This is done in the following way: for each link component $K_i$, we choose coprime integers $(p_i,\,q_i)$ such that $p_i/q_i=r_i$, and glue a solid torus $Y_i$ to $X_L$ along the boundary component $T_i$ in such a way that the meridian of the solid torus is identified with the element $p_i\mu _i+q_i\lambda _i^{s}\in H_1(T_i;\mathbb {Z})$. In particular, $H_1(S_L(r);\mathbb {Z})$ can be described as a quotient of $H_1(X_L;\mathbb {Z})$.

Remark 4.8 From the definition of surgery, it is clear that a rational framing $r=(r_1,\,\dotsc,\, r_k)$ with $r_i=p_i/q_i$ as above is compatible with $\alpha$ if and only if $\alpha (p_i\mu _i+q_i\lambda _i^{s})=1$ for all $i$. Using (4.2), we see that in terms of the coefficients of the framed linking matrix we have

\[ r \text{ is compatible with } \alpha \iff \prod_{j=1}^{k}\alpha(\mu_j)^{q_i\Lambda_{ij}}=1 \ \forall i . \]

Note that, in general, given a representation $\alpha$, there might be no rational framing that is compatible with it.

4.2. Coloured links and signatures

A $n$-colouring on a link $L$, for $n\in \mathbb {N}$, is a partition of its components into $n$ non-empty sublinks. Given a $k$-component link $L=K_1\cup \cdots \cup K_k$, we identify the colouring with a surjective function $c\colon \{1,\, \dotsc,\, k\}\to \{1,\, \dotsc,\, n\}$. The latter is the set of colours, and for every $1\leq s\leq n$, we define

\[ L_s:= {\displaystyle\bigcup\limits_{c(j)=s}^{}} K_j \]

to be the sublink of colour $s$. The pair $(L,\,c)$ is called an $n$-coloured link. We shall systematically omit $c$ (which has to be considered as fixed) and simply call $L$ an $n$-coloured link.

Using a generalization of the concept of Seifert surfaces (called C-complexes), Cimasoni and Florens defined a multivariable version of the Levine–Tristram signature of a link [Reference Cimasoni and Florens9]. Given an $n$-coloured link $L$, their multivariable signature is a function

\[ \sigma_L\colon \mathbb{T}_*^{n} \to \mathbb{Z}, \]

which coincides for $n=1$ with the Levine–Tristram signature function. We recall now the following four-dimensional description of the multivariable signature. We first need a definition.

A bounding surface for $L$ can be obtained for example by pushing the interior of a C-complex into the interior of $D^{4}$ (see e.g. [Reference Conway, Friedl and Toffoli10, Section 3] for details). Given a bounding surface $F=F_1\cup \cdots \cup F_n\subseteq D^{4}$ for a link $L$, we can take a small tubular neighbourhood $N(F_i)$ of each surface $F_i$ and define the exterior of $F$ in $D^{4}$ as the four-manifold with boundary

\[ W_F:=D^{4}\setminus (N(F_1)\cup\cdots \cup N(F_n)). \]

It is easy to show that $H_1(W_F;\mathbb {Z})$ is freely generated by the meridians of the surfaces $F_1,\, \dotsc,\, F_n$ (the meridian of $F_i$ being the image in $H_1(W_F;\mathbb {Z})$ of any of the meridians of $L_i$). The following description of the Cimasoni–Florens signature is known [Reference Conway, Nagel and Toffoli12, Proposition 3.5].

Proposition 4.11 Let $L$ be a an $n$-coloured link in $S^{3}$ and let $F=F_1\cup \cdots \cup F_n$ be a bounding surface for $L$. Let $\omega =(\omega _1,\, \dotsc,\, \omega _n)\in \mathbb {T}_*^{n}$, and let $\alpha \colon H_1(W_F;\mathbb {Z})\to U(1)$ be the representation that sends the meridian of $F_i$ to $\omega _i$. Then, we have

\[ \sigma_L(\omega)= \sigma_\alpha(W_F). \]

We conclude this part with a couple of definitions.

Definition 4.12 Let $L=K_1\cup \cdots \cup K_k$ be an $n$-coloured link. The coloured Seifert framing on $L$ is the integer framing $f_L=(f_1,\,\dotsc,\, f_k)$ given by

\[ f_i:={-} \sum_{\substack{j\ne i \text{ s.t.}\\ c(j)=c(i)}}\ {\operatorname{lk}}(K_i,K_j). \]

In other words, it is the framing obtained by providing each coloured sublink $L_i$ with its Seifert framing (see e.g. 4.5).

Observe that the coloured Seifert framing on $L$ needs not coincide with the Seifert framing of the underlying link. In fact, its framed longitudes correspond to the intersection of $\partial X_L$ with a C-complex. From the definition of the coloured Seifert framing, it is immediate to see that the coefficients of the associated framed linking matrix satisfy

(4.4)\begin{equation} \forall \: 1\leq i \leq k: \quad \displaystyle\sum_{\substack{j\text{ s.t.}\\ c(j)=c(i)}}\Lambda_{ij}=0. \end{equation}

Definition 4.13 Given an $n$-coloured link $L$, a representation $\alpha \colon H_1(X_L;\mathbb {Z})\to U(1)$ is said to be coloured if it sends meridians of the same colour to the same value, i.e. if

\[ c(i)=c(j) \implies \alpha(\mu_i)= \alpha(\mu_j). \]

Given an element $\omega =(\omega _1,\, \dotsc,\, \omega _n)\in \mathbb {T}^{n}$, the coloured representation $\alpha$ defined by $\alpha (\mu _i):=\omega _{c(i)}$ is said to be the representation associated to $\omega$.

It is immediate to see that the above association gives a natural bijection between elements $\omega =(\omega _1,\, \dotsc,\, \omega _n)\in \mathbb {T}^{n}$ and coloured representations $\alpha \colon H_1(X_L;\mathbb {Z})\to U(1)$.

4.3. Multivariable signatures as rho invariants

As the untwisted signature of $W_F$ is $0$ (see for example [Reference Conway, Nagel and Toffoli12, Proposition 3.3]), Proposition 4.11 is equivalent to the formula

\[ \sigma_L(\omega)= \sigma_\alpha(W_F) - \sigma(W_F). \]

In particular, by the Atiyah–Patodi–Singer theorem, we have

(4.5)\begin{equation} \sigma_L(\omega)={-}\rho_\alpha(\partial W_F). \end{equation}

We will give now a more explicit description of $\partial W_F$, and see how to replace it with a manifold which is independent of the choice of $F$.

We start by recalling the following construction, which is a special case of plumbing. Let $\Gamma$ be a graph whose set of vertices is $\{1,\,\dotsc,\, n\}$ and such that:

• • each vertex $i$ is decorated by a compact, oriented surface $\Sigma _i$ (or, equivalently, by pair of natural numbers $[g_i,\,r_i]$ corresponding to the genus and number of boundary components of $\Sigma _i$);

• • each edge is decorated by a number $\varepsilon =\pm 1$.

In the following, we shall refer to a graph with the decorations described above as a plumbing graph. We construct then an oriented three-manifold $P_\Gamma$ by the following process.

1. (1) For each edge with endpoints $i$ and $j$, remove a small open disk from $\Sigma _i$ and one from $\Sigma _j$. Let $\Sigma _1',\, \dotsc,\, \Sigma _j'$ be the resulting surfaces.

2. (2) For each edge with endpoints $i$ and $j$ and decoration $\varepsilon =\pm 1$, glue the three-manifolds $\Sigma _i'\times S^{1}$ and $\Sigma _j'\times S^{1}$ along the boundary components coming from the two corresponding removed disks, according to the diffeomorphism $\varphi \colon S^{1}\times S^{1} \to S^{1}\times S^{1}$ given by $\varphi (x,\,y)= (y^{\varepsilon },\,x^{\varepsilon })$.

3. (3) Set

   \[ P_\Gamma:=\Big(\bigsqcup_{i=1}^{n} \Sigma_i'\times S^{1} \Big)\,/\!\sim, \]
   where $\sim$ is the equivalence relation given by the above gluings.

This construction coincides with that of [Reference Conway, Nagel and Toffoli12, Section 4]. With respect to the general plumbing construction [Reference Neumann26], it corresponds to the special case of all Euler numbers equal to $0$, and our definition of plumbing graph also reflects this specialization. The boundary of $P_\Gamma$ is a disjoint union of $r=\sum \nolimits _i r_i$ tori. These tori maintain a preferred product structure, and we can describe the boundary of $P_\Gamma$ as

(4.6)\begin{equation} \partial P_\Gamma = \bigsqcup_{i=1}^{n}\partial \Sigma_i \times S^{1}. \end{equation}

As in [Reference Conway, Nagel and Toffoli12], we give the following definition.

Example 4.15 Let $F=F_1\cup \cdots \cup F_n\subseteq D^{4}$ be a bounding surface for $L$. The boundary of $W_F$ then is given up to orientation-preserving diffeomorphism as

(4.7)\begin{equation} \partial W_F=X_L\cup_\partial ({-}P_{\Gamma_F}), \end{equation}

where $\Gamma _F$ is a graph whose vertices $\{1,\, \dotsc,\, n\}$ are decorated by the surfaces $F_i$'s, and whose edges correspond to the intersection points between them, with the sign of the intersection as decoration (see [Reference Conway, Nagel and Toffoli12, Example 4.12]). Observe that the total weights of $\Gamma _F$ are given by

\[ p_{\Gamma_F}(s,t)=F_s\cdot F_t = {\operatorname{lk}} (L_s,L_t). \]

The boundary of $-P_{\Gamma _F}$, which can be described as in (4.6), is identified in the gluing (4.7) with $\partial X_L$ as follows. The boundary piece $-\partial F_s\times S^{1}$ is identified with the boundary tori of colour $s$ in $\partial X_L$, in such a way that:

1. (1) the classes of the $S^{1}$-factors are glued to the meridians;

2. (2) the classes of the boundary components of $F_s$ are glued to the framed longitudes associated with the coloured Seifert framing.

As we have seen in (4.5), the multivariable signature of $L$ can be expressed up to sign as the rho invariant of $\partial W_F$. In turn, as it appears from Example 4.15, $\partial W_F$ can be described as the union of $X_L$ with some plumbed three-manifold which depends on the choice of a bounding surface for $L$. In the next construction, we will build a plumbing graph $\Gamma _L$ associated with $L$. The associated closed three-manifold $Y_L$, obtained by gluing $P_{\Gamma _L}$ with $X_L$, will play the role of $\partial W_F$ with the advantage of being unequivocally determined by the link $L$.

Construction 4.17 Given an $n$-coloured link $L$, we construct the associated plumbing graph $\Gamma _L$ in the following way:

• • the set of vertices is $\{1,\,\dotsc n\}$, and the vertex $i$ is decorated by a genus-0 surface with $\left |{L_i}\right |$ boundary components;

• • between each pair of distinct vertices $i,\,j$ there are exactly $\left |{\operatorname { {lk}} (L_i,\,L_j)}\right |$ edges, and they are all decorated by $\varepsilon =\operatorname { {sign}}(\operatorname { {lk}} (L_i,\,L_j))$.

In other words, we are plumbing spheres with the appropriate number of punctures along the smallest graph $\Gamma$ whose total weights satisfy the condition $p_{\Gamma }(i,\,j)= \operatorname { {lk}}(L_i,\,L_j)$. Then, we form a closed three-manifold $Y_L$ as

\[ Y_L:= X_L\cup_\partial(- P_{\Gamma_L}). \]

The identification along the boundary is defined in the exact same way as the one arising in Example 4.15, i.e. identifying the classes corresponding to the $S^{1}$-factors of $-\partial P_{\Gamma _L}$ with the meridians of $L$ of the appropriate colour, and the classes corresponding to the boundary components of the punctured sphere with the longitudes associated with the coloured Seifert framing.

Using some of the ideas of [Reference Conway, Nagel and Toffoli12] together with the cut-and-paste formula of § 2, we will now show that the multivariable signature can be written as the rho invariant of $Y_L$.

Proposition 4.18 Let $L$ be an $n$-coloured link. Let $\omega \in \mathbb {T}_*^{n},$ and let $\alpha \colon H_1(X_L;\mathbb {Z})\to U(1)$ be the associated coloured representation. Then, $\alpha$ can be extended to a representation of $H_1(Y_L;\mathbb {Z})$ and, for any choice of an extension, we have

\[ \rho_\alpha(Y_L) ={-} \sigma_L(\omega). \]

Proof. Let $F=F_1\cup \cdots \cup F_n\subseteq D^{4}$ be a bounding surface for $L$. Then, as $\alpha$ is a coloured representation, it extends to a representation $H_1(W_F;\mathbb {Z})\to U(1)$ that, for all colour $s$, sends the meridian of $F_s$ to $\omega _s\in U(1)$. By (4.5), we have hence

(4.8)\begin{equation} \rho_\alpha(\partial W_F)={-} \sigma_L(\omega). \end{equation}

As we have seen in Example 4.15, we have $\partial W_F=X_L\cup _\partial (-P_{\Gamma _F})$, where $\Gamma _F$ is a plumbing graph determined by $F$. We perform now cut-and-paste by replacing $P_{\Gamma _F}$ with $P_{\Gamma _L}$. Schematically, this is

(4.9)\begin{equation} X_L\cup_\partial ({-}P_{\Gamma_F}) \quad \leadsto \quad X_L\cup_\partial ({-}P_{\Gamma_L}) \ \sqcup \ P_{\Gamma_L} \cup_\partial({-}P_{\Gamma_F}). \end{equation}

The manifold $P_{\Gamma _L}\cup _\partial (-P_{\Gamma _F})$ can be seen as the plumbing along the graph $\Gamma$ whose vertices $\{1,\, \dotsc,\, n\}$ are decorated by the closed surfaces $\Sigma _s\cup _\partial (-F_s)$ and whose set of edges is the union of all edges of $\Gamma _L$ and $\Gamma _F$, with the decorations of the edges of $\Gamma _F$ changed of sign. In particular, (4.9) can be rewritten as

(4.10)\begin{equation} \partial W_F \quad \leadsto \quad Y_L \ \sqcup \ P_\Gamma. \end{equation}

Observe that, by construction, for each pair of vertices $\{s,\,t\}$ we have

(4.11)\begin{equation} p_{\Gamma_L}(s,t) = p_{\Gamma_F}(s,t), \end{equation}

so that

\[ p_{\Gamma}(s,t)= p_{\Gamma_L}(s,t)- p_{\Gamma_F}(s,t)=0 \]

i.e. $\Gamma$ is balanced.

We will now prove that the representation $\alpha$ can be extended to $H_1(Y_L;\mathbb {Z})$. In the gluings, the boundary of $X_L$ is identified with the boundaries of $P_{\Gamma _F}$ and $P_{\Gamma _L}$, leading to natural maps

(4.12)\begin{equation} \varphi\colon H_1(\partial X_L;\mathbb{Z})\to H_1(P_{\Gamma_F};\mathbb{Z}) , \quad \psi\colon H_1(\partial X_L;\mathbb{Z})\to H_1(P_{\Gamma_L};\mathbb{Z}) \end{equation}

induced by the inclusions. Standard Mayer–Vietoris computations, together with the equality (4.11), show that $\ker \varphi$ and $\ker \psi$ coincide, as both are generated by the following elements (compare with [Reference Conway, Nagel and Toffoli12, Lemma 4.7]):

1. (i) the differences $\mu _i-\mu _j$ with $c(i)=c(j)$;

2. (ii) for each colour $s$, the element

   \[ \sum_{c(i)=s}\ \lambda_i - \sum_{t\ne s}{\operatorname{lk}}(L_s,L_t)\mu_{j_t}, \]
   where $\mu _{j_t}$ is any meridian of colour $t$.

As $\alpha$ extends to $H_1(W_F;\mathbb {Z})$, it is also defined on $H_1(\partial W_F;\mathbb {Z})=H_1(P_{\Gamma _F};\mathbb {Z})$. In particular, $\alpha$ has to be trivial on $\ker \varphi$ (this can be verified using the explicit description of the generators). From the fact that $\ker \varphi =\ker \psi$, we see then that $\alpha$ also admits an extension to $H_1(P_{\Gamma _L};\mathbb {Z})$, because $U(1)$ is divisible.

Pick any extension $\alpha \colon H_1(P_{\Gamma _L};\mathbb {Z})\to U(1)$ of $\alpha$. We can now apply the cut-and-paste formula of Theorem 3.9 to (4.10), obtaining

(4.13)\begin{equation} \rho_\alpha(\partial W_F)=\rho_\alpha(Y_L) + \rho_\alpha(P_{\Gamma})- \tau( V_{P_{\Gamma_L}},V_{X_L},V_{P_{\Gamma_F}}) \end{equation}

(the Maslov triple index in twisted homology is $0$ because all $\omega _i$'s are non-trivial by assumption; see Remark 4.2). As we have seen, the plumbing graph $\Gamma$ is balanced. As a consequence, we have $\rho _\alpha (P_{\Gamma })=0$ by a computation of Conway, Nagel and the author [Reference Conway, Nagel and Toffoli12, Proposition 4.10]. The Lagrangians $V_{P_{\Gamma _F}}$ and $V_{P_{\Gamma _L}}$ are identified under the gluing to

\[ V_{P_{\Gamma_F}}=\ker\varphi\otimes \mathbb{C}, \quad V_{P_\Gamma}=\ker \psi\otimes \mathbb{C}, \]

where $\varphi$ and $\psi$ are the maps defined in (4.12). From the fact that $\ker \varphi =\ker \psi$ it follows then that $V_{P_{\Gamma _F}}=V_{P_\Gamma }$, and hence we have $\tau (V_{X_L},\,V_{P_{\Gamma _F}},\, V_{P_{\Gamma _L}})=0$. The equality (4.13) gets thus rewritten as

\[ \rho_\alpha(\partial W_F)=\rho_\alpha(Y_L). \]

Substituting this into (4.8), the proof is complete.

We introduce now the following definitions.

Of course, being component-to-colour algebraically split is a stronger condition than being colour-to-colour algebraically split. Any onecoloured link is component-to-colour algebraically split, and so is any link with vanishing linking numbers, no matter what colouring it is assigned.

Remark 4.20 If $L$ is colour-to-colour algebraically split, then $Y_L$ has a very simple description. In fact, in this case, $\Gamma _L$ is a graph with no edges, and thus the associated plumbed three-manifold is

\[ P_{\Gamma_L}=\bigsqcup_{s=1}^{n} \Sigma_s\times S^{1}, \]

where $\Sigma _s$ is a sphere with $\left |{L_s}\right |$ punctures. As a consequence of Proposition 4.18, the multivariable signature is hence (up to sign) just the rho invariant of the closed three-manifold obtained by gluing these products $\Sigma _s\times S^{1}$ to the link exterior $X_L$. This holds, in particular, if $L$ is onecoloured, so that we can always express the Levine–Tristram signature of $L$ as

\[ \sigma_L(\omega)={-}\rho_\alpha(X_L\cup_\partial (-\Sigma\times S^{1})), \]

where $\Sigma$ is a punctured sphere. In the case of a knot, this gives the well-known description of the Levine–Tristram signature as the rho invariant of the manifold obtained by $0$-framed surgery. In the next two sections, we will study the relationship between rho invariants and the Dehn surgery in higher generality.

4.4. Integral surgery

We will now study the value of the rho invariant of manifolds obtained by integral surgery on a link $L$. We start by recalling the following result of Cimasoni and Florens [Reference Cimasoni and Florens9, Theorem 6.7].

Theorem 4.21 Cimasoni–Florens

Let $L$ be a $k$-coloured $k$-component link. Let $q\in \mathbb {N}$ a positive integer and let $n_1,\,\dotsc,\, n_k\in \{1,\,\dotsc,\, p-1\}$ be integers, each of which is coprime with $q$. Let $\omega =(e^{2\pi in _1/q},\, \dotsc,\, e^{2\pi i n_k/q})\in (S^{1}\setminus \{1\})^{n}$, and let $\alpha \colon H_1(X_L;\mathbb {Z})\to U(1)$ be the associated coloured representation. Let $g$ be a compatible integral framing on $L$. Then, we have

\[ \rho_\alpha(S_L(g))={-}\sigma_L(\omega) +\sum_{i< j}\Lambda_{ij} + {\operatorname{sign}} \Lambda_g -{\displaystyle\frac{2}{q^{2}}}\sum_{i=1}^{k} (q-n_i)n_j \Lambda_{ij}. \]

We would like to remove the restrictions on the values of $\omega$ in the statement of Theorem 4.21. In order to be able to do so, we need to assume some conditions about the linking numbers between components of different colours.

The main result of this section is the following, which holds in the case of colour-to-colour algebraically split links.

Theorem 4.24 Let $L$ be an $n$-coloured link which is colour-to-colour algebraically split. Let $\omega \in \mathbb {T}_*^{n}$, and let $\alpha \colon H_1(X_L;\mathbb {Z})\to U(1)$ be the associated coloured representation. Let $g$ be a compatible integral framing on $L$. Then, we have

\[ \rho_\alpha(S_L(g))={-}\sigma_L(\omega) + {\operatorname{sign}} \Lambda_g - 2\sum_{s=1}^{n} h_s\theta_s(1-\theta_s), \]

where, for each colour $s,$ $\theta _s\in (0,\,1)$ is such that $\omega _s=e^{2\pi i \theta _s},$ and $h_s$ is the sum of all the coefficients of the framed linking matrix of the sublink $L_s$.

Proof. Set $P:=P_{\Gamma _L}$. Let $k$ be the number of components of $L$. We perform the cut-and-paste illustrated schematically by

(4.14)\begin{equation} X_L\cup_\partial Y \quad \leadsto \quad X_L\cup_\partial ({-}P) \:\:\sqcup \:\: P\cup_\partial Y, \end{equation}

where $Y=Y_1\sqcup \cdots \sqcup Y_k$ is a disjoint union of $k$ solid tori, glued along the framing $g$, and $P$ is glued as prescribed by Construction 4.17. As $L$ is colour-to-colour algebraically split, by Remark 4.20, we have

\[ P=\bigsqcup_{s=1}^{n} \Sigma_s\times S^{1}, \]

where, for each colour $s$, $\Sigma _s$ is a sphere with $\left |{L_s}\right |$ punctures. The manifold $P\cup _\partial Y$ is thus the disjoint union of the $n$ closed manifolds obtained by capping all of the $\Sigma _s\times S^{1}$'s appropriately with solid tori.

Claim 1 Up to an orientation-preserving diffeomorphism, we have

\[ P\cup_\partial Y={-}\bigsqcup_{s=1}^{n} L(h_s,1), \]

in such a way that the element $1\in \mathbb {Z}/h_s=\pi _1(L(h_s,\,1))$ is given by the class $[S^{1}]\in H_1(\Sigma _s\times S^{1};\mathbb {Z})$.

We postpone for the moment the proof of this claim. As a consequence of Claim 1, (4.14) can be rewritten up to orientation-preserving diffeomorphism as

\[ S_L(g) \quad \leadsto \ Y_L \ \sqcup \ -\bigsqcup_{s=1}^{n} L(h_s,1). \]

Observe that the restriction of $\alpha$ to $H_1(\Sigma _s\times S^{1})$ sends the class $[S^{1}]$ to $\omega _s$, as the boundary circles of the form $\{p_i\}\times S^{1}$ are identified with the meridians of the link. The representation $\alpha$ extends thus to $L(h_s,\,1)$ in such a way that, for each colour $s$, the element $1\in \mathbb {Z}/h_s=H_1(L(h_s,\,1);\mathbb {Z}))$ is sent to $\omega _s$. We can now apply Theorem 3.9 and get

(4.15)\begin{equation} \rho_\alpha(S_L(g)) =\rho_\alpha(Y_L)- \displaystyle\sum_{s=1}^{n}\rho (L(h_s,1),\omega_s) - \tau(V_P, V_{X_L}, V_Y) \end{equation}

(as usual, there is no Maslov triple index in twisted cohomology because of Remark 4.2). The first summand in the right-hand term of (4.15) is minus the multivariable signature of $L$ thanks to Proposition 4.18. Using Corollary 2.11 to describe the rho invariant of the lens spaces at hand, and swapping the second and third variables of the Maslov triple index (see Proposition 3.2 (i)), we can rewrite (4.15) as

(4.16)\begin{equation} \rho_\alpha(S_L(g)) ={-}\sigma_L(\omega)- \displaystyle\sum_{s=1}^{n} (2h_s\theta_s(1-\theta_s)- {\operatorname{sgn}}(h_s)) + \tau(V_P, V_Y, V_{X_L}). \end{equation}

Hence, in order to conclude, we need to identify the Maslov triple index term in (4.16). The rest of the proof is devoted to this.

Instead of trying to calculate the Maslov triple index directly, we use the cocycle property of our three Lagrangians together with $\mathcal {M}:=\operatorname { {Span}}_\mathbb {C}\{\mu _1,\,\dotsc,\, \mu _k\}$ to simplify this task. Namely, using Proposition 3.2 (ii), we find

(4.17)\begin{equation} \tau(V_P, V_Y, V_{X_L}) = \tau (\mathcal{M},V_P, V_Y)- \tau(\mathcal{M}, V_P, V_{X_L}) + \tau(\mathcal{M}, V_Y, V_{X_L}). \end{equation}

We set the following notation. Let $\lambda _1,\,\dotsc,\, \lambda _k$ be the framed longitudes associated with the surgery framing $g=(g_1,\,\dotsc,\, g_k)$, and let $\lambda _1,\,\dotsc,\, \lambda _k$ bet the framed longitudes associated with the coloured Seifert framing $f_L=(f_1,\,\dotsc,\, f_k)$. By definition, then, we have

(4.18)\begin{equation} \lambda_i'=\lambda_i + (f_i-g_i)\mu_i \quad \text{for all } i=1,\dotsc , k. \end{equation}

We also use the notation $\Lambda _{ij}$ for the coefficients of the framed linking matrix $\Lambda _g$, and $\Lambda _{ij}'$ for those of the frame linking matrix $\Lambda _{f_L}$ (these two matrices only differ on the diagonal). We have the following description of the four Lagrangian subspaces appearing in the right-hand term of (4.17):

\begin{align*} \mathcal{M}& ={\operatorname{Span}}_\mathbb{C}\{\mu_1,\dotsc , \mu_k\},\\ V_P& ={\operatorname{Span}}_\mathbb{C}\{\mu_i-\mu_j\, \:\vert \: c(i)=c(j)\} \oplus {\operatorname{Span}}_\mathbb{C}\{ v_1, \dotsc , v_n\}, \quad \text{where } v_s=\displaystyle\sum_{c(i)=s}\lambda_i',\\ V_Y& ={\operatorname{Span}}_\mathbb{C}\{\lambda_1,\dotsc, \lambda_k\} ,\\ V_{X_L}& = {\operatorname{Span}}_\mathbb{C}\{w_1, \dotsc, w_k\}, \quad \text{where } w_i=\lambda_i - \sum_{j=1}^{k}\Lambda_{ij} \mu_j= \lambda_i' - \sum_{j=1}^{k}\Lambda_{ij}' \mu_j. \end{align*}

We compute now the three summands separately. We will prove the following.

Claim 2 $\tau (\mathcal {M},\,V_P,\, V_Y)=-\sum \nolimits _{s=1}^{n}\operatorname { {sgn}}(h_s)$.

Claim 3 $\tau (\mathcal {M},\, V_P,\, V_{X_L})=0$.

Claim 4 $\tau (\mathcal {M},\, V_Y,\, V_{X_L})=\operatorname { {sign}}(\Lambda _g)$.

These three claims, together with (4.16) and (4.17), lead to the desired formula. In order to conclude, we are only left with proving Claim 1 to 4.

Proof of Claim 2. Write $\tau (\mathcal {M},\,V_P,\, V_Y)=\tau (V_Y,\,\mathcal {M},\,V_P)$. Clearly, we have

\[ (V_Y +\mathcal{M})\cap V_P=V_P. \]

In fact, the generators of $V_P$ of the form $\mu _i-\mu _j$ are in $\mathcal {M}$, and the generators $v_s$ can be written as

(4.19)\begin{equation} v_s= \displaystyle\sum_{c(i)=s}\lambda_i + \sum_{c(i)=s}(f_i-g_i)\mu_i, \end{equation}

where the first summand is in $V_Y$, and the second summand is in $\mathcal {M}$. Let

\[ \psi\colon V_P\times V_P \to \mathbb{C} \]

be the Hermitian form associated with the triple $(V_Y,\,\mathcal {M},\,V_P)$, whose signature is $\tau (V_Y,\,\mathcal {M},\,V_P)$. The generators of the form $\mu _i-\mu _j$ are clearly in the radical of $\psi$, as they belong to the Lagrangian $\mathcal {M}$. As a consequence, it is enough to study $\psi$ on the span of $v_1,\, \dotsc,\, v_n$. Using the definition of $\psi$ and the decomposition (4.19), we compute

\[ \psi(v_s,v_t)= \left(\sum_{c(i)=s}\lambda_i\right)\cdot \left(\sum_{c(j)=t}(f_j-g_j)\mu_j\right)= \begin{cases} \displaystyle \sum_{c(i)=s}(f_i-g_i), & \text{if } s=t,\\ 0, & \text{otherwise}. \end{cases} \]

As a consequence, we have

\[ {\operatorname{sign}} \psi = \sum_{s=1}^{n}{\operatorname{sgn}} \left(\sum_{c(i)=s}(f_i-g_i) \right). \]

By definition of the coloured Seifert framing, together with the fact that $g_i=\Lambda _{ii}$, we can now compute that, for each colour $s$, we have

\[ \sum_{c(i)=s}(f_i-g_i)=\sum_{c(i)=s}\left(-\sum_{\substack{c(j)=s\\j\ne i}}{\operatorname{lk}}(K_i,K_j)\right)-\sum_{c(i)=s}g_i={-} \ \sum_{c(i)=c(j)=s} \ \Lambda_{ij}={-}h_s. \]

Putting these computations together, we find the equation in the statement of the claim.

Proof of Claim 3. The space $(\mathcal {M}+V_P)\cap V_{X_L}$ is the $n$-dimensional subspace generated by the terms

(4.20)\begin{equation} z_s:=\displaystyle\sum_{c(i)=s}w_i ={-}\sum_{c(i)=s}\sum_{j=1}^{k}\Lambda_{ij}'\mu_j + \sum _{c(i)=s}\lambda_i' \end{equation}

where the first summand is in $\mathcal {M}$ and the second summand is in $V_P$. Let

\[ \varphi \colon {\operatorname{Span}}_\mathbb{C}\{z_1, \dotsc , z_n\} \times {\operatorname{Span}}_\mathbb{C}\{z_1, \dotsc , z_n\} \to \mathbb{C} \]

be the Hermitian form associated with the triple $(\mathcal {M},\, V_P,\, V_{X_L})$. Then, from the decomposition (4.20), we can compute

\[ \varphi(z_s,z_t)= \left(-\sum_{c(i)=s}\sum_{j=1}^{k}\Lambda_{ij}'\mu_j \right)\cdot \left(\sum _{c(i)=t}\lambda_i'\right) =\sum_{\substack{c(i)=s\\ c(j)=t}} \Lambda_{ij}'. \]

For $s=t$, this is $0$ by definition of the coloured Seifert framing. For $s\ne t$, instead, it is equal to $\operatorname { {lk}}(L_s,\,L_t)$, which is $0$ because the link is colour-to-colour algebraically split. In particular, the form $\varphi$ is trivial and the Maslov triple index is $0$ as claimed.

Proof of Claim 4. This follows immediately from Lemma 4.6 (iii), as $V_Y=\mathcal {L}_g$.

Proof of Claim 1. As we have observed, $P_{\Gamma _L}$ is the disjoint union of manifolds of the form $\Sigma _s\times S^{1}$, where $\Sigma _s$ is a punctured sphere. By construction, in the gluing $X_L\cup _\partial Y$, the meridian $m_i$ of the solid torus $Y_i$ is identified with the framed longitude $\lambda _i$ of the surgery framing $g$. On the other hand, when $-P_{\Gamma _L}$ is glued to $X_L$, a boundary component $C_i\times S^{1}\subseteq \partial \Sigma _s\times S^{1}$ is identified with the boundary torus $K_i$ in such a way that, homologically, the class $[C_i]$ coincides with the framed longitude $\lambda _i'$ of the coloured Seifert framing, and $[S^{1}]$ coincide with the meridian $\mu _i$. The “by-product” gluing $P_{\Gamma _L}\cup _\partial Y$ is the result of capping the boundary $\Sigma _s\times S^{1}$ with solid tori. As a consequence of this and (4.18), these cappings are given by the identifications

\[ m_i = \lambda_i = \lambda_i'+(g_i-f_i)\mu_i =[C_i]+(g_i-f_i)[S^{1}]. \]

In particular, for each $s$, this construction leads to the lens space $L(-h'_s,\, 1)$ or equivalently to $-L(h_s',\,1)$, with

\[ h'_s:= \sum_{c(i)=s} (g_i-f_i). \]

It is also easy to see that the element $1\in \mathbb {Z}/h_s'$ corresponds to $[S^{1}]$ as desired. The proof is concluded by proving that $h_s'=h_s$. This follows from the definition of the coloured Seifert framing and the sequence of equalities

\[ \sum_{c(i)=s} (g_i-f_i)= \sum_{c(i)=s}g_i + \sum_{c(i)=s}\sum_{\substack{\:j\ne i \text{ s.t.\ }\\ c(j)=s}} {\operatorname{lk}}(K_i,K_j)=\ \sum _{c(i)=c(j)=s}\ \Lambda_{ij}=h_s. \]

Remark 4.25 Applying Theorem 4.24 in the onecoloured setting, where the first hypothesis is always satisfied, we get the following formula relating the rho invariant of the manifold obtained by surgery and the Levine–Tristram signature:

(4.21)\begin{equation} \rho_\alpha(S_L(g))={-}\sigma_L(\omega) + {\operatorname{sign}} \Lambda_g - 2 (\displaystyle\sum_{i,j}\Lambda_{ij})\theta(1-\theta), \end{equation}

where $\theta \in (0,\,1)$ is such that $\omega =e^{2\pi i\theta }$ and the sum is over the coefficients of $\Lambda _g$. For $\theta \in \mathbb {Q}$, this coincides with a formula of Casson and Gordon [Reference Casson and Gordon7, Lemma 3.1].

Example 4.26 Let $r,\,s$ be positive coprime integers, and let $T(r,\,s)$ denote the $(r,\,s)$-torus knot. The $(rs-1)$-Dehn surgery on $T(r,\,s)$ gives a manifold which is orientation-preserving diffeomorphic to the lens space $L(rs-1,\,s^{2})$ [Reference Moser24, Proposition 3.2]. Let $\zeta :=e^{2\pi i/(rs-1)}$, and let $0\leq k \leq rs-2$. Keeping track of the induced map on the fundamental group under this diffeomorphism, by (4.21) (applied with $\omega =\zeta ^{k}$) we obtain

\[ \rho(L(rs-1, s^{2}), \zeta^{krs^{2}})={-}\sigma_{T(r,s)}(\zeta^{k}) +1 -{\displaystyle\frac{2k(rs-1-k)}{rs-1}}. \]

It might be interesting to compare this formula to other known computations for the Levine–Tristram signature of torus knots (see e.g. the paper of Borodzik and Oleszkiewicz [Reference Borodzik and Oleszkiewicz4]).

Suppose now that the $n$-coloured link $L$ is component-to-colour algebraically split. Observe that, under this assumption, the coloured Seifert framing coincides with the usual Seifert framing, i.e. with the coloured Seifert framing associated with the onecolouring of the same underlying link. As explained by the next result, this framing has the important property of being compatible with all $U(1)$-representations of $H_1(X_L;\mathbb {Z})\to U(1)$.

Proof. By Remark 4.8, we need to prove that

(4.22)\begin{equation} \prod_{j=1}^{k}\alpha(\mu_j)^{\Lambda_{ij}}=1 \quad \quad \text{for all } i. \end{equation}

Let $\omega \in \mathbb {T}^{n}$ be the element determined by the relations $\alpha (\mu _i)=\omega _{c(i)}$ for all $i$. We can write then

(4.23)\begin{equation} \prod_{j=1}^{k}\alpha(\mu_j)^{\Lambda_{ij}}=\prod_{s=1}^{n} \omega_s^{\sum_{c(j)=s}\Lambda_{ij}}. \end{equation}

Since $L$ is component-to-colour algebraically split, for every $s$ different from $s_i:=c(i)$ we have

\[ \sum_{c(j)=s}\Lambda_{ij}={\operatorname{lk}}(K_i,L_s)=0. \]

As a consequence, (4.23) can be rewritten as

(4.24)\begin{equation} \prod_{j=1}^{k}\alpha(\mu_j)^{\Lambda_{ij}}= \omega_{s_i}^{\sum_{c(j)=s_i}\Lambda_{ij}}. \end{equation}

By (4.4), moreover, the exponent in the right-hand term of (4.24) is $0$, and thus (4.22) is satisfied.

Corollary 4.28 Let $L$ be an $n$-coloured link which is component-to-colour algebraically split. Let $\omega \in \mathbb {T}_*^{n},$ and let $\alpha \colon H_1(X_L;\mathbb {Z})\to U(1)$ be the associated coloured representation. Then, $\alpha$ extends to $H_1(S_L(f_L);\mathbb {Z})$ and we have

\[ \rho_\alpha(S_L(f_L))={-}\sigma_L(\omega) + {\operatorname{sign}} \Lambda_{f_L}. \]

Proof. The representation $\alpha$ extends to $H_1(S_L(f_L);\mathbb {Z})$ thanks to Lemma 4.27, so that we can apply Theorem 4.24. The desired formula follows then from the observation that, for the coloured Seifert framing, thanks to (4.4) for any colour $s$ we have

\[ h_s\stackrel{\text{def}}{=}\sum_{c(i)=c(j)=s}\Lambda_{ij}=\sum_{c(i)=s} \sum_{c(j)=s}\Lambda_{ij}=\sum_{c(i)=s}0=0. \]

4.5. Rational surgery

We now want to study the rho invariant of the closed manifold obtained by surgery along a rational framing on a link. We start from the case of a knot, as the statement and the proof are a bit simpler in this setting. Observe that, for a knot $K$, a representation $\psi \colon H_1(X_K)\to U(1)$, extends to $M_K(p/q)$ if and only if $\psi (\mu )^{p}=1$, i.e. if and only if $\omega :=\psi (\mu )$ is a $p$^th root of unity.

Proposition 4.30 Let $K$ be a knot, and let $\alpha \colon H_1(X_K;\mathbb {Z})\to U(1)$ be the representation defined by $\alpha (\mu )=\omega,$ with $\omega$ being a $p$^th root of unity. Then, we have

\[ \rho_\alpha(S_K(p/q))={-}\sigma_K(\omega) - \rho(L(p,q),\omega). \]

Proof. We perform cut-and-paste on $S_K(p/q)$ in the following way: we cut out the solid torus $Y=D^{2}\times S^{1}$ of the filling, and we replace it with another copy $Y'=D^{2}\times S^{1}$, this time glued along the $0$ framing. It is convenient to actually glue $-Y'$ instead of $Y'$: in such a way, we can define an orientation-reversing diffeomorphism between $-\partial Y'$ and $\partial X_K$ which gives the identifications

\[ m'=\lambda, \quad l' =\mu \]

between the standard basis $(m',\,l')$ of $H_1(Y';\mathbb {Z})$ and the basis $(\mu,\, \lambda )$ of $H_1(\partial X_L;\mathbb {Z})$. On the other hand, the meridian $m$ of $Y$ is identified with $p\mu +q\lambda$. Schematically, we write

(4.25)\begin{equation} X_K\cup_\partial Y \quad \leadsto \quad X_K\cup_\partial ({-}Y') \ \sqcup \ Y'\cup_\partial Y. \end{equation}

The union of solid tori $Y'\cup _\delta Y$ is now given along a diffeomorphism which gives the identification of $m$ with $qm'+pl'$, so that the resulting manifold is the lens space $L(p,\,-q)\cong -L(p,\,q)$ (see § 2.4). In particular, (4.25) can be rewritten as

\[ S_K(p/q) \quad \leadsto \quad S_K(0) \ \sqcup \ -L(p,q). \]

Observe that the generator $1\in \mathbb {Z}/p=\pi _1(L(p,\,q))$ corresponds by construction to the longitude $l'$. In turn, $l'$ is glued in the surgery with the meridian $\mu$ of $K$. As a consequence, the extension of the representation $\alpha$ to $H_1(L(p,\,q);\mathbb {Z})$ is the representation $\mathbb {Z}/p\to U(1)$ given by $1\mapsto \omega$. Before applying the cut-and-paste formula, we observe that the result is trivially true for $\omega =1$. We shall hence suppose $\omega \ne 1$. Thanks to Theorem 3.9, we can now compute

(4.26)\begin{equation} \rho_\alpha(S_K(p/q))= \rho_\alpha(S_K(0)) - \rho(L(p,q),\omega) -\tau(V_{Y'}, V_{X_K}, V_Y) \end{equation}

(as usual, the Maslov triple index in twisted homology is $0$ thanks to Remark 4.2 and the assumption $\omega \ne 1$). Now, we know that the first summand in the right-hand term of (4.26) is minus the Levine–Tristram signature (see Remark 4.20), while the second summand coincides with the one of the statement. As a consequence, to complete the proof, it is enough to show that $\tau (V_{Y'},\, V_{X_K},\, V_Y)=0$. We are going to describe the Lagrangians explicitly. Considering the identifications given by the gluings, in terms of the basis $(\mu,\,\lambda )$ of $H_1(\partial X_K;\mathbb {C})$ we have

\[ V_{Y'}={\operatorname{Span}}_\mathbb{C}(\lambda), \quad V_{X_K}={\operatorname{Span}}_\mathbb{C}(\lambda),\quad V_Y={\operatorname{Span}}_\mathbb{C}(p\mu +q\lambda). \]

As the first two subspaces coincide, the Maslov triple index is $0$, and the proof is complete.

We shall now prove the general version of Proposition 4.30, which holds for all component-to-colour algebraically split links. As we have seen, a special case of these are onecoloured links. In particular, this gives a general formula relating the Levine–Tristram signature of a link $L$ with the Atiyah–Patodi–Singer rho invariant of the closed three-manifold obtained by rational surgery on $L$.

Theorem 4.31 Let $L$ be an $n$-coloured, $k$-component link that is component-to-colour algebraically split. Let $\omega \in \mathbb {T}_*^{n},$ and let $\alpha \colon \pi _1(X_L)\to U(1)$ be the associated coloured representation. Let $r$ be a compatible rational framing on $L$. Then, we have

\[ \rho_\alpha(S_L(r))={-}\sigma_L(\omega) + {\operatorname{sign}} \Lambda_r - \sum_{i=1}^{k} (\rho(L(p_i,q_i),\omega_{c(i)}) + {\operatorname{sgn}}(p_i/q_i)) , \]

where $p_i,\,q_i$ are coprime integers such that $r_i-f_i=p_i/q_i$ (here, $f_i$ is the $i$-th coefficient of the Seifert framing).

Proof. We generalize the cut-and-paste construction of Proposition 4.30, removing the union of solid tori $Y$ coming from the $r$-framed filling of $X_L$, and replacing them with a union of solid tori $Y'$ glued along the coloured Seifert framing. As the difference between these framings is now given by the $k$-tuple $(p_1/q_1,\, \dotsc,\, p_k/q_k)$, the same argument used in the proof of Proposition 4.30 (repeated now for each link component) implies that this cut-and-paste can be written as

\[ S_L(r) \quad \leadsto S_L(f_L) \ \sqcup \ \left(\bigsqcup_{i=1}^{k} -L(p_i,q_i) \right). \]

In particular, Theorem 3.9 gives in this case

\[ \rho_\alpha(S_L(r)) = \rho_\alpha(S_L(f_L)) - \sum_{i=1}^{k} \rho_\alpha(L(p_i,q_i)) - \tau(V_{Y'}, V_{X_L}, V_Y). \]

Applying Corollary 4.28 and rearranging the variables of the Maslov triple index with Proposition 3.2 (i), we can rewrite the last equation as

(4.27)\begin{equation} \rho_\alpha(S_L(r)) ={-}\sigma_L(\omega) + {\operatorname{sign}} \Lambda_{f_L} - \displaystyle\sum_{i=1}^{k} \rho(L(p,q), \omega_{c(i)}) + \tau(V_{Y'}, V_Y, V_{X_L}). \end{equation}

The computation of the Maslov triple index is more involved than in the case of knots. As we did in the proof of Theorem 4.24, instead of trying to calculate it directly, we use the cocycle property of our three Lagrangians together with $\mathcal {M}:=\{\mu _1,\,\dotsc,\, \mu _k\}$ to simplify this task. Namely, using Proposition 3.2 (ii), we find

(4.28)\begin{equation} \tau(V_{Y'}, V_Y, V_{X_L}) = \tau (\mathcal{M},V_{Y'}, V_Y)- \tau(\mathcal{M}, V_{Y'}, V_{X_L}) + \tau(\mathcal{M}, V_Y, V_{X_L}). \end{equation}

Claim 1 $\tau (\mathcal {M},\,V_{Y'},\, V_Y)= -\sum \nolimits _{i=1}^{k} \operatorname { {sgn}}(p_i/q_i)$.

Claim 2 $\tau (\mathcal {M},\, V_{Y'},\, V_{X_L})= \operatorname { {sign}}\Lambda _{f_L}$ and $\tau (\mathcal {M},\, V_Y,\, V_{X_L})=\operatorname { {sign}} \Lambda _r$.

Thanks to (4.28), the two claims (whose proof we postpone for the moment) combine to give

\[ \tau(V_{Y'}, V_Y, V_{X_L}) ={-}\sum_{i=1}^{k} {\operatorname{sgn}}(p_i/q_i)- {\operatorname{sign}}\Lambda_{f_L} + {\operatorname{sign}} \Lambda_r. \]

Substituting this value into (4.27), we obtain the formula in the statement of the theorem.

Proof of Claim 1. Let $\lambda _1,\,\dotsc,\, \lambda _k$ be the framed longitudes corresponding to the framing $r$, and let $\lambda _1',\,\dotsc,\, \lambda _k'$ be the framed longitudes of the Seifert framing. Then, it is clear that

\[ V_Y={\operatorname{Span}}_\mathbb{C}\{\lambda_1, \cdots , \lambda_k\}, \quad \quad V_{Y'}={\operatorname{Span}}_\mathbb{C}\{\lambda_1', \cdots , \lambda_k'\}. \]

Moreover, by definition, we have $\lambda _i=\lambda _i^{s}+r_i\mu _i$ and $\lambda _i'=\lambda _i^{s}+f_i\mu _i$, so that the two sets of longitudes are related by

\[ \lambda_i=\lambda_i'+(r_i-f_i)\mu_i= \lambda_i'+ {\displaystyle\frac{p_i}{q_i}} \mu_i. \]

In particular, the three Lagrangians in the first summand of (4.28) all split according to the symplectic decomposition

\[ H_1(\partial X_L;\mathbb{C})=\bigoplus_{i=1}^{k} {\operatorname{Span}}_\mathbb{C}\{\mu_i,\lambda_i\}. \]

As the pair $(\mu _i,\,\lambda _i')$ is a symplectic basis for $(H_1(T_i;\mathbb {C}),\, \cdot )$ (see Example 3.3), it is immediate to compute

\[ \tau (\mathcal{M},V_{Y'}, V_Y)= \sum_{i=1}^{k} \tau\Big(\mu_i, \lambda_i', \, \lambda_i' + {\displaystyle\frac{p_i}{q_i}} \mu_i\Big)={-}\sum_{i=1}^{k} {\operatorname{sgn}}(p_i/q_i). \]

Proof of Claim 2. The claim follows immediately from Lemma 4.6 (iii), as we have $V_Y=\mathcal {L}_r$ and $V_{Y'}=\mathcal {L}_{f_L}$.