**Setting:** There are two objects in knot theory that are commonly referred to as the Casson-Gordon invariants: the invariant $\sigma$, and the invariant $\tau$ (see for example A. Conway’s notes *Algebraic Concordance and Casson-Gordon Invariants* [3] for an introduction to these invariants). When it comes to the $\sigma$-invariant, usual references in the literature include Casson and Gordon’s original papers *Cobordism of Classical Knots* [1], where the invariant appears as $\sigma(M, \chi)$, and *On Slice Knots in Dimension Three* [2], where the invariant appears as $\sigma_r(M, \chi)$. I am currently trying to understand the relation between these two different formulations of the $\sigma$-invariant, but I am having some difficulties with it.

To be more precise, let me quickly outline the construction of the invariant $\sigma(M, \chi)$, as found in [3] on p.14, (resp. [1] on p. 183), and of $\sigma_r(M, \chi)$, as found in [2] on p. 41/42 (readers that are already familiar with the definitions and/or cited papers might want to skip the next two paragraphs).

**Definition of $\sigma(K, \chi)$:** Given a compact $4$-manifold $W$ and a morphism $\psi: \pi_1(W) \rightarrow \mathbb{Z}_m$, let $\widetilde W \rightarrow W$ be the associated $\mathbb{Z}_m$-covering. Then $H_2(\widetilde W; \mathbb{Z})$ is a $\mathbb{Z}[\mathbb{Z}_m]$-module. By mapping the generator of $\mathbb{Z}_m$ to $\omega := e^{\frac{2\pi i}{m}}$, we get a map $\mathbb{Z}_m \rightarrow \mathbb{Q}(\omega)$, which endows $\mathbb{Q}(\omega)$ with a $(\mathbb{Q}(\omega), \mathbb{Z}[\mathbb{Z}_m])$-bimodule structure. Set

\begin{equation} H_*(W; \mathbb{Q}(\omega)) = \mathbb{Q}(\omega) \otimes_{\mathbb{Z}[\mathbb{z}_m]} H_*(\widetilde W; \mathbb{Z}). \end{equation}

Then the $\mathbb{Q}(\omega)$-vector space $H_2(W; \mathbb{Q}(\omega))$ is endowed with a $\mathbb{Q}(\omega)$-valued Hermitian form $\lambda_{\mathbb{Q}(\omega)}$ (whose definition is analogous to the ordinary intersection form on $H_2(W; \mathbb{Z})$). Define the signature of $W$ twisted by $\psi$ as

\begin{equation} \textrm{sign}^\psi(W) := \textrm{sign}(\lambda_{\mathbb{Q}(\omega)}). \end{equation}

Now, given a closed $3$-manifold $M$ and a homomorphism $\chi: \pi_1(M) \rightarrow \mathbb{Z}_m$, bordism theory implies that there exists a non-negative integer $k$, a $4$-maifold $W$ and a homomorphism $\psi: \pi_1(W) \rightarrow \mathbb{Z}_m$ such that $\partial(W, \psi) = k(M, \chi)$. Define the invariant $\sigma(M, \chi)$ as

\begin{equation} \sigma(M, \chi) := \frac{1}{k}(\textrm{sign}^\psi(W) - \textrm{sign}(W)) \in \mathbb{Q}. \end{equation}

**Definition of $\sigma_r(M, \chi)$:** Let $M$ be a closed $3$-manifold and $\chi: H_1(M) \rightarrow \mathbb{Z}_m$ an epimorphism. Again, $\chi$ induces an $m$-fold cyclic covering $\widetilde M \rightarrow M$ with a canonical generator of the group of covering translations, corresponding to $1 \in \mathbb{Z}_m$. Suppose that for some positive integer $k$ there exists an $mk$-fold cyclic branched covering of $4$-manifolds $\widetilde W \rightarrow W$, branched over a surface $F \subset \textrm{int }W$, such that $\partial(\widetilde W \rightarrow W) = k(\widetilde M \rightarrow M)$, and such that the covering translation of $\widetilde W$ that induces rotation through $2\pi/m$ on the fibers of the normal bundle of $\widetilde F$ restricts on each component of $\partial\widetilde W$ to the canonical covering translation of $\widetilde M$ determined by $\chi$. Let this covering translation induce $\tau$ on $H = H_2(\widetilde W) \otimes \mathbb{C}$. Further, let $\cdot_H$ denote the intersection form of $H_2(\widetilde W)$ extended to $H$ (which is Hermitian, but in general not non-singular). Then $(H, \cdot)$ decomposes as an orthogonal direct sum of eigenspaces of $\tau$, corresponding to the eigenvalues $\omega^r$, $0 \leq r < m$, where $\omega := e^{\frac{2\pi i}{m}}$. Let $\varepsilon_r(\widetilde W)$ denote the signature of $\cdot_H$ restricted to the eigenspace corresponding to the eigenvalue $\omega^r$. Define, for $0 < r < m$, the invariant $\sigma_r(M, \chi)$ as

\begin{equation} \sigma_r(M, \chi) := \frac{1}{k}\left(\textrm{sign}(W) - \varepsilon_r(\widetilde W) - \frac{2[F]^2r(m-r)}{m^2}\right) \in \mathbb{Q}, \end{equation}

where $[F]$ denotes the self-intersection number of the branching surface $F$.

**Problem:** On pages 185-186 in [1], Casson and Gordon explain the relation of $\sigma(M, \chi)$ to the Atiyah-Singer G-signature. In particular, they conclude on page 186 that (with the notation from above),

\begin{equation} k\sigma(M, \chi^r) + \textrm{sign}(W) = \varepsilon_r(\widetilde W), \end{equation}

which reads for $r = 1$ in particular as

\begin{equation} k\sigma(M, \chi) + \textrm{sign}(W) = \varepsilon_1(\widetilde W). \end{equation}

Further, Proposition 2.19 on p. 18 in [3] states that the formula for computing $\sigma_r(M, \chi)$ in terms of a surgery description of $M$ (Lemma 3.1 on p. 42 in [2]) is also valid for $\sigma(K, \chi)$. From this, I deduced that we (should) have a relation like $\sigma(M, \chi^r) = -\sigma_r(M, \chi)$ for $0 < r < m$. However, there is the summand $\frac{2[F]^2r(m-r)}{m^2}$ appearing in the definition of $\sigma_r(M, \chi)$, and I don't see where or in what form this summand appears in the definition of $\sigma(M, \chi)$ (I do see however why it appears in the definition of $\sigma_r(M, \chi)$, namely because of the fact that for a branched covering $\widetilde W \rightarrow W$ of *closed* 4-manifolds, there is the formula $\varepsilon_r(\widetilde W) = \textrm{sign}(W) - \frac{2[F]^2r(m-r)}{m^2}$ (Lemma 2.1 on p. 40 in [2])). So my question is the following:

**Question:** *Is the relation $\sigma(M, \chi^r) = -\sigma_r(M, \chi)$ correct? If so, where does the summand $\frac{2[F]^2r(m-r)}{m^2}$ appear in $\sigma(M, \chi)$? If not, is there another relation that holds for $\sigma(M, \chi)$ and $\sigma_r(M, \chi)$?*

I assume that my question has something to do with the branching set $F$ as I don't see it mentioned in [1] or [3]. However, I'm too unexperienced in the subject to come to a conclusion on my own. Thus, any elaboration would be greatly appreciated. Thanks in advance!

**References:**

[1] A. J. Casson; C. Mc A. Gordon, *Cobordism of Classical Knots*, À la Recherche de la Topologie Perdue, Progress in Mathematics, Volume 62, pp. 181 - 199, Birkhäuser Boston, 1986. With an appendix by P. M. Gilmer (available here)

[2] A. J. Casson; C. Mc A. Gordon, *On Slice Knots in Dimenstion Three*, Proceedings of Symposia in Pure Mathematics, Volume 32, pp. 39 - 53, American Mathematical Society, 1978 (available here).

[3] A. Conway, *Algebraic Concordance and Casson-Gordon Invariants*, notes of a reading group held in Geneva, Spring 2017 (available here).