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Let $Y$ be a compact manifold and let $\pi_1(Y) \to \mathbb{Z}^n= \langle t_1,\ldots,t_n\rangle$ be a homomorphism. Extend it to the group rings $\mathbb{Z}[\pi_1(Y)] \to \mathbb{Z}[ t_1,\ldots,t_n]$ and evaluate it in a certain $\omega \in (S^1)^n \subset \mathbb{C}^n$. We obtain a homomorphism $$ \psi: \mathbb{Z}[\pi_1(Y)] \to \mathbb{C}$$ which endows $\mathbb{C}$ of a $(\mathbb{C},\mathbb{Z}[\pi_1(Y)])$-bimodule structure. I will indicate this bimodule as $\mathbb{C}^\omega$.

What I call $\omega$-twisted homology of $Y$ will be the homology of the complex $$ \mathbb{C}^\omega \otimes_{\mathbb{Z}[ \pi_1(Y)]} C^{ \text{CW}}_*(\widetilde{Y}) $$ and I denote it with $H_*(Y;\mathbb{C}^\omega)$, where $\widetilde{Y}$ is the universal cover.

The cohomology of the cochain complex $$\text{Hom}_{\text{Mod}-\mathbb{Z}[\pi_1(Y)]}( \text{inv}(C_{*}(\widetilde{Y})), \mathbb{C}^\omega)$$ is the $\omega$-twisted cohomology of $Y$. Here $ \text{inv}(C_{*}(\widetilde{Y}))$ indicates $C_{*}(\widetilde{Y})$ with the same additive structure, but the action of $\mathbb{Z}[\pi_1(Y)]$ is now on the right because we precompose it with $g \mapsto g^{-1}$ for $g \in \pi_1(Y)$.

If $Y$ is of even dimension $2k$ there is an intersection form on $H_k(Y;\mathbb{C}^\omega)$, defined more or less as usual: $$\phi:H_k(Y;\mathbb{C}^\omega) \to H_k(Y, \partial Y;\mathbb{C}^\omega) \xrightarrow{\text{PD}} H^k(Y;\mathbb{C}^\omega) \xrightarrow{\text{ev}} \text{inv}(\text{Hom}_{\mathbb{C}}(H_k(W;\mathbb{C}^\omega),\mathbb{C})) $$

The Poincaré Duality is an isomorphism in this context as well, and it is defined starting from the following isomorphism: denote $Y'$ the space $Y$ endowed with the dual cell decomposition w.r.t. $Y$. Then there is a chain complex isomorphism: $$ C_{n-*}(\widetilde{Y}) \to \text{Hom}_{\text{Mod}-\mathbb{Z}[\pi_1(Y)])}(\text{inv}(C_*(\widetilde{Y'}, \widetilde{\partial Y'})), \mathbb{Z}[\pi_1(Y)])$$

$c' \mapsto [-,c']$ and $$[c,c']= \sum_{\gamma \in \pi_1(Y)} (c \cdot \gamma c') \gamma $$

where $(c \cdot \gamma c')$ is the integer intersection number of $c$ and $\gamma c'$.

I am trying to adapt the proof of Wall for the non-additivity theorem of signatures (from his article "Non Additivity of the Signature" of 1969) to the case of homology with twisted coefficients in the $(\mathbb{Z}[\pi_1(Y)],\mathbb{C})$-bimodule $\mathbb{C}$. Most of the proof works exactly the same thanks to the properties of twisted homology, but I am really having difficulties in adapting the final geometric argument to this setting. The setting of the theorem is:

Let $Y$ be an oriented connected compact $4k$-manifold and let $X_0$ be an oriented compact $4k-1$-manifold, properly embedded into $Y$ so that $\partial X_0= X_0 \cap \partial M$. Suppose that $X_0$ splits $Y$ into two manifolds $Y_-$ and $Y_+$. For $\varepsilon= \pm$, denote by $X_\varepsilon$ the closure of $\partial Y_\varepsilon \setminus X_0$, which is a compact $4k-1$-manifold. Let $Z$ denote the compact $4k-2$-manifold $$Z= \partial X_0 = \partial X_+ = \partial X_-. $$ The manifolds $Y_+$ and $Y_-$ inherit an orientation from $Y$. Orient $X_0$, $X_+$ and $X_-$ such that $$ \partial Y_+ = X_+ \cup (-X_0)$$ and $$ \partial Y_- = X_0 \cup (-X_-)$$ and orient $Z$ such that $$Z= \partial X_- = \partial X_+ = \partial X_0. $$

I would like to prove Novikov-Wall non additivity theorem:

In the situation above, $$\text{sign}_\omega(Y)= \text{sign}_\omega(Y_+) + \text{sign}_\omega(Y_-) + \text{Maslov}(L_-,L_0, L_+) $$ where $L_\varepsilon= \ker (H_{2k-1}(Z; \mathbb{C}^\omega) \to H_{2k-1}(X_\varepsilon ; \mathbb{C}^\omega)) $ for $\varepsilon=-, +, 0$.

Here $\text{sign}_\omega$ means the signature of the twisted intersection form.

In the final part of the proof we need to calculate the signature (which should give us the Maslov index term) of the twisted intersection form restricted to a subspace $L$ of $H_{2k}(Y, \partial Y; \mathbb{C}^\omega)$ which is isomorphic to $$\frac{L_0 \cap ( L_-+L_+)}{(L_0 \cap L_+)+ (L_0\cap L_-)} .$$ The idea is to calculate the signature of the intersection form on $L$ by expressing it in terms of $Z$'s own skew-hermitian intersection pairing.

Here is where I get stuck. I understand how to represent an element $b \in L_0 \cap ( L_-+L_+)$ with a $2k$-cycle $ \xi + \eta + \zeta$ in $Y$, where $\xi, \eta, \zeta$ belong respectively to $Z_{2k}(X_+,Z; \mathbb{C}^\omega), Z_{2k}(X_0,Z; \mathbb{C}^\omega), Z_{2k}(X_-,Z; \mathbb{C}^\omega)$ and I know that given $b, b' \in L_0 \cap ( L_-+L_+)$ I want to calculate the twisted intersection form of $\xi + \eta + \zeta $ and $\xi'+\eta'+ \zeta'$, however, I have no idea how to compute it.

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  • $\begingroup$ I'm not completely sure I understand how you're defining signatures with twisted coefficients. The way I usually see such things involves some sort of self-dual sheaf of coefficients. Can you say more about the definition of your twisted intersection form? $\endgroup$ – Greg Friedman Aug 21 '20 at 21:40
  • $\begingroup$ I edited the question, adding some information. I hope it is more clear now! $\endgroup$ – Alice M. Aug 25 '20 at 9:42
  • $\begingroup$ Thanks. So then isn't the intersection form you're looking for on Z given by essentially the same sort of formula you give on Y, which is pretty explicit? $\endgroup$ – Greg Friedman Aug 27 '20 at 5:51
  • $\begingroup$ There is of course an intersection form on Z, but what you want to do is to express the form on the subspace L in terms of the intersection form on Z, but they are not going to be the same. $\endgroup$ – Alice M. Aug 29 '20 at 15:58
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In a paper of W. Neumann, I noticed a reference to the thesis of W. Meyer, Die Signatur von lokalen Koeffizientensystemen und Faserbündeln. Neumann says that Meyer discusses some details Wall's non-additivity result with local coefficients. I don't have access to Meyer's work (it's in the series Bonner Math. Schriften). You could try Meyer's paper, Die Signatur von Flächenbündeln. Math. Ann. 201 (1973), 239–264 but there's not much on the subject in there.

Perhaps this will be of some use.

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