is the cyclic group of order $n$, the map induced by $W$ can be written $$\label{cobordmap}
F^{{{\mathbb F}}[C_n]}_W(\zeta^k\cdot U^{-j}\cdot{\mbox{\bf x}}) = \sum_{{\mbox{\bf y}}\in{\mathbb{T}_\alpha}\cap {\mathbb{T}_{{{\boldsymbol \gamma}^{ 1}}}}}\sum_{\psi\in\pi_2({\mbox{\bf x}},\Theta,{\mbox{\bf y}})} \#{{\mathcal M}}(\psi)\, \zeta^{n_p(\partial\psi)+k}\cdot U^{n_w(\psi) - j}\cdot {\mbox{\bf y}}$$ yielding a map on homology $$F^{{{\mathbb F}}[C_n]}_W: {\underline}{HF}^+(-Z_0; {{\mathbb F}}[C_n])\to {\underline}{HF}^+(-Z; {{\mathbb F}}[C_n]).$$ When we view $W$ as a cobordism from $Z$ to $Z_0$, then for any coefficient module $\mathbb{A}$ over ${{\mathbb F}}[H^1(Z)]$, the induced module over ${{\mathbb F}}[H^1(Z_0)]$ is $$\mathbb{A}\otimes_{{{\mathbb F}}[H^1(Z)]}\mathbb{K}(W)\cong\mathbb{A}\otimes_{{{\mathbb F}}[H^1(Z)]}{{\mathbb F}}[H^1(Z_0)]\cong \mathbb{A}[T,T^{-1}],$$ with isomorphisms induced by the splitting $H^1(Z_0)\cong H^1(Z)\oplus {{\mathbb Z}}$, and where the additional variable $T$ corresponding to a generator of the ${{\mathbb Z}}$ summand. For ${{\mathbb F}}$ coefficients, we thus have a chain map $$\label{cobordmap2}
F^{{{\mathbb F}}}_W( U^{-j}\cdot{\mbox{\bf x}}) = \sum_{{\mbox{\bf y}}\in{\mathbb{T}_\alpha}\cap {\mathbb{T}_{{{\boldsymbol \gamma}^{ 1}}}}}\sum_{\psi\in\pi_2({\mbox{\bf x}},\Theta,{\mbox{\bf y}})} \#{{\mathcal M}}(\psi)\, T^{n_p(\partial\psi)}\cdot U^{n_w(\psi) - j}\cdot {\mbox{\bf y}}$$ which induces a map $$F^{{{\mathbb F}}}_W: {HF}^+(Z)\to {\underline}{HF}^+(Z_0; {{\mathbb F}}[T,T^{-1}]).$$

We will ultimately need to use the map in the case that $-Z_0 = -Y_0(K)$ and $-Z = -(Y_{-1/n}(K)) = (-Y)_{1/n}(K)$, where $K$ is the connected binding of an open book in a $3$-manifold $Y$ supporting a contact structure $\xi$ as above. Note that $Z_0$ is indeed obtained by zero surgery on a 