About natural identifications in knot theory

Question

Let's consider a knot $K$ in a general closed oriented 3-manifold $Y$. And for technical simplicity assume $K$ is rationally nullhomologous, ie, $[K]\in H_1(Y)$ is a torsion element. Now choose a framing $\lambda$ of $K$ so that we can do surgery along $K$ with this framing to obtain a 3-manifold $Y(K)=Y_\lambda(K)$ and a 4-dimensional cobordism $W=W_\lambda(K)$ from $Y$ to $Y(K)$.

1. It is well-known that we have a short exact sequence of the form $0\to \mathbb{Z}\left<\mu\right>\to H_1(Y\setminus \nu(K))\to H_1(Y)\to 0$, where $\mu\subset Y\setminus \nu(K)$ is the meridian of the knot $K$ and $\nu(K)$ is a tubular neighborhood of $K\subseteq Y$. By Poincare duality, it can also be written as $0\to \mathbb{Z}\to H^2(Y\setminus \nu(K),\partial \nu(K))\to H^2(Y)\to 0$.

2. And there is another short exact sequence computing the cohomology of $W$: From the M-V sequence applied to $W=Y\times[0,1]\cup (2\textrm{-handle})$, $\cdots \to H^1(Y\times [0,1])\oplus H^1(2\textrm{-handle})\to H^1(\nu(K))\cong \mathbb{Z}\to H^2(W)\to H^2(Y\times [0,1])\oplus H^2(2\textrm{-handle})\to H^2(\nu(K))=0$.

We can simplify further: there is an obvious identification $H^*(Y\times [0,1])=H^2(Y)$, and the leftmost map $H^1(Y)\to H^1(\nu(K))$ is zero as $K$ is torsion in $H_1(Y)$. (Elaboration: letting the inclusion be $\iota:K\to Y$, $\left<\iota^* \alpha, a\right>=\left<\alpha,\iota_* a\right>=0$ for any $\alpha\in H^1(Y)$ and $a\in H_1(K)$. Thus $\iota^*\alpha=0$ for any $\alpha$, meaning $\iota^*=0$.)

3. On the other hand, we have a natural identification of $H^2(Y\setminus \nu(K),\partial \nu(K))$ with $H^2(W)$: From the excision, $H^2(Y\setminus \nu(K),\partial \nu(K))\cong H^2(Y,\nu(K))\cong H^2(Y\times [0,1],\nu(K)\times\left\{1\right\})\cong H^2(Y\times [0,1]\cup (2\textrm{-handle}),(2\textrm{-handle}))=H^2(W,(2\textrm{-handle}))\cong H^2(W)$

(where the final identity follows from the fact that any handles are contractible.)

Furthermore, comparing the previous two short exact sequences, we have the commutativity

That is, we have a natural identification of the form

So far so good... Now I'd like to pursue a better understanding of this identification. E.g.)

1. Note that $\mathbb{Z}$ in the first row is generated by the meridian $\mu$. That is, the image of $\mu$ in $H^2(Y\setminus \nu(K),\partial \nu(K))$ is simply the Poincare dual $PD(\mu)$. Then we can consider the corresponding element in $H^2(W)$. What is a geometric interpretation of this element $H^2(W)$? For example, from Poincare duality, there is a surface $(F,\partial F)\subseteq (W,\partial W)$ whose Poincare dual is this element. What would this $F$ be?

2. Note that from the surgery construction we have that $H_2(W,Y)\cong \mathbb{Z}$ and the generator is the core of the 2-handle (whose boundary is elongated by vertical band to lie inside $Y\times \left\{0\right\}\cong Y$. Denote this generator by $C$. Then $C$ can be interpreted as an element in $H^2(W)\cong H_2(W,\partial W)$ through natural inclusion. Can we compare this $C$ with $PD(\mu)$? That is, is $C$ a multiple of $PD(\mu)$?