I've been reading Surgery on Contact 3-Manifolds and Stein Surfaces by Ozbagci and Stipsicz, and have gotten stuck on the following exercise on p. 44. Below $Y_1, Y_2$ are closed 3-manifolds, and $Q$ denotes intersection form:
It is only a little more complicated to investigate homologies in cobordisms. Suppose that $W$ is a given cobordism from $Y_1$ to $Y_2$. Fix a 4-manifold $X$ with $\partial X = Y_1$, and suppose that it is given by attaching 2-handles to $D^4$ along a framed link $L$. For the sake of simplicity, suppose furthermore that $W$ is given by a single 2-handle attachment to $Y_1$. Denote the 4-manifold $X\cup W$ by $X'$.
Exercises 2.3.4. (a) Determine the homology class in $H_2(X';\mathbb{Z})$ generating $H_2(W,\partial W;\mathbb{Z})$. (Hint: Consider a primitive homology class $\alpha\in H_2(X';\mathbb{Z})$ such that $Q_{X'}(\alpha,\beta)=0$ for all $\beta\in H_2(X;\mathbb{Z})\subset H_2(X';\mathbb{Z}).$)
(b) Determine the self-intersection $Q_W(\alpha,\alpha)$ of this generator.
(c) Find a surface in $W$ representing the above $\alpha \in H_2(W;\mathbb{Z})$. (Hint: Use the above computation to represent $\alpha\in H_2(X';\mathbb{Z})$ with a surface. By adding extra handles make sure that the surface is disjoint from the cores of all the 2-handles defining $X$. Now show that the surface is in $W$.) Notice that different presentations of $Y_1$ as $\partial X^4$ might provide different estimates on the genus of a surface representing $\alpha$.
To fix notation let $K_i$ denote the components of $L$, let $K$ denote the attaching circle of the additional 2-handle, and let the corresponding framing coefficients be $n_i,n$. Here's what I know so far:
- $H_2(W,Y_1)\cong\mathbb{Z}$, generated by the core of the 2-handle attached along $K$,
- $H_2(W,Y_2)\cong \mathbb{Z}$, generated by the cocore of the same 2-handle [represented in the diagram by a small meridional disk of $K$],
- By looking at the LES of the triple $(X',X\cup Y_2, X)$, we have an exact sequence $$H_2(X)\to H_2(X', Y_2)\to H_2(W,\partial W)\to 0.$$ To describe the map $\varphi\colon H_2(X)\to H_2(X',Y_2)$, let $\Sigma_i$ denote the closed surface obtained by taking a Seifert surface for $K_i$ in $D^4$ and attaching the core disk of the corresponding handle. Let $D_i,D$ denote small meridional disks of $K_i$, $K$. Then $H_2(X)$ is freely generated by $[\Sigma_i]$, $H_2(X',Y_2)$ is freely generated by $[D_i],[D]$, and $$\varphi([\Sigma_i]) = n_i [D_i]+\mathrm{lk}(K_i,K)[D] +\sum_{j\neq i}\mathrm{lk}(K_i,K_j)[D_j].$$
I am generally lost about how to approach the exercise, but more specifically I am confused about the following:
- I am not sure what the approach in the hint for part (a) is. It's not clear to me why such an $\alpha\in H_2(X')$ should be a generator for $H_2(W,\partial W)$.
- I am not sure why $H_2(W,\partial W)$ is cyclic. From what I have above, it seems that $H_2(W,\partial W)\cong \mathbb{Z}^2$ if $L$ and $K$ are unlinked knots and $L$ has $0$ framing. [If $Y_1$ or $Y_2$ is a homology sphere, I think one does have $H_2(W,\partial W)\cong\mathbb{Z}$.] This makes me think that I've made a mistake somewhere, or maybe that the exercise should say $H_2(W,Y_1)$ or $H_2(W,Y_2)$ in place of of $H_2(W,\partial W)$. In the case of $H_2(W,Y_1)$, for part (a) it seems that one can just take a Seifert surface for $K$ in $D^4$ capped off by the core of the corresponding 2-handle.
- The exercise seems to suggest that $H_2(W)\to H_2(W,\partial W)$ is surjective, and that for a generator of $ H_2(W,\partial W)$ the choice of lift $\alpha\in H_2(W)$ does not affect $Q_W(\alpha,\alpha)$. I'm not sure why this is the case either.
Thank you in advance for your help [and for reading this long question].
