ScalingStacks

Let us now specialize the addition of 3-handles to the case where the initial manifold $W=W_{2}$ is a union of $0$-, $1$- and 2-handles. We will then have available to us the description of $\mathcal{S}_{0}^{N}(W_{2};L,\alpha)$ from Section 3.1.

If we attach a 3-handle to $W_{2}$, in terms of Kirby calculus, the attaching sphere $S$ can be represented as a surface $\Sigma$ (of genus $0$, and disjoint from $L$) with boundary some copies of the $K_{i}$’s (the attaching circles for 2-handles). Then $S$ is the union of $\Sigma$ and (parallel copies of) cores of the 2-handles.

We draw $J\subset S$ as a small unknot away from all $K_{i}$, and let $\Delta_{+}$ be the small disk it bounds. The other hemisphere $\Delta_{-}$ is the complement of $\Delta_{+}$ in $S$, and goes over some of the handles. We let

This is a surface on $\partial W_{1}$ whose boundary is the union of $J$ and several copies of the $K_{i}$’s. Let $s^{-}_{i}$ be the number of copies of $K_{i}$ in $\partial\Sigma_{-}$ that appear with the negative orientation, and $s^{+}_{i}$ the number of those with the positive orientation. We form the vectors

We proceed to describe the maps $\Psi_{I\times Y;\Delta_{+},\alpha^{\prime}}$ and $\Psi_{I\times Y;\Delta_{-},\alpha^{\prime}}$ in this case. By Theorem 3.2 with notation as in (10), the range $\bigoplus_{\alpha\in\langle\alpha^{\prime}\rangle}\mathcal{S}_{0}^{N}(W_{2};Z,\alpha)$ of these maps is identified with the direct sum of cabled skein lasagna modules $\bigoplus_{\alpha\in\langle\alpha^{\prime}\rangle}\underline{\mathcal{S}\mkern-3.0mu}\mkern 3.0mu_{0}^{N,\alpha}(W_{1};K,L)$. Similarly, their domain is identified with

We used here the fact that $J$ is split disjoint from all the attaching links for the 2-handles, and therefore each summand that appears in the definition of $\underline{\mathcal{S}\mkern-3.0mu}\mkern 3.0mu_{0}^{N,\alpha}(W_{1};K,L\cup J)$ splits off a $\operatorname{KhR}_{N}(J)$ factor; moreover, the equivalence relation is compatible with this splitting.

The map $\Psi_{I\times Y;\Delta_{+},\alpha^{\prime}}$ is now easy to describe. It is induced by capping $J$ with a disk, so it only affects the factor $\operatorname{KhR}_{N}(J)$, in a standard way. Precisely, we have

for all $v\in\underline{\mathcal{S}\mkern-3.0mu}\mkern 3.0mu_{0}^{N,\alpha}(W_{1};K,L)$.

To describe the second map $\Psi_{I\times Y;\Delta_{-},\alpha^{\prime}}$, consider the diagram

Here, in the top row we wrote $(k^{-},k^{+})$ for a pair $(r-\alpha^{-},r-\alpha^{+})$ as in Definition 3.1. The vertical maps from the first to the second row are induced by the inclusion of the summands into the cabled skein lasagna module; cf. Definition 3.1. The vertical maps from the second to the third row are the isomorphisms $\Phi$ from Theorem 3.2.

Ignoring the middle dashed arrow for the moment, note that the above diagram commutes. Indeed, by the definition of $\Phi$ in the proof of Theorem 3.2, the vertical compositions (from the first to the third row) are given by attaching cores of the 2-handles to lasagna fillings in $W_{1}$. Note that we are attaching more cores on the right; namely, those in the boundary of $\partial\Sigma$, counted by the vectors $s^{-}$ and $s^{+}$. The horizontal cobordism maps (as defined in Section 2.2) are given by attaching the surface $\Sigma_{-}$ (in the top row) and $\Delta_{-}$ (in the bottom row). Because $\Delta_{-}$ is the union of $\Sigma_{-}$ and the extra cores of 2-handles counted by $s^{-}$ and $s^{+}$, the diagram (12) commutes.

Since the bottom vertical arrows in the diagram are isomorphisms, let us now add the middle dashed arrow, given by the map

Because (12) commutes, we deduce that this map is induced on the skein lasagna modules by applying the cobordism maps $\Psi_{I\times\partial W_{1};\Sigma_{-},\alpha^{\prime}}$ on each summand; this justifies the notation.

Recall that $\Sigma_{-}$ is the complement of the disk $\Delta_{+}$ inside $\Sigma$. Thus, we can write the cobordism maps $\Psi_{I\times\partial W_{1};\Sigma_{-},\alpha^{\prime}}$ in terms of the maps $\Psi_{I\times\partial W_{1};\Sigma(n\bullet),\alpha^{\prime}}$ associated to the surface $\Sigma$ with $n$ dots, as in (7):

Fixing $n$, the maps $\Psi_{I\times\partial W_{1};\Sigma(n\bullet),\alpha^{\prime}}$ on various summands in the construction of the skein lasagna module induce a map:

such that

We are now ready to give a general formula for the skein lasagna module of a four-manifold decomposed into handles in terms of skein lasagna modules of 1-handlebodies. We will phrase it for an arbitrary number of handles.

Theorem 3.10 gives a description of an arbitrary skein lasagna module in terms of skein lasagna modules for links in the boundary of $W_{1}=\natural^{m}(S^{1}\times B^{3})$, and cobordism maps for surfaces in $I\times\partial W_{1}$. In the next section we will obtain a further reduction to links in $S^{3}$ and cobordism maps between them, under the additional constraint of working with field coefficients; see Theorem 4.7.