ScalingStacks

The paper [25] contains a description of the skein lasagna module for 2-handlebodies (four-manifolds $W$ made of a 0-handle and some 2-handles), where the link $L\subset\partial W$ is empty, or at least local (contained in a $3$-ball). The description is in terms of the Khovanov-Rozansky homology of cables of the attaching link $K$.

In this subsection we extend that description to the case where we attach 2-handles to any four-manifold $W$, to obtain a new manifold $W^{\prime}$. Moreover, we do not impose any restriction on the link $L\subset\partial W$. The formula is very similar to that in [25]. The role of the Khovanov-Rozansky homology $\operatorname{KhR}_{N}$ will be played by the skein lasagna module $\mathcal{S}_{0}^{N}(W;-)$, which can be thought of as a link homology for links in the boundary of $W$. (When $W=B^{4}$, we have $\mathcal{S}_{0}^{N}(W;L)=\operatorname{KhR}_{N}(L)$.)

Let $K_{1},\dots,K_{n}$ be the components of the framed link $K\subset\partial W$ along which the 2-handles are attached. The framing gives diffeomorphisms $f_{i}$ between tubular neighborhoods $\nu(K_{i})$ of each $K_{i}$ and $S^{1}\times D^{2}$. Given $n$-tuples of nonnegative integers

$$k^{-}=(k_{1}^{-},\dots,k_{n}^{-}),\ \ \ k^{+}=(k_{1}^{+},\dots,k_{n}^{+}),$$

we let $K(k^{-},k^{+})$ denote the framed, oriented cable of $K$ consisting of $k_{i}^{-}$ negatively oriented parallel strands to $K_{i}$ and $k_{i}^{+}$ positively oriented parallel strands. Here, the notion of parallelism for the strands is determined by the framing, that is,

$$K(k^{-},k^{+})=\bigcup_{i}f_{i}^{-1}(S^{1}\times\{x_{1}^{-},\dots,x_{k_{i}^{-}}^{-},x_{1}^{+},\dots,x_{k_{i}^{+}}^{+}\})$$

for fixed points $x_{1}^{-},\dots,x_{k_{i}^{-}}^{-},x_{1}^{+},\dots,x_{k_{i}^{+}}^{+}\in D^{2}.$

After attaching 2-handles to $W$ along $K$, we obtain the manifold $W^{\prime}$. Suppose we are given a framed link $L\subset\partial W^{\prime}$. Generically, we can assume that $L$ stays away from the attaching regions of the 2-handles, and therefore we can represent it as a link in $\partial W$, disjoint from (but possibly linked with) $K$. (There are various ways of isotoping $L$ off of the attaching regions; the results of the calculation will be isomorphic.) We let

$$K(k^{-},k^{+})\cup L$$

be the union of $K(k^{-},k^{+})$ and $L$, where we do the cabling on the components of $K$ by choosing the tubular neighborhoods of $K_{i}$ to be disjoint from $L$. (Note that $K(k^{-},k^{+})\cup L$ is not a split disjoint union.)

We seek to express the skein lasagna module $\mathcal{S}_{0}^{N}(W^{\prime};L)$ in terms of $\mathcal{S}_{0}^{N}(W;K(k^{-},k^{+})\cup L)$. To do this, we need to introduce a few more notions.

For each $i$, let $B_{k_{i}^{-},k_{i}^{+}}$ be the subgroup of the braid group on $k_{i}^{-}+k_{i}^{+}$ strands that consists of self-diffeomorphisms of $D^{2}$ rel boundary (modulo isotopy rel boundary) taking the set $\{x_{1}^{-},\dots,x_{k_{i}^{-}}^{-}\}$ to itself and the set $\{x_{1}^{+},\dots,x_{k_{i}^{+}}^{+}\}$ to itself. By taking the product with the identity on $S^{1}$, a braid element $b\in B_{k_{i}^{-},k_{i}^{+}}$ induces a self-diffeomorphism of $D^{2}\times S^{1}$, which can be pulled back (via $f_{i}$) to a self-diffeomorphism of $\nu(K_{i})$. This gives a group action

$$\beta_{i}:B_{k_{i}^{-},k_{i}^{+}}\to\operatorname{Aut}(\mathcal{S}_{0}^{N}(W;K(k^{-},k^{+})\cup L)).$$

Let $e_{i}\in{\mathbb{Z}}^{n}$ denote the $i^{th}$ basis vector. Two strands parallel to $K_{i}$, if they have opposite orientations, co-bound a ribbon band $R_{i}$ in $S^{3}$. By pushing $R_{i}$ into $S^{3}\times[0,1]$ so that it is properly embedded there, and taking the disjoint union with the identity cobordisms on the other strands, we obtain an oriented cobordism (still denoted $R_{i}$) from $K(k^{-},k^{+})\cup L$ to $K(k^{-}+e_{i},k^{+}+e_{i})\cup L$. For $d=0,1,\dots,N-1$, we can decorate $R_{i}$ with $d$ dots, and obtain a cobordism map

$$\psi^{[d]}_{i}:\mathcal{S}_{0}^{N}(W;K(k^{-},k^{+})\cup L)\to\mathcal{S}_{0}^{N}(W;K(k^{-}+e_{i},k^{+}+e_{i})\cup L),$$

which changes the bigrading by $(0,2d)$.

Next, recall that we have a decomposition (2) for the skein lasagna module $\mathcal{S}_{0}^{N}(W^{\prime};L)$, according to homology classes in $H_{2}^{L}(W^{\prime};{\mathbb{Z}})$. Let us see how these homology classes are related to the similar ones in $W$. Consider the tubular neighborhood $\nu(K)=\cup_{i}\nu(K_{i})$, which is a union of solid tori. Express $W^{\prime}$ as the union

$$W^{\prime}=W\cup C\cup Z,$$

where $Z$ is the union of the new 2-handles, and $C\cong\nu(K)\times[0,1]$ is a connecting cylinder between $W$ and $Z$. Let also

$$C^{\prime}=\nu(K)\times\{0,1\}\subset C.$$

We identify $\nu(K)$ with $\nu(K)\times\{0\}$ and denote $\nu(K)\times\{1\}$ by $\partial_{-}Z$ (part of the boundary $\partial Z$).

The Mayer-Vietoris sequence for $W^{\prime}$ relative to the union of $W$ and $Z\cup L$ reads

$$\cdots\to H_{*}(W^{\prime},W\cap(Z\cup L);{\mathbb{Z}})\to H_{*}(W^{\prime},W;{\mathbb{Z}})\oplus H_{*}(W^{\prime},Z\cup L;{\mathbb{Z}})\to H_{*}(W^{\prime},W\cup(Z\cup L);{\mathbb{Z}})\to\cdots$$

Observe that, by excision, $H_{3}(W^{\prime},W\cup(Z\cup L);{\mathbb{Z}})\cong H_{3}(C,C^{\prime};{\mathbb{Z}})=0$. From here we obtain an exact sequence

(8)

$$0\to H_{2}(W^{\prime},L;{\mathbb{Z}})\to H_{2}(Z,\partial_{-}Z;{\mathbb{Z}})\oplus H_{2}(W,\nu(K)\cup L;{\mathbb{Z}})\to H_{2}(C,C^{\prime};{\mathbb{Z}}).$$

Thus, an element in $H_{2}^{L}(W^{\prime};{\mathbb{Z}})\subseteq H_{2}(W^{\prime},L;{\mathbb{Z}})$ can be identified with its image in $H_{2}(Z,\partial_{-}Z;{\mathbb{Z}})\oplus H_{2}(W,\nu(K)\cup L;{\mathbb{Z}})$, which we write as a pair $(\alpha,\eta)$.

Let us further identify $H_{2}(Z,\partial_{-}Z;{\mathbb{Z}})$ with ${\mathbb{Z}}^{n}$ by letting the $i$th handle correspond to the coordinate vector $e_{i}$. Then, we write

$$\alpha=(\alpha_{1},\dots,\alpha_{n})\in{\mathbb{Z}}^{n}$$

and let $\alpha^{+}$ denote its positive part and $\alpha^{-}$ its negative part; i.e., $\alpha^{+}_{i}=\operatorname{max}(\alpha_{i},0)$ and $\alpha^{-}_{i}=\operatorname{min}(\alpha_{i},0)$. We also let $|\alpha|=\sum_{i}|\alpha_{i}|$.

Let $r\in{\mathbb{N}}^{n}$ and consider the cable $K(r-\alpha^{-},r+\alpha^{+})$. The fact that $(\alpha,\eta)\in{\mathbb{Z}}^{n}\oplus H_{2}(W,\nu(K)\cup L;{\mathbb{Z}})$ is in the kernel of the map to $H_{2}(C,C^{\prime};{\mathbb{Z}})\cong{\mathbb{Z}}^{n}$ in (8) implies the existence of a (unique) class

$$\eta^{r}\in H_{2}^{L\cup K(r-\alpha^{-},r+\alpha^{+})}(W;{\mathbb{Z}})\subseteq H_{2}(W,L\cup K(r-\alpha^{-},r+\alpha^{+});{\mathbb{Z}})$$

which is sent to $\eta$ by the natural map to $H_{2}(W,L\cup\nu(K);{\mathbb{Z}}).$

From now on, using the deformation retraction from $\nu(K)$ to $K$, let us think of $\eta$ as a class in $H_{2}(W,K\cup L;{\mathbb{Z}})$.

We will primarily be using the results from this subsection in the case where the role of $W$ is played by

$$W_{1}:=\natural^{m}(S^{1}\times B^{3}),$$

a manifold obtained from a 0-handle by attaching some 1-handles. We denote $W^{\prime}$ by $W_{2}$. Then, $H_{2}(W_{1};{\mathbb{Z}})=0$, so $H_{2}^{L}(W_{1};{\mathbb{Z}})=0$ for any null-homologous $L$, and the decomposition (2) for skein lasagna modules of links in $W_{1}$ is trivial (consists of a single summand). Moreover, in this case an element $(\alpha,\eta)\in H_{2}^{L}(W_{2};{\mathbb{Z}})\subseteq H_{2}(W_{2},L;{\mathbb{Z}})$ is uniquely determined by its image $\alpha$ in $H_{2}(W_{2},W_{1};{\mathbb{Z}})\cong{\mathbb{Z}}^{n}$. Indeed, the exact sequence

$$0=H_{2}(W_{1};{\mathbb{Z}})\to H_{2}(W_{1},L\cup\nu(K);{\mathbb{Z}})\to H_{1}(L\cup\nu(K);{\mathbb{Z}})$$

show that the component $\eta$ is determined by its image in

$$H_{1}(L\cup\nu(K);{\mathbb{Z}})=H_{1}(L;{\mathbb{Z}})\oplus H_{1}(\nu(K);{\mathbb{Z}}).$$

The part in $H_{1}(L;{\mathbb{Z}})$ has to be the fundamental class $[L]$, while the part in $H_{1}(\nu(K);{\mathbb{Z}})\cong{\mathbb{Z}}^{n}$ is the image of $\alpha$ under the isomorphisms

$$H_{2}(W_{2},W_{1};{\mathbb{Z}})\xrightarrow{\cong}H_{2}(Z,\partial_{-}Z;{\mathbb{Z}})\xrightarrow{\cong}H_{1}(\partial_{-}Z;{\mathbb{Z}})\xrightarrow{\cong}H_{1}(\nu(K);{\mathbb{Z}}).$$

Therefore, in this case the class $\eta$ is redundant (being determined by $\alpha$), so we simply drop it from the notation, writing for example $\alpha$ instead of $(\alpha,\eta)$ for the classes in $H_{2}^{L}(W_{2};{\mathbb{Z}})$. With this in mind, the isomorphism from Theorem 3.2 is written as

(10)

$$\Phi:\underline{\mathcal{S}\mkern-3.0mu}\mkern 3.0mu_{0}^{N,\alpha}(W_{1};K,L)\xrightarrow{\phantom{a}\cong\phantom{a}}\mathcal{S}_{0}^{N}(W_{2};L,\alpha).$$

In [25, Proposition 2.1] the following result was shown:

In this section we focus on the case of 3-handle attachments. We will generalize the statement of Proposition 3.4 to 3-handle attachments in the presence of boundary links and explicitly describe the kernel of the resulting maps on $\mathcal{S}_{0}^{N}$.

Consider the following setting. Let $W$ be a four-manifold with a framed link $L\subset Y=\partial W$ and an embedded $2$-dimensional sphere $S\subset Y$, disjoint from $L$. Let $Z$ be the cobordism given by attaching a 3-handle to $W$ along $S$, and let

$$W^{\prime}=W\cup Z.$$

Let $Y^{\prime}=\partial W^{\prime}$ be the outgoing boundary of $Z$, so that $\partial Z=(-Y)\cup Y^{\prime}$. Inside $Z$ we have the two-dimensional annular cobordism $A=I\times L$, from $L=\{0\}\times L$ to a new link $L^{\prime}=\{1\}\times L$. Given $\alpha^{\prime}\in H_{2}^{L}(W^{\prime};{\mathbb{Z}})\cong H_{2}^{L}(W;{\mathbb{Z}})/([S])$, let us consider the set of all $\alpha\in H_{2}^{L}(W;{\mathbb{Z}})$ whose equivalence class modulo $[S]$ is $\alpha^{\prime}$:

$$\langle\alpha^{\prime}\rangle:=\{\alpha\in H_{2}^{L}(W;{\mathbb{Z}})\mid\alpha\text{ mod }[S]=\alpha^{\prime}\}.$$

We obtain a cobordism map as in (6):

$$\Psi_{Z;A,\alpha}:\mathcal{S}_{0}^{N}(W;L,\alpha)\to\mathcal{S}_{0}^{N}(W^{\prime};L^{\prime},\alpha^{\prime}).$$

Let

$$\Psi_{Z;A,\alpha^{\prime}}:=\sum_{\alpha\in\langle\alpha^{\prime}\rangle}\Psi_{Z;A,\alpha}:\bigoplus_{\alpha\in\langle\alpha^{\prime}\rangle}\mathcal{S}_{0}^{N}(W;L,\alpha)\to\mathcal{S}_{0}^{N}(W^{\prime};L^{\prime},\alpha^{\prime}).$$

Let $J$ be the equator of $S$ (which is an unknot in $Y$). Equip $J$ with an arbitrary orientation. By pushing a hemisphere of $S$ slightly from $Y=\{0\}\times Y$ into the cylinder $I\times Y$, and taking its union with $I\times L$, we obtain a properly embedded cobordism in $I\times Y$, going from $L\cup J$ to $L$. There are two such hemispheres, which produce two cobordisms, denoted $\Delta_{+}$ and $\Delta_{-}\subset I\times Y$. We orient $\Delta_{+}$ and $\Delta_{-}$ so that their boundary orientation is the one on $J$. (Note that they are therefore “oppositely oriented,” in the sense that they do not match up to produce an orientation on $S$.) Let us identify $W\cup(I\times Y)$ with $W$ itself using a standard collar neighborhood. Then, the cobordism maps associated to $\Delta_{+}$ and $\Delta_{-}$ take the form

$$\Psi_{I\times Y;\Delta_{+},\alpha}\colon\mathcal{S}_{0}^{N}(W;L\cup J,\alpha+[\Delta_{+}])\to\mathcal{S}_{0}^{N}(W;L,\alpha),$$

$$\Psi_{I\times Y;\Delta_{-},\alpha}\colon\mathcal{S}_{0}^{N}(W;L\cup J,\alpha+[\Delta_{-}])\to\mathcal{S}_{0}^{N}(W;L,\alpha).$$

From here we get direct sum maps

$$\Psi_{I\times Y;\Delta_{+},\alpha^{\prime}}:=\bigoplus_{\alpha\in\langle\alpha^{\prime}\rangle}\Psi_{I\times Y;\Delta_{+},\alpha}$$

and

$$\Psi_{I\times Y;\Delta_{-},\alpha^{\prime}}:=\bigoplus_{\alpha\in\langle\alpha^{\prime}\rangle}\Psi_{I\times Y;\Delta_{-},\alpha}.$$

Observe that these two maps have the same domain

$$\bigoplus_{\alpha\in\langle\alpha^{\prime}\rangle}\mathcal{S}_{0}^{N}(W;L\cup J,\alpha+[\Delta_{+}])=\bigoplus_{\alpha\in\langle\alpha^{\prime}\rangle}\mathcal{S}_{0}^{N}(W;L\cup J,\alpha+[\Delta_{-}])$$

and the same range $\bigoplus_{\alpha\in\langle\alpha^{\prime}\rangle}\mathcal{S}_{0}^{N}(W;L,\alpha).$ Let

$$f:=\Psi_{I\times Y;\Delta_{+},\alpha^{\prime}}-\Psi_{I\times Y;\Delta_{-},\alpha^{\prime}}.$$

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

$$\Sigma_{-}=\Sigma\setminus\Delta_{+}\subseteq\Delta_{-}.$$

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

$$s^{-}=(s^{-}_{1},\dots,s^{-}_{n}),\ \ \ s^{+}=(s^{+}_{1},\dots,s^{+}_{n}).$$

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

	$\displaystyle\bigoplus_{\alpha\in\langle\alpha^{\prime}\rangle}\underline{\mathcal{S}\mkern-3.0mu}\mkern 3.0mu_{0}^{N,\alpha}(W_{1};K,L\cup J)$	$\displaystyle\cong\bigoplus_{\alpha\in\langle\alpha^{\prime}\rangle}\underline{\mathcal{S}\mkern-3.0mu}\mkern 3.0mu_{0}^{N,\alpha}(W_{1};K,L)\otimes\operatorname{KhR}_{N}(J)$
		$\displaystyle\cong\bigoplus_{\alpha\in\langle\alpha^{\prime}\rangle}\underline{\mathcal{S}\mkern-3.0mu}\mkern 3.0mu_{0}^{N,\alpha}(W_{1};K,L)\otimes{\mathbb{Z}}[X]/(X^{N}).$

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

(11)

$$\Psi_{I\times Y;\Delta_{+},\alpha^{\prime}}(v\otimes X^{n})=\begin{cases}v&\text{if }n=N-1,\\ 0&\text{if }n=0,1,\dots,N-2,\end{cases}$$

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

(12)

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

$$\underline{\Psi}_{I\times\partial W_{1};\Sigma_{-},\alpha^{\prime}}:=\Phi^{-1}\circ\Psi_{I\times Y;\Delta_{-},\alpha^{\prime}}\circ\Phi.$$

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):

$$\Psi_{I\times\partial W_{1};\Sigma_{-},\alpha^{\prime}}(v\otimes X^{n})=\Psi_{I\times\partial W_{1};\Sigma(n\bullet),\alpha^{\prime}}(v).$$

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:

$$\underline{\Psi}_{I\times\partial W_{1};\Sigma(n\bullet),\alpha^{\prime}}\colon\bigoplus_{\alpha\in\langle\alpha^{\prime}\rangle}\underline{\mathcal{S}\mkern-3.0mu}\mkern 3.0mu_{0}^{N,\alpha}(W_{1};K,L)\to\bigoplus_{\alpha\in\langle\alpha^{\prime}\rangle}\underline{\mathcal{S}\mkern-3.0mu}\mkern 3.0mu_{0}^{N,\alpha}(W_{1};K,L)$$

such that

(13)

$$\underline{\Psi}_{I\times\partial W_{1};\Sigma_{-},\alpha^{\prime}}(v\otimes X^{n})=\underline{\Psi}_{I\times\partial W_{1};\Sigma(n\bullet),\alpha^{\prime}}(v).$$

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.