Subsection 4.2. (0NC7) · ScalingStacks

4.2. Link homology in abstract 3-spheres

Definition 4.8. A link homology for links in $3$ -spheres is a functor

\begin{Bmatrix}\textrm{link embeddings in oriented }S\cong S^{3}\\ \textrm{link cobordisms in oriented }Y\cong S^{3}\times{[0,1]}\textrm{ up to isotopy rel }\partial\end{Bmatrix}\xrightarrow{}\begin{Bmatrix}\textrm{bigraded abelian groups}\\ \textrm{homogeneous homomorphisms}\end{Bmatrix}

0NB0

Theorem 4.9. $\mathrm{KhR}_{N}$ extends to a link homology theory for links in $3$ -spheres.

The proof occupies the remainder of this subsection.

Remark. In the proof of Theorem 4.9, we will show that the sweep-around property from Theorem 1.1 is sufficient to extend a link homology for links in $3$ -balls to $3$ -spheres, without using any special properties of $\mathrm{KhR}_{N}$ .

0NB1

Definition 4.10. Let $S$ be an oriented $3$ -manifold diffeomorphic to $S^{3}$ . For any point $p\in S\setminus L$ , we consider $L$ as a link in the $3$ -ball $S\setminus\{p\}$ and denote by $\pi\colon T(S,L)\to S\setminus L$ the bundle of bigraded abelian groups, whose fiber at the point $p\in S\setminus L$ is $\mathrm{KhR}_{N}(S\setminus\{p\},L)$ as defined in Definition 4.4.

For any path $p_{t}$ in $S\setminus L$ , we have that $L\times{[0,1]}\subset S\times{[0,1]}\setminus\{(p_{t},t)\}_{t\in{[0,1]}}\cong{\mathbb{R}}^{3}\times{[0,1]}$ . By the results of §4.1, this cobordism induces a parallel transport isomorphism

(\mathrm{KhR}_{N}(p_{t})\colon T(S,L)_{p_{0}}\to T(S,L)_{p_{1}})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\mathrm{KhR}_{N}(S\times{[0,1]}\setminus\{(p_{t},t)\}_{t\in{[0,1]}},L\times{[0,1]})

0NB2

Lemma 4.11. The parallel transport isomorphisms $\mathrm{KhR}_{N}(p_{t})$ define a flat connection on $T(S,L)$ .

0NB3

Proof. We have to show that the parallel transport isomorphisms associated to closed loops $p_{t}$ in $S\setminus L$ are identity maps. Suppose first that $p_{t}$ is a contractible loop. Then the pair $(S\times{[0,1]}\setminus\{(p_{t},t)\}_{t\in{[0,1]}},L\times{[0,1]})$ is diffeomorphic to a pair $({\mathbb{R}}^{3}\times{[0,1]},\Sigma)$ where $\Sigma$ is isotopic to an identity link cobordism, which implies that the parallel transport isomorphism $\mathrm{KhR}_{N}(p_{t})$ is the identity. This also implies that the parallel transport isomorphisms associated to isotopic paths between two points $p_{0}$ and $p_{1}$ in $S\setminus L$ are equal. Now suppose that the loop $p_{t}$ is a small meridian around a component of $L$ . Then the pair $(S\times{[0,1]}\setminus\{(p_{t},t)\}_{t\in{[0,1]}},L\times{[0,1]})$ is diffeomorphic to a pair $({\mathbb{R}}^{3}\times{[0,1]},\Sigma)$ where $\Sigma$ is a sweep-around cobordism as in (1.1). By Theorem 1.1, it follows that the parallel transport isomorphism $\mathrm{KhR}_{N}(p_{t})$ is the identity. Since $\pi_{1}(S\setminus L)$ is generated by such small meridian loops, it follows that the parallel transport isomorphism for every loop $p_{t}$ is the identity. ∎

0NB4

Definition 4.12. Let $L\subset S$ be a link embedded in a 3-sphere. Then we define the Khovanov–Rozansky homology of $L$ in $S$ to be

\mathrm{KhR}_{N}(S,L)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Gamma_{\textrm{flat}}(T(S,L)),

the bigraded abelian group of flat sections of the bundle $T(S,L)$ .

Note that every point $p\in S\setminus L$ induces a grading-preserving isomorphism

\mathrm{KhR}_{N}(S,L)\to\mathrm{KhR}_{N}(S\setminus\{p\},L)

of evaluating sections at the point $p$ .

0NB5

Definition 4.13. Consider a link cobordism $\Sigma\subset W$ in a 4-manifold $W$ diffeomorphic to $S^{3}\times{[0,1]}$ . Let $\Sigma_{\mathrm{in}}\subset W_{\mathrm{in}}$ and $\Sigma_{\mathrm{out}}\subset W_{\mathrm{out}}$ denote the boundary links in the incoming and outgoing boundary 3-spheres of $W$ . Now we define

\mathrm{KhR}_{N}(W,\Sigma)\colon\mathrm{KhR}_{N}(W_{\mathrm{in}},\Sigma_{\mathrm{in}})\to\mathrm{KhR}_{N}(W_{\mathrm{out}},\Sigma_{\mathrm{out}})

by first choosing a path $p_{t}\subset W\setminus\Sigma$ from $p_{0}\in W_{\mathrm{in}}\setminus\Sigma_{\mathrm{in}}$ to $p_{1}\in W_{\mathrm{out}}\setminus\Sigma_{\mathrm{out}}$ . Then we have $W\setminus\{(p_{t},t)\}_{t\in{[0,1]}}\cong{\mathbb{R}}^{3}\times{[0,1]}$ and we declare $\mathrm{KhR}_{N}(W,\Sigma)(\eta)$ , for a flat section $\eta\in\mathrm{KhR}_{N}(W_{\mathrm{in}},\Sigma_{\mathrm{in}})$ , to be the unique flat section of $\mathrm{KhR}_{N}(W_{\mathrm{out}},\Sigma_{\mathrm{out}})$ with value

\mathrm{KhR}_{N}(W,\Sigma)(\eta)(p_{\mathrm{out}})=\mathrm{KhR}_{N}(W\setminus\{(p_{t},t)\}_{t\in{[0,1]}},\Sigma)(\eta(p_{\mathrm{in}}))

0NB6

Lemma 4.14. $\mathrm{KhR}_{N}(W,\Sigma)$ is independent of the choice of the path $p_{t}$ .

0NB7

Proof. Let us first fix a choice of endpoints $p_{0}\in W_{\mathrm{in}}\setminus\Sigma_{\mathrm{in}}$ and $p_{1}\in W_{\mathrm{out}}\setminus\Sigma_{\mathrm{out}}$ . Then any two choices of paths $p_{t}\in$ and $p^{\prime}_{t}$ from $p_{0}$ to $p_{1}$ can be related by isotopy in $W\setminus\Sigma$ or splicing in a little loop linking a component of $\Sigma$ . As before, isotopic paths give rise to isotopic surfaces in ${\mathbb{R}}^{3}\times{[0,1]}$ , which induce equal maps. Similarly, in the case of a linking loop, we can choose a standard local model and then notice that the sweep-around property from Theorem 1.1 implies that the two paths induce the same map. Finally, the independence from the choice of endpoints $p_{0}\in W_{\mathrm{in}}\setminus\Sigma_{\mathrm{in}}$ and $p_{1}\in W_{\mathrm{out}}\setminus\Sigma_{\mathrm{out}}$ follows as in the proof of Lemma 4.6. ∎

This completes the proof of Theorem 4.9.

Scott Morrison, Kevin Walker, Paul Wedrich

Original source: arXiv:1907.12194v5