# 2021-10-18

## 15:07

$$K_1, K_2$$ are smoothly concordant iff there exists a smoothly embedded cylinder $$S^1\times I \hookrightarrow S^3\times I$$ with $${{\partial}}(S^1\times I) = K_1 {\textstyle\coprod}-K_2$$. The concordance group $$C$$ is the abelian group given by knots $$K \hookrightarrow S^3$$ under connect sum, modulo concordance.

If $$K_i \hookrightarrow Y_i \in {\mathbf{Z}}\operatorname{HS}^3$$, then the $$K_i$$ are homologically concordant if there is smoothly embedded cylinder $$S^1\times I \hookrightarrow W$$ with $${{\partial}}(W, S^1\times I) = (Y_1, K_1) {\textstyle\coprod}(Y_2, K_2)$$ with $$W$$ a homology cobordism:

• $$W \in {\mathsf{Mfd}}^4$$ compact oriented,
• $${{\partial}}W = Y_1 {\textstyle\coprod}Y_2$$,
• There are induced isomorphisms $$H_*(Y_i; {\mathbf{Z}}) { \, \xrightarrow{\sim}\, }H_*(W; {\mathbf{Z}})$$.

This yields a homological concordance group $$\widehat{C}_{\mathbf{Z}}$$.

There is an injection (?) $$C_{\mathbf{Z}}\hookrightarrow\widehat{C}_{\mathbf{Z}}$$ which is known by Levine not to be surjective. What can be said about the cokernel?

• Infinitely generated, known using d invariants and reduced Heegard-Floer homology.
• Contains a $${\mathbf{Z}}{\hbox{-}}$$subgroup using epsilon invariants and tau invariants
• Contains a $${\mathbf{Z}}^\infty$$ subgroup, and in fact a summand

See Seifert fibered space, ZHS3. These are all homology cobordant to $$S^3$$.

Proof uses CFK, a $${ \mathbf{F} }[u, v]{\hbox{-}}$$module.

A knot-like complex over $$R$$ is a complex $$C \in {\mathsf{gr}\,}_{{\mathbf{Z}}{ {}^{ \scriptscriptstyle\times^{2} } }} \mathsf{Ch}(R)$$ such that

• $$H_*(C/u)/C_{{\operatorname{tors}}_v} \conf { \mathbf{F} }[v]$$
• $$H_*(C/v)/C_{{\operatorname{tors}}_u} \conf { \mathbf{F} }[u]$$
Some examples: the knot Floer complex CFK over a knot, $$\CFK_{{ \mathbf{F} }[u, v]}(K)$$. Theorem: every such complex is locally equivalent to a unique standard complex. Concordant knots produce locally equivalent complexes $$\CFK_R(K)$$ for $$R \coloneqq{ \mathbf{F} }[u] \otimes_{ \mathbf{F} }{ \mathbf{F} }[z] / \left\langle{uv}\right\rangle$$.
Set $$\mathsf{C} \coloneqq{\operatorname{Emb}}(S^1, S^3)$$, add the monoidal structure $${\sharp}$$ for connect sum. Take “isotopy” category instead of homotopy category? The unit is $$\one = U$$, the unknot up to isotopy. What is the stabilization of $${-}{\sharp}X$$ for fixed choices of $$X$$? Or of other interesting functors? #personal/idle-thoughts
• Can do base changes $$\CFK_{{ \mathbf{F} }[u, v]}(M_n, K_n) \otimes_{{ \mathbf{F} }[u, v]} \mathcal{X} \leadsto \CFK_{\mathcal{X}}(M_n, K_n)$$ (may also need to change basis to get standard complex).