Subsection 6.3 (0PNG) · ScalingStacks

6.3 Link cobordisms and maps of cohomology groups

In this section by a surface $S$ in $\mathbb{R}^{4}$ we mean an oriented, compact surface $S,$ possibly with boundary, properly embedded in $\mathbb{R}^{3}\times[0,1].$ The boundary of $S$ is then a disjoint union

\partial S=\partial_{0}S\sqcup-\partial_{1}S

(146)

of the intersections of $S$ with two boundary components of $\mathbb{R}^{3}\times[0,1]$ :

	$\displaystyle\partial_{0}S$	$\displaystyle=$	$\displaystyle(S\cap\mathbb{R}^{3}\times\{0\})$
	$\displaystyle-\partial_{1}S$	$\displaystyle=$	$\displaystyle(S\cap\mathbb{R}^{3}\times\{1\})$

Note that $\partial_{0}S$ and $\partial_{1}S$ are oriented links in $\mathbb{R}^{3}.$

The surface $S$ can be represented by a sequence $J$ of plane diagrams of oriented links where every two consecutive diagrams in $J$ are related either by one of the Reidemeister moves I-IV (Section 4.1) or by one of the four moves depicted below (see [CS] where such representations by sequences of plane diagrams are studied in detail).

Following Carter-Saito [CS] and Fisher [Fs], we call these moves birth, death, fusion ([CS] and [Fs] deal with the non-oriented version of these moves). We call $J$ a representation of $S.$ The first diagram in the sequence $J$ is necessarily a diagram of oriented link $\partial_{0}S,$ and the last diagram is a diagram of $\partial_{1}S.$

The birth move consists of adding a simple closed curve to a diagram $D.$ Denote the new diagram by $D_{1}.$ Then $C(D_{1})=C(D)\otimes_{R}A$ and the unit map $\iota:R\to A$ of the algebra $A$ induces a map of complexes $C(D)\to C(D_{1}).$ This is the map we associate to the birth move.

The death move consists of removing a simple circle from a diagram $D_{1}$ to get a diagram $D.$ In this case the counit $\epsilon:A\to R$ induces a map of complexes $C(D_{1})\to C(D).$

Finally, to a fusion move between diagrams $D_{0}$ and $D_{1}$ we associate a map $C(D_{0})\to C(D_{1})$ corresponding to the elementary surface with one saddle point in the manner discussed in Section 4.3.

In Section 5 to each Reidemeister move between diagrams $D_{0}$ and $D_{1}$ we associated a quasi-isomorphism map of complexes $C(D_{0})\longrightarrow C(D_{1}).$

Given a representation $J$ of a surface $S$ by a sequence of diagrams, denote the first and last diagrams of $J$ by $J_{0}$ and $J_{1}$ respectively. Then to $J$ we can associate a map of complexes

\varphi_{J}:C(J_{0})\to C(J_{1})

(147)

which is the composition of maps associated to elementary transformations between consecutive diagrams of $J.$ The map $\varphi_{J}$ induces a map of cohomology groups

\theta_{J}:H^{i,j}(J_{0})\to H^{i,j+\chi(S)}(J_{1}),\hskip 21.68121pti,j\in\mathbb{Z}.

(148)

We now ready to state our main conjecture.

0PM1

Conjecture 1 If two representations $J,\widetilde{J}$ of a surface $S$ have the property that

(a) diagrams $J_{0}$ and $\widetilde{J}_{0}$ are isomorphic,

(b) diagrams $J_{1}$ and $\widetilde{J}_{1}$ and isomorphic,

then the maps $\theta_{J}$ and $\theta_{\widetilde{J}}$ are equal, up to an overall minus sign, $\theta_{J}=\pm\theta_{\widetilde{J}}.$

In other words, we conjecture that, after a suitable $\mathbb{Z}_{2}$ extension of the link cobordism category, our construction associates honest cohomology groups $H^{i}(L)$ to oriented links $L$ in $\mathbb{R}^{3}$ (and not just isomorphism classes of groups) and associates homomorphisms between these groups to isotopy classes of oriented surfaces embedded in $\mathbb{R}^{3}\times[0,1].$ In the categorical language, we expect to get a functor from the category of ( $\mathbb{Z}_{2}$ -extended) oriented link cobordisms to the category of bigraded $R$ -modules and module homomorphisms.

Suppose that the above conjecture is true. Then, in the case of a closed oriented surface $S$ embedded in $\mathbb{R}^{4},$ the map $\theta_{S}$ of cohomology groups is a homomorphism from $R$ to itself (since $\partial S=\emptyset$ and the cohomology of the empty link is equal to the ground ring $R$ ). This homomorphism has degree $\chi(S)$ and is automatically $0$ when $\chi(S)<0.$ Thus, the conjectural invariants are zero whenever $S$ has empty boundary and the Euler characteristic of $S$ is negative. If, again, $\partial S=\emptyset$ and the Euler characteristic of $S$ is nonnegative (when $S$ is connected, $S$ is then necessarily a 2-sphere or a 2-torus), the homomorphism $\theta_{S}:R\to R$ is determined by $\theta_{S}(1)=kc^{\frac{\chi(S)}{2}}$ and amounts to an integer number $k.$ Hence, we expect to have integer-valued invariants of closed oriented surfaces with non-negative Euler characteristic, embedded in $\mathbb{R}^{4}.$

Mikhail Khovanov

Original source: arXiv:math/9908171v2