Ribbon concordance

Let $F_0$ and $F_1$ be knotted surfaces of the same genus. We say that $F_1$ is ribbon concordant to $F_0$ if there is a concordance $C$ (a properly embedded orientable submanifold diffeomorphic to $F_0\times I$) in ${\mathbb{R}}^4\times [0,1]$ between $F_1\subset{\mathbb{R}}^4\times\{1\}$ and $F_0\subset{\mathbb{R}}^4\times\{0\}$ such that the restriction to $C$ of the projection ${\mathbb{R}}^4\times[0,1]\rightarrow[0,1]$ is a Morse function with critical points of index 0 and $1$ only. We write $F_1\geq F_0$.

Figure: Ribbon concordance

Note that if $F_1\geq F_0$, then there is a set of $n$ $1$-handles on a split union of $F_0$ and $n$ trivial sphere-knots, for some $n\geq 0$, such that $F_1$ is obtained by surgeries along these handles (Fig. ). Ribbon concordance was first defined in [Gor81]. It is defined in general for knots in any dimension. In [CSS03*], quandle cocycle invariants were used as obstructions to ribbon concordance for surfaces, that is, to give examples $F_0$ and $F_1$ that are not related by ribbon concordance.