# How to think about concordance?

Two continuous maps $f, g: A \to B$ are homotopic if there exists a continuous map $H:A \times [0,1] \to B$ such that $H(x,0)$ = $f(x)$ and $H(x,1) = g(x)$. I understand that one should think of $H$ as a continuous deformation of the map $f$ to the map $g$, and there are gifs on the wikipedia page and all is right with the world.

Two continuous maps $f, g: A\to B$ are called concordant if there exists a continuous map an embedding $J: A\times[0,1] \to B\times[0,1]$ such that $J(x,0)$ = $f(x)\times\{0\}$ and $J(x,1) = g(x)\times\{1\}$. Is there an intuitive way to understand concordance? Or are there some simple, illuminating examples of maps that are concordant but not homotopic?

(edit: corrected the definition of concordance.)

• No, the injection $B \to B \times [0,1]$ is a homotopy equivalence (with homotopy inverse the projection map). So with this (probably incorrect assuredly non-standard) definition of concordance, the two are equivalent. If you mean the usual definition of concordance used in geometric topology (where the homotopy is required to be an embedding), concordance is a stronger condition than homotopy but weaker than isotopy and there are standard examples in say Rolfsen and any other comparable book. – PVAL-inactive Nov 28 '17 at 3:10

Every two knots are homotopic, since they are just different embeddings of $S^1$ in $S^3$. So as a knot theorist, we use ambient isotopy as our equivalence, which is a homotopy where every intermediate map is a homeomorphism. (This is equivalent to the other definitions you might see.) But for either of these two ways of making maps, we can view this happening in S^3 over time.
Now for concordance. We say to knots are concordant in exactly the same way as you define above. But when we think about this in a geometric way, we don't view this as happening in $S^3$ over time anymore. What we do is let the two knots lie in 4 dimensional space $(x,y,z,w)$ where $w\in [0,1]$, one at $w=0$ and the other at $w=1$. And in this space, if we can find a surface that connects the two knots that only intersects the 3 dimensional space at $w=0$ and 1 where the the knots do. For knots, this surface is necessarily a cylinder.