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.)

  • 2
    $\begingroup$ 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. $\endgroup$ – PVAL-inactive Nov 28 '17 at 3:10

This is really just a comment that is too long. Also, I will approach this from the view of knot theory.

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.

When I first saw this definition, I naively thought this would mean every knot was concordant to every other knot, but this is not true. My erroneous thought was that in 4 dimensions, all knots are trivial, which is what concordance sees. While my thought is true for a knot in 4 dimensions, the difference is that the surface cannot lie in the 3 dim space either knot does. So there are knots which are not concordant to the unknot and knots which are. But all knots are homtopic to the unknot, as I stated above.

I think that PVAL-inactive said it best to actually answer your question in the comments, but I hope I gave some intuition.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.