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Consider the closed two dimensional disk and two fixed points $A$ and $B$ on the boundary. I am looking for a reference for the following fact: a simple topological curve with end points $A$ and $B$ is unique up to an ambient isotopy, relative to the boundary of the disk.

There is a result that says that two simple closed curves in a compact manifold are isotopic if and only if they are homotopic. Does this apply to compact manifolds with boundary too, like the closed disk? Is the isotopy an ambient isotopy? If so, then I guess one can use that to prove the above statement in the above paragraph.

Thanks!

Consider the closed disk and two fixed points $A$ and $B$ on the boundary. I am looking for a reference for the following fact: a simple topological curve with end points $A$ and $B$ is unique up to an ambient isotopy, relative to the boundary of the disk.

There is a result that says that two simple closed curves in a compact manifold are isotopic if and only if they are homotopic. Does this apply to compact manifolds with boundary too, like the closed disk? Is the isotopy an ambient isotopy? If so, then I guess one can use that to prove the above statement in the above paragraph.

Thanks!

Consider the closed two dimensional disk and two fixed points $A$ and $B$ on the boundary. I am looking for a reference for the following fact: a simple topological curve with end points $A$ and $B$ is unique up to an ambient isotopy, relative to the boundary of the disk.

There is a result that says that two simple closed curves in a compact manifold are isotopic if and only if they are homotopic. Does this apply to compact manifolds with boundary too, like the closed disk? Is the isotopy an ambient isotopy? If so, then I guess one can use that to prove the above statement in the above paragraph.

Thanks!

Source Link
guest84
guest84
  • 31
  • 2

Relative isotopy of simple curves in a disk

Consider the closed disk and two fixed points $A$ and $B$ on the boundary. I am looking for a reference for the following fact: a simple topological curve with end points $A$ and $B$ is unique up to an ambient isotopy, relative to the boundary of the disk.

There is a result that says that two simple closed curves in a compact manifold are isotopic if and only if they are homotopic. Does this apply to compact manifolds with boundary too, like the closed disk? Is the isotopy an ambient isotopy? If so, then I guess one can use that to prove the above statement in the above paragraph.

Thanks!