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If $\Sigma_1$ and $\Sigma_2$ are two compact topological surfaces with boundary and $\phi, \psi : \Sigma_1 \hookrightarrow \Sigma_2$ are two orientation-preserving embeddings that are homotopic, then they are also isotopic. This is a consequence of a result of Maxime Fortier Bourque on conformal embeddings, but it feels like it ought to have been proved earlier. Is there an earlier reference?

If $\Sigma_1$ is an annulus, then this is really a question about simple curves, and it appears in a paper by Hass, Freedman, and Scott. If $\Sigma_1$ and $\Sigma_2$ are closed surfaces (so that $\phi$ and $\psi$ are homeomorphisms), this is a consequence of the Dehn-Nielsen-Baer theorem.

Maxime Fortier Bourque, MR 3358269 The holomorphic couch theorem, ISBN: 978-1321-75413-1.

Michael Freedman, Joel Hass, and Peter Scott, MR 671777 Closed geodesics on surfaces, Bull. London Math. Soc. 14 (1982), no. 5, 385--391.

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First, for simple closed curves, this was known long before Freedman-Hass-Scott. For closed surfaces, it was first proved by Baer in

Baer, R., Kurventypen auf Flächen. J. reine angew. Math., 156 (1927), 231–246.

and

Baer, R., Isotopie von Kurven auf orientierbaren, geschlossenen Flächen und ihr Zusammenhang mit der topologischen Deformation der Flächen. J. reine angew. Math., 159 (1928), 101–111.

For general surfaces (not necessarily compact, and also possibly with boundary), it was proved by Epstein in

Epstein, D. B. A. Curves on 2-manifolds and isotopies. Acta Math. 115 1966 83–107.

Epstein's paper is also the first one to prove a general homotopy implies isotopy theorem for essentially all surfaces (though as you indicated this was proved earlier by Baer for closed surfaces following work of Dehn and Nielsen; the references are the ones above). There are a couple of silly counterexamples to this that you have to rule out (e.g. the homeomorphism of an annulus that flips the boundary components is homotopic to the identity but not isotopic to the identity).

Now, on to your question. I believe that the embedding question can be easily reduced to the case of homeomorphisms of surfaces (and thus should be mostly attributed to Epstein, though I am not aware of a reference that points this out). The idea is as follows:

  1. You first reduce to the case where the images of your embeddings $\phi,\psi\colon \Sigma_1 \rightarrow \Sigma_2$ are contained in the interior of $\Sigma_2$.

  2. For the next step, let's regard $\Sigma_1$ as a subsurface of $\Sigma_2$ via $\phi$. Thus $\phi$ is the restriction of the identity map $\Sigma_2 \rightarrow \Sigma_2$ to $\Sigma_1$. It is then easy to see that you can extend $\psi$ to a homeomorphism $\Psi\colon \Sigma_2 \rightarrow \Sigma_2$ that is homotopic to the identity (which you can certify by ensuring that $\Psi$ acts as the identity on the fundamental group). There are a few cases to check here, but it isn't so hard.

  3. You then quote homotopy implies isotopy for $\Psi$ and deduce that $\Psi$ is isotopic to the identity. The desired isotopy between $\psi$ and $\phi$ is the restriction of this isotopy to $\Sigma_1$.

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    $\begingroup$ Thanks! That's pretty much what I wanted, although getting to the homeomorphism in step 2 is a little vague still. (As I'm sure you know, there are often corner cases in these things that are easy to forget about.) $\endgroup$ – Dylan Thurston Sep 2 '16 at 17:32
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    $\begingroup$ @DylanThurston: Yeah, there are a lot of tedious cases you have to go through (which is why I didn't write it out), but I think I managed to enumerate them all on my scratch paper before I posted this. As I said, I don't know a good reference for this. If you need it for something, I can try to write out a formal proof (just email me!). $\endgroup$ – Andy Putman Sep 2 '16 at 17:55
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    $\begingroup$ No thanks, I think that's fine. It is indeed a consequence of the (much harder) result of Fortier Bourque that I mentioned above. So one almost-proof and one over-proof should do it. $\endgroup$ – Dylan Thurston Sep 3 '16 at 16:40

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