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Homotopy class of a homeomorphism

I was reading a book and find this statement:

''It is well kown that each homeomorphism $f:S\rightarrow R$ between compact surfaces is homotopic to a diffeomorphism''

I would know some references of this affirmation to see one proof.

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Even more is true, any homotopy-equivalence is homotopic to a diffeomorphism. If the surface has boundary then you have to also require the homotopy-equivalence restricts to a homotopy-equivalence of the boundaries. A common reference is Zieschang, Vogt and Coldeway, Surfaces and planar discontinuous groups. – Ryan Budney Nov 28 2011 at 23:38
Statements like this abound in homotopy theory. For instance, any map $f$ is homotopic to an injection. Or a surjection. Or a smooth map. Or pretty much whatever you want (not all at the same time, though). The first two have to do with making the kernel or cokernel contractible. I'm not sure about the others, and I'd like to see an answer. – David White Nov 28 2011 at 23:40
Isn't this question almost the same as mathoverflow.net/questions/35198/… ? – Daniel Moskovich Nov 28 2011 at 23:52

3 Answers

You may find a proof of this using hierarchies in some notes of Lackenby. See Theorem 12.1. I think this sort of argument is probably due to Waldhausen, since his proof of homotopy rigidity of Haken 3-manifolds is dependent on this, so you could have a look at his paper too.

I think there are probably many other proofs of this fact. One possible proof is to endow $S$ and $R$ with hyperbolic metrics, and use the Douady-Earle map.

Edit: I was trying to answer the stronger question of whether a homotopy equivalence is homotopic to a homeomporphism (your use of the term homotopic threw me), which is answered in the above references. Another strengthening is to ask whether a homeomorphism is isotopic to a diffeomorphism? In other words, does a surface have a unique differential structure? In the context of PL structures, this uniqueness was answered by Rado (see Moise's book). I think it's also known that PL and differential structures are equivalent. This is discussed in Thurston's book (Theorem 3.10.9).

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Didn't Munkres prove a homeomorphism is isotopic to a diffeomorphism? – Steve D Nov 29 2011 at 11:17
He proved uniqueness of differentiable structures on 3-manifolds. projecteuclid.org/… – Ian Agol Nov 29 2011 at 17:00

This fact is known as "the classification of surfaces". For a particularly non elementary view see http://www.math.uchicago.edu/~shmuel/tom-readings/ranicki-intro, chapter 4.

EDIT A particular elementary proof is here.

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Well, "homeomorphic surfaces are diffeomorphic" is weaker than "homeomorphisms are homotopic to diffeomorphisms". But, then again, any two true statements are equivalent. ;) – Richard Kent Nov 29 2011 at 2:00
Any true statements may or may not be equivalent, but I was specifically answering the OP's statement. Look at the reference. – Igor Rivin Nov 29 2011 at 10:22
@Igor - I guess you are referring to Theorem 4.32, part (ii), on page 61 of the pdf file. But Ranicki doesn't give a proof. Instead he refers to chapter 9 of Hirsch's book "Differential Topology". Skimming chapter 9 it seems that the exact statement the OP wants is not there? – Sam Nead Nov 29 2011 at 10:55
@Sam: Ranicki doesn't give a full proof of classification of surfaces, that's true, but among other things, the subject of when homotopy equivalences are homotopic to diffeos is central to surgery theory, so once you understand classification of surfaces, the answer to the OP is immediate. The whole machine of surgery is a bit of an overkill, hence the "non elementary" comment. – Igor Rivin Nov 29 2011 at 12:32

A homeomorphism (of surfaces) is isotopic to a PL homeomorphism. See Theorem A4 of Epstein's "Curves on 2-manifolds and isotopies". He gives a proof. I haven't read the paper recently, but I recall that it is fairly self-contained.

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