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.
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. 


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 3manifolds 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 DouadyEarle 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). 


This fact is known as "the classification of surfaces". For a particularly non elementary view see http://www.math.uchicago.edu/~shmuel/tomreadings/ranickiintro, chapter 4. EDIT A particular elementary proof is here. 


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

