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. 
 A: 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). 
A: 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.
A: 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.  
