Difference between the diffeomorphism classification of a manifold $M$ and the set of equivalences of homotopy smoothings $hS(M)$ In Lopez de Medrano "Involutions on manifolds", a homotopy smoothing of a Poincaré space $X$ is a homotopy equivalence $f:M^n\rightarrow X$, where $M^n$ is a smooth $n$-dim. manifold (everything is oriented and orientation-preserving). Two homotopy smoothings $f_i:M_i^n\rightarrow X$, $i=0,1$, are equivalent if there exists a diffeomorphism $\phi:M^n_0\rightarrow M^n_1$ such that $f_1\circ \phi\simeq f_0$. Denote by $hS(X)$ the set of equivalence classes of homotopy smoothings on $X$.
Now suppose $X$ is already a smooth, oriented $n$-dim. manifold. Under which conditions does $hS(X)$ actually correspond to a diffeomorphism classification, i.e. when is the homotopy condition $f_1\circ \phi\simeq f_0$ always satisfied?
Do you know a space $X$ where $hS(X)$ does not correspond to the diffeomorphism classification?
 A: Assuming that $X$ is a smooth manifold, your question can be reformulated as: Under which conditions every self-homotopy-equivalence $X\to X$ is homotopic to a diffeomorphism? 
I will say that $X$ satisfying this property is smoothly rigid. I will say that $X$ is rigid if every self-homotopy equivalence is homotopic to a homeomorphism. 
Here are some positive and negative answers:


*

*In dimensions 2 and 3 there is no difference between smooth rigidity and rigidity since TOP=DIFF in these dimensions. 

*If $X$ is 2-dimensional and closed (compact and has empty boundary) then $X$ is rigid. Ditto for the case when $X$ is 2-dimensional and has abelian fundamental group. However, if $X$ is noncompact, oriented connected, has nonebelian fundamental group (say, $X$ is the triply punctured sphere) and is different from the once punctured torus, then $X$ is not rigid.  
For this reason, I will restrict to closed manifolds. 


*There are non-rigid 3-manifolds, say, lens spaces. David Gabai gives $L(8,1)$ as an example:  


Gabai, David, On the geometric and topological rigidity of hyperbolic 3-manifolds, Bull. Am. Math. Soc., New Ser. 31, No. 2, 228-232 (1994).  
Nevertheless, closed aspherical 3-manifolds are known to be rigid. (This is due to many people, starting with Waldhausen and concluding with Perelman.) 


*Starting from dimension 4, there are examples of homeomorphisms which are not homotopic to diffeomorphisms. See for instance here for some examples of 4-manifolds as well as among exotic 7-dimensional spheres. 

