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?



Difference between the diffeomorphism classification of a manifold $M$ and the set of equivalences of homotopy smoothings $hS(M)$
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:


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


2. 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**](http://dx.doi.org/10.1090/S0273-0979-1994-00523-3), 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.) 


3. Starting from dimension 4, there are examples of homeomorphisms which are not homotopic to diffeomorphisms. See for instance [here](https://math.stackexchange.com/questions/1068805/is-every-self-homeomorphism-homotopic-to-a-diffeomorphism) for some examples of 4-manifolds as well as among exotic 7-dimensional spheres.


