# 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?

• You are effectively asking for a smooth manifold $M$ such that there exists a self-homotopy-equivalence $M\to M$ which is not homotopic to a diffeomorphism. If one allows for noncompact $M$, then such examples exist already among surfaces (say, the triply-punctured 2-sphere). If you want compact examples, see my answer here. – Moishe Kohan Apr 28 '20 at 17:39
• One approach to the diffeomorphism classification of closed manifolds in a given homotopy type is to compute the action of the monoid of homotopy self-equivalences on the structure set. You can find a discussion of these matters in arxiv.org/abs/0912.4874. For example in section 10 you can find a homotopy self-equivalence of $S^7\times CP^3$ with nontrivial invariant. In Remark 5.4 we discuss a self-homotopy equivalence with trivial normal invariant that is not homotopic to a diffeomorphism. – Igor Belegradek Apr 28 '20 at 18:22
• It's probably worth mentioning Mostow rigidity (hyperbolic manifolds are smoothly rigid, and in fact hot-equiv implies isometric, for dim $\ge 3$) and the Borel conjecture (that aspherical manifolds are topologically rigid). – Kevin Casto Apr 28 '20 at 21:46

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.

1. 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.)

1. 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.
• Thanks for your answer! I am actually looking for conditions on $X$ in dimension greater than 4 such that $hS(X)$ corresponds to the different diffeomorphism types. For example, how could I determine if $hS(\mathbb{R}P^n)$ gives the diffeomorphism classification? – Kafka91 Apr 29 '20 at 7:54
• @Kafka91: The only way to interpret your question "such that ℎ𝑆(𝑋) corresponds to the different diffeomorphism types" is that you are asking for (sufficient or necessary?) conditions on a (smooth manifold) $X$ such that $hS(X)$ is (naturally) bijective to the set of diffeomorphism classes of manifolds homotopy-equivalent to $X$. As I said in my answer, this happens if and only if every self-homotopy-equivalence $X\to X$ is homotopic to a diffeomorphism. As for the specific question about projective spaces, I do not know, but you should update your question. – Moishe Kohan Apr 29 '20 at 15:45
• @Kafka91: the group of homotopy self-equivalences of real projective spaces is computed in Corollary 6 of "Coverings of fibrations" Becker and Gottlieb, Compositio Mathematica, Volume 26 (1973) no. 2, p. 119-128, numdam.org/item/CM_1973__26_2_119_0. It is trivial in even dimensions and $\mathbb Z_2$ in odd dimensions $>1$. – Igor Belegradek May 1 '20 at 18:51
• @IgorBelegradek Thank you very much! – Kafka91 May 5 '20 at 11:26