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Every topological 3-manifold carries a smooth structure, unique up to diffeomorphism. In addition, "up to homotopy", the maps between them are the same: every continuous map is homotopic to a smooth map, and any homotopy between smooth maps is homotopic to a smooth homotopy (true for any pair of closed smooth manifolds. Probably the inclusion of mapping spaces $\text{Smooth}(M,N) \hookrightarrow \text{Cont}(M,N)$ is a homotopy equivalence, given the right topologies; all one really needs is to be able to homotope continuous maps that are smooth on the boundary to smooth maps, even if the domain is a manifold with corners.)

For closed orientable surfaces of genus $g>1$, $\text{Diffeo}(\Sigma) \hookrightarrow \text{Homeo}(\Sigma)$ is a homotopy equivalence; one can essentially see this by showing that each is homotopy equivalent to the discrete group $\text{Out}(\pi_1)$. (One can check by hand that it's a homotopy equivalence for $\Sigma = S^2$ or $T^2$; but no longer to a discrete group.)

What can we say for the same situation in closed 3-manifolds? Is the inclusion $\text{Diffeo}(M) \hookrightarrow \text{Homeo}(M)$ a homotopy equivalence? Does it at least induce a bijection on $\pi_0$ (i.e., induces an isomorphism on the smooth and continuous mapping class groups?) If not, can we restrict to some subclass (hyperbolic?) for which either of these is true?

(By the computation of mapping class groups of tori, or noting the existence of exotic spheres, this fails miserably for $n \geq 5$. Presumably it also fails miserably for $n=4$.)

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    There are involutions of the 3-sphere which are not conjugate to any smooth involution. I don't think such a thing can be connected by a path to a smooth diffeomorphism. – PVAL-inactive Mar 18 '15 at 03:02
  • @PVAL It must be, since the mapping class group of $S^3$ is $\Bbb Z/2\Bbb Z$ (the nontrivial element just a reflection), both smoothly and continuously. Hatcher proved that $O(4) \hookrightarrow \text{Diffeo}(S^3)$ is a homotopy equivalence. I think the inclusion $\text{Diffeo}(S^3) \hookrightarrow \text{Homeo}(S^3)$ is a homotopy equivalence. –  Mar 18 '15 at 03:14
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    This is written up in some old papers of Hatcher's (back then they depended on some conjectures but now they don't). Diffeomorphisms of a 3-manifold have the homotopy-type of homeomorphisms, yes. When you go to the equivariant world (as Mike suggests) things are different, but provided everything is "locally linear" then you continue to get equivalences. The involutions Mike talks about have non-manifold fixed point sets. The relevant Hatcher paper is his "Linearization in 3-manifold topology" paper, I think it might be in an ICM proceedings. – Ryan Budney Mar 19 '15 at 18:28
  • @RyanBudney Thanks for the reference. Hatcher references a book of Cerf's, which proves the claimed result assuming Smale's conjecture. I'll find where precisely it's proved in the book sometime next week. –  Mar 19 '15 at 23:08
  • I want to point out that my earlier thought is incorrect (not the fact I stated but the second sentence). I didn't delete it so that Mike Miller's comment made sense. I am not sure why it was upvoted. – PVAL-inactive Mar 25 '15 at 20:02
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    Nonetheless I appreciated your comment, @PVAL, and Ryan's later point that if one talks about spaces of equivariant automorphisms we no longer have an equivalence between smooth and continuous stuff. –  Mar 25 '15 at 20:04

2 Answers2

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The key reference here is Jean Cerf, "Groupes d’automorphismes et groupes de difféomorphismes des variétés compactes de dimension 3", Bull. Soc. Math. France (1959). The full text is available here.

Let $M$ be a closed 3-manifold, $G$ its group of self-homeomorphisms, $H$ its group of self-diffeomorphisms. (All orientation preserving for convenience.) He proves that $\pi_n(G,H) = 0$ for all $n \geq 0$, thus that the inclusion $H \hookrightarrow G$ is a weak homotopy equivalence - assuming Smale's conjecture (now theorem, due to Hatcher) that the inclusion $SO(4) \hookrightarrow \text{Diff}^+(S^3)$ is a homotopy equivalence. (In fact, he shows that the inclusion is a homotopy equivalents, but the arguments are a bit more delicate.)

For a manifold $M$ with boundary, $H$ and $G$ as above should be the automorphisms restricting to the identity on the boundary. The idea is to show that, if you have a decomposition $M = M_1 \cup M_2$, with $M_1 \cap M_2$ a properly embedded surface, having $H_i$ be $(n-1)$-connected in $G_i$ (for both $i$) is equivalent to having $H$ be $(n-1)$-connected in $G$.

Now we want to induct. Say that a manifold homeomorphic to $D^3$ is order 0; and a manifold that decomposes into order $(n-1)$-pieces is order $n$ if we can't decompose it into pieces smaller than order $(n-1)$. But by Smale's theorem, it's true that $\pi_i(G,H) = 0$ for all $i \geq 0$ when $M = D^3$. Because every 3-manifold has finite order (look at Heegaard decompositions), this proves the theorem.

As for why the third paragraph is true, this is Cerf's "Lemma 0", at the very end of his paper.

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    Thanks to Ryan Budney in the comments above for pointing me in the direction of this paper. –  Mar 25 '15 at 20:25
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Warning: This should be a comment on Mike Miller's answer but I'm not allowed to comment.

The paper cited by Miller is a conference proceeding announcing Cerf's result but there are many more details in Chapter III of the published version of Cerf's thesis: Topologie de certains espaces de plongements Bull. Soc. Math. France 89 1961 227–380 (also in French).

It's also worth mentioning a potentially confusing historical point. In his 1959 announcement, Cerf cites a forthcoming paper by Smale proving what we now call Smale's conjecture. It seems that, by 1961, it became clear Smale's proof didn't work and Cerf's results are accordingly downgraded to theorems conditional on Smale's conjecture. Then Hatcher proved Smale's conjecture in his 1983 paper so we can now use Cerf's results about this question.

One last comment: Cerf's results in those papers also include information on the local situation. For instance, for any neighborhood $U$ of the identity in $Homeo(M)$, there is a smaller neighborhood $V$ such that any diffeomorphism in $V$ is smoothly isotopic to the identity through diffeomorphisms staying in $U$. This does not follow from injectivity of $\pi_0(Diff) \to \pi_0(Homeo)$.

Patrick Massot
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