# exotic differentiable structures on manifolds in dimensions 5 and 6

It's a result of low-dimensional topology that in dimensions 3 and lower, two manifolds are homeomorphic if and only if they are diffeomorphic. Milnor's 7-spheres give nice counterexamples to this result in dimension 7, and exotic $\mathbb{R}^4$'s give nice counterexamples in dimension 4. But I don't know about dimensions 5 and 6. Is the result true or false in dimensions 5 and 6? And, if false, what are some classic counterexamples, and do stronger constraints -- say compactness or closedness -- happen to make it true?

• Well, the famous examples in dimension 4 are the exotic $\mathbb{R}^4$'s disovered by Friedman and Kirby. en.wikipedia.org/wiki/Exotic_R4
– HJRW
Apr 22, 2010 at 2:20
• Yes, thanks, realized that just after I wrote the post. Apr 22, 2010 at 2:38

It is false in dimension 5 and 6. Spheres happen to be standard, but some other (compact and closed) manifolds happen to admit different smooth (and PL) structures.

Simple example are tori. For example, $\mathbb T^5$ admits 3 different PL structures that give rise to 3 different differentiable structures. See, e.g., Hsiang, Shaneson "Fake tori" or Wall's book on surgery.

• Wikipedia tells me that the result actually does hold for spheres in dimensions 5 and 6, contrary to what you say: en.wikipedia.org/wiki/… That's the main reason I asked the question. Or perhaps by calling the spheres "standard" you didn't mean they were standard counterexamples, but only that they carry unique differential structures. Otherwise, thanks! Apr 22, 2010 at 4:37
• Yes, that's right, spheres in dimensions 5 and 6 are diffeomorphic to the standard $S^5$ ($S^6$). This is what I meant by calling them standard. I would like to point out that my and Igor's answer do not contradict each other. They complement each other very nicely. Apr 22, 2010 at 14:37
• While I'm at it, I might as well ask whether there's a simple compact or closed counterexample in dimension 4. As far as I know, this question for the 4-sphere is unresolved: I doubt it would be any simpler for any other closed 4-manifold, but perhaps I'm wrong? Apr 22, 2010 at 15:13
• My understanding is that it is rather large industry. There're exotic smooth structures on compact simply connected 4-manifolds. For example $\Bbb{CP}^2\#6\overline{\Bbb{CP}^2}$ (not sure about 6) admits exotic structure. People are constantly making progress making the example "smaller" in second homology. Some names here are Park, Stipsicz, Szabo, Akhmedov. Apr 22, 2010 at 15:34
• Andrey, 6 is correct; you could replace it by any $n\geq 2$ (cf. Fintushel-Stern's latest...). Moreover the number of different smooth structures is in each case countably infinite, while for simply connected manifolds of higher dimension it is always finite. Apr 22, 2010 at 15:52

Any PL-manifold of dimension $\le 7$ is smoothable, and the smooth structure is unique in dimensions $5,6$. See e.g. remark 6.7 in Rudyak's paper for details.

EDIT: To explain the above, the smooth structures on a PL manifold $M$ of dimension $\ge 5$ are in 1-1 correspondence with $[M, PL/O]$, homotopy classes of maps from $M$ to the space $PL/O$, which is $6$-connected. This implies the claim in the previous paragraph. Similarly, PL structures on a topological manifold $M$ of dimension $\ge 5$ are in 1-1 correspondnece with $[M,TOP/PL]$, and $TOP/PL$ is $K(\mathbb Z_2,3)$. Thus $[M,TOP/PL]$ is simply $H^3(M;\mathbb Z_2)$, the third cohomology group with $\mathbb Z_2$ coefficients, and if $H^3(M;\mathbb Z_2)$ is nonzero, then $M$ admits more than one PL structure. See Madsen-Milgram "Classifying spaces for surgery and cobordism of manifolds".

• But a given topological manifold can host different PL structures, right? So if there are several nonisomorphic PL structures on a manifold $M$, then there are as many nonisomorphic differentiable structures, right? I was just figuring out why your answer is not in contradiction with Gogolev's answer. Apr 22, 2010 at 4:34
• Yes, that's right. Apr 22, 2010 at 10:43