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If a topological group $G$ is also a topological manifold, it is well-known (Hilbert's 5th Probelm) that there is a unique analytic structure making it a Lie group.

However, for a compact Lie group $G$, do we know if the underlying topological manifold supports any other exotic smooth structures (necessarily not a Lie group)?

Even a more specific example: Up to diffeomorphism, we have $SO(8)=SO(7)\times S^7$. If we replace the smooth structure on $S^7$ by an exotic one, do we get an exotic smooth structure on $SO(8)$?

Thank you!

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    $\begingroup$ There are uncountably many different smooth structures on the Lie group $\mathbb{R}^4$. $\endgroup$
    – Liviu Nicolaescu
    Commented Dec 11, 2015 at 23:24
  • $\begingroup$ @LiviuNicolaescu, you are right, let me add the compactness $\endgroup$
    – Piojo
    Commented Dec 11, 2015 at 23:25
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    $\begingroup$ @QiaochuYuan Think of $S^7$ as unit octonions and $SO(7)$ as rotations preserving the unit octonion, then every element in $SO(8)$ can be uniquely represented as a left multiplication by a unit octonion composed with an element in $SO(7)$. This decomposition trivializes the bundle. $\endgroup$
    – Piojo
    Commented Dec 12, 2015 at 3:42
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    $\begingroup$ Huh. Well, I learned something today. $\endgroup$
    – Qiaochu Yuan
    Commented Dec 12, 2015 at 5:26
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    $\begingroup$ As Lie group, $SO(4)$ is even a semidirect $SU(2)\rtimes SO(3)$, where the action is induced by conjugation (since $SO(3)=PSU(2)$). $\endgroup$
    – YCor
    Commented Dec 12, 2015 at 8:48

2 Answers 2

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There are two smooth structures on a five-torus, aren't there?

S^7 is the units in the octonions. Maybe left multiplication furnishes a section of the principal SO(7)-bundle SO(8)-->S^7.

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  • $\begingroup$ Thanks! Can you give a reference on smooth structures on five-torus? Do you any thought on the example I mentioned? $\endgroup$
    – Piojo
    Commented Dec 12, 2015 at 4:21
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    $\begingroup$ $T^5$ has two PL structures, hence it has two smooth structures. See (Example 3.10 of indiana.edu/~jfdavis/teaching/m623/dp.pdf) $\endgroup$
    – Zhiyun Cheng
    Commented Dec 12, 2015 at 6:39
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According to the introduction to the following paper of Farrell and Jones, if $n>4$ and $\Sigma^n$ is any exotic homotopy sphere, then $T^n\#\Sigma^n$ is not diffeomorphic to $T^n$. So lots of higher-dimensional tori have exotic structures. Farrell and Jones cite Section 15A of Wall's book on surgery for this result, but I couldn't pull it easily out of there. In any case, here's the paper reference:

Farrell, F. T.; Jones, L. E. Examples of expanding endomorphisms on exotic tori. Invent. Math. 45 (1978), no. 2, 175–179.

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