# Reference for a proof of a Theorem by Joseph Wolf

We know that Lie Groups are parallelizable, I was looking for a version of the converse and came across this: https://books.google.com/books?id=w4bhBwAAQBAJ&pg=PA115 in Introduction to Smooth Manifolds by John M. Lee (1st edition, Springer, 2003).

As says in the link : a remarkable theorem by Joseph Wolf says that the only compact,simply connected manifolds that are parallelizable are products of Lie groups and copies of $$S^7$$

But, I am not being able to find this proof, If anyone can give a proof or a reference that would be much appreciated. Thanks in advance.

$$\textbf{Update}$$: As discussed in the comments the following result in Lee's book isn't exactly correct. Anyway these are the Wolf's papers mentioned in the link:

[Wol71] Joseph A. Wolf. On the geometry and classification of absolute parallelisms I. J. Differential Geometry, 6:317-342, 1971. projecteuclid, author's website;

[Wol72] Joseph A. Wolf. On the geometry and classification of absolute parallelisms II. J. Differential Geometry, 7:19-44, 1972. projecteuclid, author's website

The original question about conditions which put a multiplicative or a group structure on a parallelizable manifold still remains unanswered. So any help would be appreciated.

• I don't get it... the source you link to specifically references [Wol71] and [Wol72]. Check there. – Marty Jan 4 at 17:06
• In the case you do not have access to the part of the book with the bibliography (for example, if you only have access through Google Books), I have posted below a full quote and also references [Wol71] and [Wol72] mentioned in the above comment. (Of course, it's possible that other users will provide some additional references.) – Martin Sleziak Jan 4 at 17:13
• Actually, this is a misreading of Wolf's actual result, which was about absolute parallelisms consistent with a given metric, not general parallelizable manifolds. The claim as made above about them being products of Lie groups and copies of $S^7$ is certainly not correct. For example, the exotic 7-spheres of Milnor are not products, and, yet, they are parallelizable. I think this may be why Lee removed this claim in the second edition. – Robert Bryant Jan 4 at 17:54
• I see... thanks for the comment – PSG Jan 4 at 17:56
• @PSG: Yes, there are such conditions, but they are not trivial. Wolf's results are, as I wrote, about 'abolute parallelisms consistent with a (pseudo-Riemannian) metric'. Wolf's work on this extended (and slightly corrected) the original work of Cartan and Schouten that proved, in the Riemannian case, that compact simply connected examples of such were products of Lie groups and $S^7$. The key word here is 'consistent', for the definition of which, see Wolf's cited papers. – Robert Bryant Jan 4 at 18:06