See Smooth manifold with non-trivial inertia group? (wrt homotopy spheres) for the definition of $\Theta_n$ and inertia subgroups.

I'm wondering what can be said about Lie groups. If $M^n$ is an n-dimensional manifold with Lie group structure and $\Sigma^n$ is a homotopy n-sphere, is there a Lie group structure on $M\#\Sigma$ that is in some sense compatible with the original structure? If this new group structure is isomorphic to the old structure, this implies Lie group isomorphism, correct?

What I would like to see is that there is a canonically induced Lie group structure on $M\#\Sigma$ and that this structure is isomorphic to that of $M$, and hence the inertia group for Lie groups is the full $\Theta^n$


There is no natural Lie group structure on a connected sum of a Lie group and an exotic sphere. Where would it possibly come from?

Also, the implication you are trying to derive is wrong. Note that $S^3$ is a Lie group and hence so is a product of several $S^3$s. I'm not an expert on surgery theory but it's well-known that the inertia group of a product of spheres is trivial which means that the opposite of the statement you are after holds: the connected sum of a product of spheres with an exotic sphere is never orientably diffeomorphic to the original manifold. So for example the inertia group of $S^3\times S^3\times S^3$ is trivial while $\Theta_9$ has order 8.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.