See http://mathoverflow.net/questions/79991/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