Inertia group I (Ex)
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Let be a closed smooth oriented -manifold. The inertia group of is defined to be the following subgroup of :
where denotes orientation preserving isomorphism.
Recall that by the -cobordism theorem every exotic sphere in dimension and higher is a twisted double
for some orientation preserving diffeomorphism . (This is also true in dimension since there are no exotic -spheres). Set
Tex syntax errorand identify . Show that if and only if there is an orientation preserving diffeomorphism with .
Hint: You may assume a theorem of Cerf which states that all orientation preserving embeddings of the -disc into an oriented -manifold are ambient isotopic.