HOMOTOPY INERTIA GROUPS AND TANGENTIAL STRUCTURES

摘要

We show that if M and N have the same homotopy type of simply connected closed smooth m-manifolds such that the integral and mod-2 cohomologies of M~(2n) vanish in odd degrees, then their homotopy inertia groups are equal. Let M~(2n) be a closed (n - 1) -connected 2n-dimensional smooth manifold. We show that, for n = 4， the homotopy inertia group of M~(2n) is trivial and if n = 8 and H~n(M~(2n); Z) = Z, then the homotopy inertia group of M~(2n) is also trivial. We further compute the group C(M~(2n)) of concordance classes of smoothings of M~(2n) for n = 8. Finally, we show that if a smooth. manifold N is tangentially homotopy equivalent to M~8 , then N is diffeomorphic to the connected sum of M~8 and a homotopy 8-sphere.