Homotopy inertia groups and tangential structures
JP Journal of Geometry and Topology
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 vanish in odd degrees, then their homotopy inertia groups are equal. Let M2n be a closed (n − 1) -connected 2n-dimensional smooth manifold. We show that, for n = 4, the homotopy inertia group of M2n is trivial and if n = 8 and Hn( M2n; Z) ~= Z, then the homotopy inertia group of M2n is also trivial. We further compute the group C (M2n) of concordance classes of smoothings of M2n for n = 8. Finally, we show that if a smooth manifold N is tangentially homotopy equivalent to M8, then N is diffeomorphic to the connected sum of M8 and a homotopy 8-sphere.
Kasilingam, Ramesh, "Homotopy inertia groups and tangential structures" (2017). Journal Articles. 2774.