Homotopy Inertia Groups and Tangential Structures

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 $M^{2n}$ 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 $H^n(M^{2n};\mathbb{Z})\cong \mathbb{Z}$, the homotopy inertia group of $M^{2n}$ is also trivial. We further compute the group $\mathcal{C}(M^{2n})$ 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.