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.
Categories: math.GT