We show that if $M$ and $N$ have the same homotopy type of simply connectedclosed 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. Weshow 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 groupof $M^{2n}$ 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 showthat 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.
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机译：我们证明，如果$ M $和$ N $具有相同的同伦类型，即简单连接的闭合光滑$ m $流形，使得$ M $的整数和mod- $ 2 $同调以奇数度消失，则它们的同伦惯性群为等于。令$ M ^ {2n} $为闭合的（n-1）$连通的$ 2n $维光滑流形。我们显示，对于$ n = 4 $，$ M ^ {2n} $的同伦惯性群是微不足道的，如果$ n = 8 $和$ H ^ n（M ^ {2n}; \ mathbb {Z}）\ cong \ mathbb {Z} $，$ M ^ {2n} $的同伦惯性群也是微不足道的。我们进一步针对$ n = 8 $计算$ M ^ {2n} $平滑度的一致性类的组\\ mathcal {C}（M ^ {2n}）$。最后，我们证明，如果光滑流形$ N $在切向同构中等于$ M ^ 8 $，则$ N $对与之相连的和$ M ^ 8 $和一个同构性$ 8 $球微分。
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