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The following paper may be helpful:

Wall, C. T. C. Classification of $(n−1)$-connected $2n$-manifolds. Ann. of Math. (2) 75 1962 163–189.

Let me quote one result from this paper:

Theorem 5. If $\pi_{n-1}(SO)=0$ ,$n\geq 3$,and $M_1$ and $M_2$ are differential $(n-1)$ connected $2n$-manifolds of the same homotopy type,then for some manifold $T$ homeomorphic (and so combinatorially equivalent) to $S^{2n}$, $M_1$ is diffeomorphic to $M_2\sharp T$. If $n=3,6$, $M_1$ is diffeomorphic to $M_2$.

This tells you that there is no differentiable fake $S^n\times S^n$ for $n\equiv 6\mod8$

The following paper may be helpful:

Wall, C. T. C. Classification of $(n−1)$-connected $2n$-manifolds. Ann. of Math. (2) 75 1962 163–189.

Let me quote one result from this paper:

Theorem 5. If $\pi_{n-1}(SO)=0$ ,$n\geq 3$,and $M_1$ and $M_2$ are differential $(n-1)$ connected $2n$-manifolds of the same homotopy type,then for some manifold $T$ homeomorphic (and so combinatorially equivalent) to $S^{2n}$, $M_1$ is diffeomorphic to $M_2\sharp T$. If $n=3,6$, $M_1$ is diffeomorphic to $M_2$.

This tells you that there is no smoothable fake $S^n\times S^n$ for $n\equiv 6\mod8$

I think a relevant reference which is worthwhile to have a look is:

Wall, C. T. C. Classification of $(n−1)$-connected $2n$-manifolds. Ann. of Math. (2) 75 1962 163–189.

Let me quote one result from this paper:

Theorem 5. If $\pi_{n-1}(SO)=0$ ,$n\geq 3$,and $M_1$ and $M_2$ are differential $(n-1)$ connected $2n$-manifolds of the same homotopy type,then for some manifold $T$ homeomorphic (and so combinatorially equivalent) to $S^{2n}$, $M_1$ is diffeomorphic to $M_2\sharp T$. If $n=3,6$, $M_1$ is diffeomorphic to $M_2$.

This tells you that there is no differentiable fake $S^n\times S^n$ for $n\equiv 6\mod8$

The following paper may be helpful:

Wall, C. T. C. Classification of $(n−1)$-connected $2n$-manifolds. Ann. of Math. (2) 75 1962 163–189.

Let me quote one result from this paper:

Theorem 5. If $\pi_{n-1}(SO)=0$ ,$n\geq 3$,and $M_1$ and $M_2$ are differential $(n-1)$ connected $2n$-manifolds of the same homotopy type,then for some manifold $T$ homeomorphic (and so combinatorially equivalent) to $S^{2n}$, $M_1$ is diffeomorphic to $M_2\sharp T$. If $n=3,6$, $M_1$ is diffeomorphic to $M_2$.

This tells you that there is no differentiable fake $S^n\times S^n$ for $n\equiv 6\mod8$

I think a relevant reference which is worthwhile to have a look is:

Wall, C. T. C. Classification of (n−1)-connected 2n-manifolds. Ann. of Math. (2) 75 1962 163–189.

Let me quote one result from this paper:

Theorem 5. If $\pi_{n-1}(SO)=0$ ,$n\geq 3$,and $M_1$ and $M_2$ are differential $(n-1)$ connected $2n$-manifolds of the same homotopy type,then for some manifold $T$ homeomorphic (and so combinatorially equivalent) to $S^{2n}$, $M_1$ is diffeomorphic to $M_2\sharp T$. If $n=3,6$, $M_1$ is diffeomorphic to $M_2$.

This tells you that there is no differentiable fake $S^n\times S^n$ for $n\equiv 6\mod8$

I think a relevant reference which is worthwhile to have a look is:

Wall, C. T. C. Classification of $(n−1)$-connected $2n$-manifolds. Ann. of Math. (2) 75 1962 163–189.

Let me quote one result from this paper:

Theorem 5. If $\pi_{n-1}(SO)=0$ ,$n\geq 3$,and $M_1$ and $M_2$ are differential $(n-1)$ connected $2n$-manifolds of the same homotopy type,then for some manifold $T$ homeomorphic (and so combinatorially equivalent) to $S^{2n}$, $M_1$ is diffeomorphic to $M_2\sharp T$. If $n=3,6$, $M_1$ is diffeomorphic to $M_2$.

This tells you that there is no differentiable fake $S^n\times S^n$ for $n\equiv 6\mod8$