I am looking for some explanation how

  • $\mathbb{HP}^2$,

  • exotic 7-spheres, and

  • Bott manifolds

are related? And how the construction of a Bott manifold is related to $\mathbb{HP}^2$ and exotic 7-spheres?

p.s. The description I found is Kervaire-Milnors work on homotopy spheres and the description by Johannes Ebert. But this description is too dense and too quick to me, could you provide more details in a slow manner?

Johannes Ebert said: "A textbook reference is Kosinski: ''Differential manifolds''. In section IX.8 (Theorem 8.7), you find the statement that there exists an 8-dimensional manifold which is almost parallelizable (i.e. parallelizable away from a point) whose signature is $8 \cdot 28$. Because this $M$ is almost parallelizable, $p_1 (TM)=0$, and from the formulae for the $\hat A$-class and the $L$-class, you get that $\hat{A} =1$. Typically, one wants that the signature is zero, and this you can achieve by connected sum with $8 \cdot 28$ copies of $\overline{HP^2}$. How is $M$ constructed? You take the $E_8$-plumbing manifold $V$. It is a $3$-connected $8$-manifold which is parallelizable, which has signature $8$ and whose boundary is a homotopy sphere. In fact, $\partial V$ generates the group $\Theta_7 \cong Z/28$ of exotic $7$-spheres. Now you form the boundary connected sum of $28$ copies of $V$; the boundary of the resulting manifold is the standard $S^7$, and you glue in a copy of $D^8$ to obtain $M$."

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