Construction of a Bott manifold I have been searching the literature for a construction of a simply connected spin manifold of dimension 8 with A-genus 1. I am not sure, but I think this is called a Bott manifold.
Can anybody help me with a reference? Are there in the literature index computations for this type of manifold? Thank you for your help! 
 A: One construction is somehow hidden in Kervaire-Milnors work on homotopy spheres. 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 achive 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 $M$.
