Classifying 8-dimensional E-manifolds

A manifold M is called an E-manifold if it has homology only in even dimensions, ie. $H_{2k+1}(M;Z) = 0$ for all k. Examples include complex projective spaces and complete intersections. We consider 8-dimensional simply-connected E-manifolds. Those that have Betti numbers $b_2 = r$ and $ b_4 = 0$, and fixed second Stiefel-Whitney class $w_2 = w$ form a group $\Theta(r;w$), which acts on the set of E-manifolds with $b_2 = r$ and $w_2 = w$. The classification of E-manifolds based on this action consists of 3 steps: computing $\Theta(r;w)$, classifying the set of orbits and finding the stabilizers. I will present results for each of these steps, as well as the special case of 3-connected 8-manifolds, where a complete classification is obtained, based on Wall's earlier results.