Loop homotopy of 6–manifolds over 4–manifolds

Abstract

Let M be the 6-manifold M arising as the total space of the sphere bundle of a rank 3 vector bundle over a simply connected closed 4-manifold. We show that, after looping, M is homotopy equivalent to a product of loops on spheres in general. This particularly implies a cohomological rigidity property of M after looping. Furthermore, passing to rational homotopy we show that such an M is Koszul.