Loop homotopy of $6$-manifolds over $4$-manifolds

Abstract

Let $M$ be the $6$-manifold $M$ 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 the cohomology rigidity property of $M$ after looping. Furthermore, passing to the rational homotopy, we show that such $M$ is Koszul in the sense of Berglund.

16 pages; correct the proof of Theorem 1.1

References in corpus (2)

• Loop space decompositions of $(2n-2)$-connected $(4n-1)$-dimensional Poincaré Duality complexes
• Suspension homotopy of $6$-manifolds