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

TL;DR

This paper investigates the homotopy properties of certain 6-manifolds constructed over 4-manifolds, revealing their loop space structure and cohomology rigidity, and establishing their Koszul property in rational homotopy theory.

Contribution

It demonstrates that looping these 6-manifolds simplifies their homotopy type to a product of spheres and proves their cohomology rigidity and Koszul property.

Findings

Looped 6-manifolds are homotopy equivalent to products of loops on spheres.

Looping induces cohomology rigidity in these manifolds.

Such manifolds are Koszul in rational homotopy theory.

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.