On fibrations with formal elliptic fibers

TL;DR

This paper establishes a criterion linking the formality of the base and total space in fibrations with formal elliptic fibers, with applications to quaternion Kähler manifolds and twistor fibrations.

Contribution

It proves that in certain fibrations with elliptic fibers, the formality of the base and total space are equivalent, and the fibration map is formal, extending to geometric cases like quaternion Kähler manifolds.

Findings

Base is formal iff total space is formal in the specified fibrations.

Fibration map is formal under the given conditions.

Positive quaternion Kähler manifolds are formal and their twistor fibrations are formal.

Abstract

We prove that for a fibration of simply-connected spaces of finite type $F\hookrightarrow E\to B$ with $F$ being positively elliptic and $H^*(F,\qq)$ not possessing non-trivial derivations of negative degree, the base $B$ is formal if and only if the total space $E$ is formal. Moreover, in this case the fibration map is a formal map. As a geometric application we show that positive quaternion K\"ahler manifolds are formal and so are their associated twistor fibration maps.