Formality of 7-dimensional 3-Sasakian manifolds

TL;DR

This paper characterizes the formality of 7-dimensional simply connected 3-Sasakian manifolds, showing it depends on the second Betti number, and provides examples of formal Sasakian manifolds with higher Betti numbers.

Contribution

It establishes a precise criterion for the formality of 7-dimensional 3-Sasakian manifolds based on their second Betti number and offers new examples of formal Sasakian manifolds.

Findings

3-Sasakian manifolds are formal iff b2<2

Existence of formal Sasaki-Einstein manifolds with b2≥2

Examples of formal Sasakian manifolds with b2≥2

Abstract

We prove that any simply connected compact 3-Sasakian manifold, of dimension seven, is formal if and only if its second Betti number is $b_2<2$. In the opposite, we show an example of a 7-dimensional Sasaki-Einstein manifold, with second Betti number $b_2\geq 2$, which is formal. Therefore, such an example does not admit any 3-Sasakian structure. Examples of 7-dimensional simply connected compact formal Sasakian manifolds, with $b_2\geq 2$, are also given.