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On Formality of Some Homogeneous Spaces

Faculty of Mathematics and Computer Science, University of Warmia and Mazury, Słoneczna 54, 10-710 Olsztyn, Poland
Symmetry 2019, 11(8), 1011; https://doi.org/10.3390/sym11081011
Received: 17 July 2019 / Revised: 31 July 2019 / Accepted: 2 August 2019 / Published: 5 August 2019
(This article belongs to the Special Issue Geometry of Submanifolds and Homogeneous Spaces)
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Abstract

Let G / H be a homogeneous space of a compact simple classical Lie group G. Assume that the maximal torus T H of H is conjugate to a torus T β whose Lie algebra t β is the kernel of the maximal root β of the root system of the complexified Lie algebra g c . We prove that such homogeneous space is formal. As an application, we give a short direct proof of the formality property of compact homogeneous 3-Sasakian spaces of classical type. This is a complement to the work of Fernández, Muñoz, and Sanchez which contains a full analysis of the formality property of S O ( 3 ) -bundles over the Wolf spaces and the proof of the formality property of homogeneous 3-Sasakian manifolds as a corollary.
Keywords: formality; 3-Sasakian manifold; homogeneous space formality; 3-Sasakian manifold; homogeneous space
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Tralle, A. On Formality of Some Homogeneous Spaces. Symmetry 2019, 11, 1011.

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