Equivariant formality of the isotropy action on $\mathbb{Z}_2\oplus \mathbb{Z}_2$-symmetric spaces

TL;DR

This paper proves that $( ext{Z}_2 imes ext{Z}_2)$-symmetric spaces are equivariantly formal and formal in Sullivan's sense, extending known results from symmetric spaces and providing a new proof approach.

Contribution

It establishes equivariant formality for a new class of symmetric spaces and offers an alternative proof method for symmetric spaces.

Findings

$( ext{Z}_2 imes ext{Z}_2)$-symmetric spaces are equivariantly formal.

These spaces are also formal in Sullivan's sense.

Provides a new proof technique for symmetric spaces.

Abstract

Compact symmetric spaces are probably one of the most prominent class of formal spaces, i.e. of spaces where the rational homotopy type is a formal consequence of the rational cohomology algebra. As a generalisation, it is even known that their isotropy action is equivariantly formal. In this article we show that $(\mathbb{Z}_2\oplus \mathbb{Z}_2)$-symmetric spaces are equivariantly formal and formal in the sense of Sullivan, in particular. Moreover, we give a short alternative proof of equivariant formality in the case of symmetric spaces with our new approach.