This paper contains a comparison of several definitions of equivariant formality for actions of torus groups. We develop and prove some relations between the definitions. Focusing on the case of the circle group, we use $S^1$-equivariant minimal models to give a number of examples of $S^1$-spaces illustrating the properties of the various definitions.

Keywords:

Equivariant homotopy, circle action, minimal model, rationalization, formality Equivariant homotopy, circle action, minimal model, rationalization, formality

MSC Classifications:

55P91, 55P62, 55R35, 55S45 show english descriptions Equivariant homotopy theory [See also 19L47]
Rational homotopy theory
Classifying spaces of groups and $H$-spaces
Postnikov systems, $k$-invariants 55P91 - Equivariant homotopy theory [See also 19L47]
55P62 - Rational homotopy theory
55R35 - Classifying spaces of groups and $H$-spaces
55S45 - Postnikov systems, $k$-invariants

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