Equivariant Formality for Actions of Torus Groups

Abstract

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 S1-equivariant minimal models to give a number of examples of S1-spaces illustrating the properties of the various definitions.

On the equivariant formality of Kähler manifolds with torus group actions

“…These can be used to define formality as in [13], and this definition is equivalent to the one just discussed [14]. However, currently these models have only been defined for spaces whose fixed sets are connected as well as simply connected, and so are not well suited for studying Kähler manifolds.…”

“…In this paper, we extend this result and consider actions of torus groups T. Working with complex coefficients, we prove that simply connected compact Kähler manifolds with holomorphic T-actions are equivariantly formal. This result uses the definition of equivariant formality from [14], which is based on the algebraic models for T-spaces developed in [11,12].…”

The topology of toric origami manifolds

“…Historically, the term "formal" has been used in rational homotopy theory, and so equivariantly formal has multiple interpretations. Scull describes the relationships between these interpretations [25]. To avoid further confusion, we will not use this term in the remainder of this paper.…”