Algebraic models for equivariant homotopy theory over Abelian compact Lie groups

Abstract

We describe some algebraic models for equivariant rational and p-adic homotopy theory over Abelian compact Lie groups.

On the equivariant formality of Kähler manifolds with torus group actions

“…In defining equivariant algebraic models for the rational homotopy of spaces with actions of torus groups, we use the diagram category defined in [11]. Because we are dealing with actions of connected torus groups, we do not need the full generality of the diagrams defined in [11,Definition 1.2], but can instead use the simplified diagrams discussed in [11,Sect.…”

“…Because we are dealing with actions of connected torus groups, we do not need the full generality of the diagrams defined in [11,Definition 1.2], but can instead use the simplified diagrams discussed in [11,Sect. 4].…”

“…This Cartan diagram can be used in place of the equivariant model Ω(M) of [11], as shown by the following.…”

“…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].…”

“…In addition, compactness assures that all T-spaces have finitely many orbit types and that the rational cohomology of each fixed point subspace X H = {x ∈ X | hx = x for all h ∈ H} is of finite type for all closed subgroups H ⊆ T. We will also assume that all T-spaces are based, with basepoint fixed by the T-action, and T-simply connected in the sense that the connected components of the fixed point subspaces X H are all simply connected. These will be standing assumptions in what follows; note that we are not assuming that the fixed point sets are also connected, as needed for the minimal models of [12], but instead are allowing disconnected fixed sets as in [11], as is necessary for the study of Kähler manifolds. Section 2 gives a quick sketch of the background material on equivariant formality.…”