The Toral Rank Conjecture and variants of equivariant formality

TL;DR

This paper introduces two new variants of equivariant formality, explores their properties using advanced algebraic tools, and demonstrates their implications for the toral rank conjecture in various geometric contexts.

Contribution

It proposes MOD-formal actions and actions of formal core, linking equivariant formality with rational homotopy theory and proving the toral rank conjecture for these actions.

Findings

New definitions of equivariant formality are characterized using rational homotopy tools.

Actions with these properties satisfy the toral rank conjecture, extending previous results.

Applications include symplectic manifolds, rationally elliptic spaces, and non-negatively curved manifolds.

Abstract

An action of a compact Lie group is called equivariantly formal, if the Leray--Serre spectral sequence of its Borel fibration degenerates at the E_2-term. This term is as prominent as it is restrictive. In this article, also motivated by the lack of junction between the notion of equivariant formality and the concept of formality of spaces (surging from rational homotopy theory) we suggest two new variations of equivariant formality: "MOD-formal actions" and "actions of formal core". We investigate and characterize these new terms in many different ways involving various tools from rational homotopy theory, Hirsch--Brown models, $A_\infty$-algebras, etc., and, in particular, we provide different applications ranging from actions on symplectic manifolds and rationally elliptic spaces to manifolds of non-negative sectional curvature. A major motivation for the new definitions was that an almost free action of a torus $T^n\curvearrowright X$ possessing any of the two new properties satisfies the toral rank conjecture, i.e. $dim H^*(X;Q)\geq 2^n$. This generalizes and proves the toral rank conjecture for actions with formal orbit spaces.