Torus equivariant algebraic models and compact realization

TL;DR

This paper characterizes the algebraic models of torus-equivariant spaces using differential graded algebras, providing criteria for realization as rational equivariant cohomology and classifying certain equivariant cohomology algebras.

Contribution

It introduces a complete algebraic description of $T$-spaces via differential graded algebras and characterizes when these models correspond to finite $T$-CW-complexes.

Findings

Algebraic models correspond to finite $T$-CW-complexes satisfying Borel localization.

GKM graph cohomology can be realized by finite $T$-CW-complexes.

Classifies equivariant cohomology algebras of finite $S^1$-CW-complexes with discrete fixed points.

Abstract

Let $T$ be a compact torus. We prove that, up to equivariant rational equivalence, the category of $T$-simply connected, $T$-finite type $T$-spaces with finitely many isotropy types is completely described by certain finite systems of commutative differential graded algebras with consistent choices of degree $2$ cohomology classes. We show that the algebraic systems corresponding to finite $T$-CW-complexes are exactly those which satisfy the necessary condition imposed by the Borel localization theorem along with certain finiteness conditions. We derive an algebraic characterization of when an algebra over a polyonmial ring is realized as the rational equivariant cohomology of a finite $T$-CW-complex. As further applications we prove that any GKM graph cohomology is realized by a finite $T$-CW-complex and classify equivariant cohomology algebras of finite $S^1$-CW-complexes with discrete fixed points.