Torus equivariant algebraic models and compact realization

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.