Free Torus Action Research Articles

Minimal Models of Some Differential Graded Modules

Minimal models of chain complexes associated with free torus actions on spaces have been extensively studied in the literature. In this paper, we discuss these constructions using the language of operads. The main goal of this paper is to define a new Koszul operad that has projections onto several of the operads used in these minimal model constructions.

Finiteness and realization theorems for Alexandrov spaces with bounded curvature

Every closed Alexandrov space with a lower and upper curvature bound (in the triangle comparison sense) is a space of bounded curvature (in the sense of Berestovskii and Nikolaev). These spaces are topological manifolds and admit a canonical smooth structure such that, in its charts, the metric is induced by a Riemannian metric tensor with low regularity. In this note, we show that there are finitely many diffeomorphism types of closed, simply connected n-dimensional Alexandrov spaces with bounded curvature and upper diameter bound, provided $$2\le n\le 6$$. We also show that an analogous diffeomorphism finiteness result for such spaces does actually hold for general dimension n if, in addition, the second homotopy group is finite. Moreover, we prove that all closed, simply connected n-dimensional Alexandrov spaces with bounded curvature and upper diameter bound can be realized as quotients of finitely many closed, simply connected smooth manifolds with finite second homotopy group by some free torus action. These results extend well-known finiteness and realization theorems in Riemannian geometry to the more general setting of Alexandrov geometry. Indeed, our results may be simply viewed as concrete instances of the more general phenomenon that virtually any result that holds for a given class of closed Riemannian manifolds with bounded curvature and diameter may be extended to the Lipschitz closure of this class.

Multiplicative Aspects of the Halperin–Carlsson Conjecture

We use the multiplicative structure of the Koszul resolution to give short and simple proofs of some known estimates for the total dimension of the cohomology of spaces which admit free torus actions and some analogous results for filtered differential modules over polynomial rings. We also point out the possibility of improving these results in the presence of a multiplicative structure on the so-called minimal Hirsch–Brown model for the equivariant cohomology of the space.

Spaces of Polytopes and Cobordism of Quasitoric Manifolds

Our aim is to bring the theory of analogous polytopes to bear on the study of quasitoric manifolds, in the context of stably complex manifolds with compatible torus action. By way of application, we give an explicit construction of a quasitoric representative for every complex cobordism class as the quotient of a free torus action on a real quadratic complete intersection. We suggest a systematic description for omnioriented quasitoric manifolds in terms of combinatorial data, and explain the relationship with non-singular projective toric varieties (otherwise known as toric manifolds). By expressing the first and third authors' approach to the representability of cobordism classes in these terms, we simplify and correct two of their original proofs concerning quotient polytopes; the first relates to framed embeddings in the positive cone, and the second involves modifying the operation of connected sum to take account of orientations. Analogous polytopes provide an informative setting for several of the details.

Free torus actions and two-stage spaces

We prove the toral rank conjecture of Halperin in some new cases. Our results apply to certain elliptic spaces that have a two-stage Sullivan minimal model, and are obtained by combining new lower bounds for the dimension of the cohomology and new upper bounds for the toral rank. The paper concludes with examples and suggestions for future work.

On the cohomology of a polarization which contains the generators of a free torus action

In [Trans. Am. Math. Soc. 230 (1977) 235], Rawnsley studies a strongly integrable subtangent bundle F on a smooth manifold M which contains the generators of a free circle action on M and a line bundle L→ M with a flat F-connection ∇. It is proved that the cohomology groups of the sheaf S F of sections of L which are covariantly constant along F can be injectively mapped in the cohomology groups of the restriction of S F on the Bohr–Sommerfeld set Y of the action on M. In the present paper, we discuss similar results for free torus actions as well as some consequences in the context of the geometric quantization of a symplectic manifold with a Hamiltonian action of a torus T k .