arXiv subjects

Andre Henriques

Publications and source records attributed to Andre Henriques.

21 records · Page 2Linked to original sources

arXiv2004

T-equivariant cohomology of cell complexes and the case of infinite Grassmannians

In 1998, Goresky, Kottwitz, and MacPherson showed that for certain spaces X equipped with a torus action, the T-equivariant cohomology ring of X can be described by combinatorial data obtained from its orbit decomposition. Thus, their theory transforms calculations of the equivariant topology of X to those of the combinatorics of the orbit decomposition. Since then, many authors have studied this interplay between topology and combinatorics. In this paper, we generalize the theorem of Goresky, Kottwitz, and MacPherson to the (possibly infinite-dimensional) setting where X is any equivariant cell complex with only even-dimensional cells and isolated T-fixed points, along with some additional technical hypotheses on the gluing maps. This generalization includes many new examples which have not yet been studied by GKM theory, including homogeneous spaces of a loop group LG.

math.DG↗

arXiv2003

Presentations of Noneffective Orbifolds

It is well-known that an effective orbifold M (one for which the local stabilizer groups act effectively) can be presented as a quotient of a smooth manifold P by a locally free action of a compact lie group K. We use the language of groupoids to provide a partial answer to the question of whether a noneffective orbifold can be so presented. We also note some connections to stacks and gerbes.

math.AT↗

arXiv2001

Orbispaces and Orbifolds from the Point of View of the Borel Construction, a new Definition

``An orbifold is a space which is locally modeled on the quotient of a vector space by a finite group.'' This sentence is so easily said or written that more than one person has missed some of the subtleties hidden by orbifolds. Orbifolds were first introduced by Satake under the name ``V-manifold'' and rediscovered by Thurston who called them ``orbifolds''. Both of them used only faithful actions to define their orbifolds (these are the so called reduced orbifolds). This intuitive restriction is very unnatural mathematically if we want for example to study suborbifolds. Orbispaces are to topological spaces what orbifolds are to manifolds. They have been defined by Haefliger via topological groupoids. He defined the homotopy and (co)homology groups of an orbispace to be those of the classifying space of the topological groupoid. However, his definition of morphisms is complicated and technical. Ruan and Chen have tried to reformulate the definitions, but they don't seem treat the non-reduced case in a satisfactory way. Our idea is to define orbispaces and orbifolds directly via their classifying spaces. This approach allows us to easily define morphisms of orbispaces and orbifolds with all the desired good properties.

math.GT↗