Equivariant Cohomology in Algebraic Geometry Appendix A: Algebraic Topology

Introduction to Equivariant Cohomology in Algebraic Geometry

Introduced by Borel in the late 1950’s, equivariant cohomology encodes information about how the topology of a space interacts with a group action. Quite some time passed before algebraic geometers picked up on these ideas, but in the last twenty years, equivariant techniques have found many applications in enumerative geometry, Gromov-Witten theory, and the study of toric varieties and homogen...

The Index of Operators on Foliated Bundles

We compute the equivariant cohomology Chern character of the index of elliptic operators along the leaves of the foliation of a flat bundle. The proof is based on the study of certain algebras of pseudodifferential operators and uses techniques for analizing noncommutative algebras similar to those developed in Algebraic Topology, but in the framework of cyclic cohomology and noncommutative geo...

Equivariant Algebraic Topology

This paper develops the introductory theory of equivariant algebraic topology. We first define G-CW complexes and prove some basic homotopy-theoretic results Whitehead’s theorem, cellular and CW approximation, and the Freudenthal suspension theorem. We then define ordinary (Bredon) homology and cohomology theories and give an application to Smith theory. Our treatment of this material closely f...

An Algebraic Model for Rational S-equivariant Stable Homotopy Theory

Greenlees defined an abelian category A whose derived category is equivalent to the rational S1-equivariant stable homotopy category whose objects represent rational S1equivariant cohomology theories. We show that in fact the model category of differential graded objects in A models the whole rational S1-equivariant stable homotopy theory. That is, we show that there is a Quillen equivalence be...

Equivariant Cohomology in Algebraic Geometry Lecture One: Preview