Equivariant map

About: Equivariant map is a research topic. Over the lifetime, 9205 publications have been published within this topic receiving 137115 citations.

The relation between the Baum-Connes Conjecture and the Trace Conjecture

Abstract: We prove a version of the L2-index Theorem of Atiyah, which uses the universal center-valued trace instead of the standard trace. We construct for G-equivariant K-homology an equivariant Chern character, which is an isomorphism and lives over the ring ℤ⊂λ G ⊂ℚ obtained from the integers by inverting the orders of all finite subgroups of G. We use these two results to show that the Baum-Connes Conjecture implies the modified Trace Conjecture, which says that the image of the standard trace K0(C* r (G))→ℝ takes values in λ G . The original Trace Conjecture predicted that its image lies in the additive subgroup of ℝ generated by the inverses of all the orders of the finite subgroups of G, and has been disproved by Roy [15].

Equivariant de Rham theory and graphs

Abstract: Goresky, Kottwitz and MacPherson have recently shown that the computation of the equivariant cohomology ring of a G-manifold can be reduced to a computation in graph theory. This opens up the possibility that many of the fundamental theorems in equivariant de Rham theory may, on closer inspection, turn out simply to be theorems about graphs. In this paper we show that for some familiar theorems, this is indeed the case.

Equivariant Seiberg-Witten Floer Homology

Abstract: In this paper we construct, for all compact oriented three- manifolds, a U(1)-equivariant version of Seiberg-Witten Floer homology, which is invariant under the choice of metric and perturbation. We give a detailed analysis of the boundary structure of the monopole moduli spaces, compactified to smooth manifolds with corners. The proof of the independence of metric and perturbation is then obtained via an analysis of all the relevant obstruction bundles and sections, and the corresponding gluing theorems. The paper also contains a discussion of the chamber structure for the Seiberg-Witten invariant for rational homology 3-spheres, and proofs of the wall crossing formula, obtained by studying the exact sequences relating the equivariant and the non-equivariant Floer homologies and by a local model at the reducible monopole.

Isomorphisms between left and right adjoints

Abstract: There are many contexts in algebraic geometry, algebraic topology, and homological algebra where one encounters a functor that has both a left and right ad- joint, with the right adjoint being isomorphic to a shift of the left adjoint specified by an appropriate "dualizing object". Typically the left adjoint is well understood while the right adjoint is more mysterious, and the result identifies the right adjoint in fa- miliar terms. We give a categorical discussion of such results. One essential point is to differentiate between the classical framework that arises in algebraic geometry and a deceptively similar, but genuinely different, framework that arises in algebraic topol- ogy. Another is to make clear which parts of the proofs of such results are formal. The analysis significantly simplifies the proofs of particular cases, as we illustrate in a sequel discussing applications to equivariant stable homotopy theory.