We show how the symmetry of attractors of equivariant dynamical systems can be observed by equivariant projections of the phase space. Equivariant projections have long been used, but they can give misleading results if used improperly and have been considered untrustworthy. We find conditions under which an equivariant projection generically shows the correct symmetry of the attractor.

Let G be a countable discrete group and let M be a smooth proper cocompact G-manifold without boundary. The Euler operator defines via Kasparov theory an element, called the equivariant Euler class, in the equivariant KO -homology of M. The universal equivariant Euler characteristic of M, which lives in a group UG(M), counts the equivariant cells of M , taking the component structure of the var...

We suggest Novikov type inequalities in the situation of a compact Lie groups action assuming that the given closed 1-form is invariant and basic. Our inequalities use equivariant cohomology and an appropriate equivariant generalization of the Novikov numbers. We test and apply our inequalities in the case of a finite group. As an application we obtain Novikov type inequalities for a manifold w...

In this paper, we compute the first two equivariant heat kernel coefficients of the Bochner Laplacian on differential forms. The first two equivariant heat kernel coefficients of the Bochner Laplacian with torsion are also given. We also study the equivariant heat kernel coefficients of nonmininmal operators on differential forms and get the equivariant Gilkey-Branson-Fulling formula. MSC: 58J0...

For a complete toric variety, we obtain an explicit formula for the localized equivariant Todd class in terms of the combinatorial data – the fan. This is based on the equivariant Riemann-Roch theorem and the computation of the equivariant cohomology and equivariant homology of toric varieties. ∗This research was supported in part by NSF grant DMS-9504522 and DMS-9803593

Let G be a countable discrete group and let M be a smooth proper cocompact G-manifold without boundary. The Euler operator defines via Kasparov theory an element, called the equivariant Euler class, in the equivariant KO-homology of M. The universal equivariant Euler characteristic of M, which lives in a group UG(M), counts the equivariant cells of M , taking the component structure of the vari...

(1.1) This paper concerns three aspects of the action of a compact group K on a space X . The ®rst is concrete and the others are rather abstract. (1) Equivariantly formal spaces. These have the property that their cohomology may be computed from the structure of the zero and one dimensional orbits of the action of a maximal torus in K. (2) Koszul duality. This enables one to translate facts ab...

We construct for an equivariant cohomology theory for proper equivariant CW -complexes an equivariant Chern character, provided that certain conditions about the coefficients are satisfied. These conditions are fulfilled if the coefficients of the equivariant cohomology theory possess a Mackey structure. Such a structure is present in many interesting examples.

We outline the theory of equivariant transversality for maps equivariant with respect to a compact Lie group. We indicate some applications to generic equivariant bifurcation theory.

Suppose one is given a discrete group G, a cocompact proper Gmanifold M , and a G-self-map f : M → M . Then we introduce the equivariant Lefschetz class of f , which is globally defined in terms of cellular chain complexes, and the local equivariant Lefschetz class of f , which is locally defined in terms of fixed point data. We prove the equivariant Lefschetz fixed point theorem, which says th...