Equivariant maps between cohomology spheres (original) (raw)
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Equivariant maps between representation spheres
Simon Stevin, 2017
Let G be a compact Lie group. We prove that if V and W are orthogonal G-representations such that V G = W G = {0}, then a G-equivariant map S(V) → S(W) exists provided that dim V H ≤ dim W H for any closed subgroup H ⊆ G. This result is complemented by a reinterpretation in terms of divisibility of certain Euler classes when G is a torus.
SpringerBriefs in Mathematics, 2019
Equivariant Cohomology 9.1 Introduction Equivariant cohomology was designed to allow the study of spaces which are the quotient of a manifold M by the action of a compact group G. This can be accomplished by studying the fixed point sets of subgroups of G, notably the maximal torus T. The Cartan model replaces the study of infinite-dimensional manifolds by families of differential forms on finite-dimensional G-manifolds parametrized by an element X in the Lie algebra of G with polynomial dependence on X. A version of de Rham cohomology can be developed for the Cartan model. The localization theorem of Atiyah-Bott and Berline-Vergne describes the evaluation of such an equivariantly closed differential form on the fundamental class of the manifold. In this chapter, we first define homotopy quotients in Sect. 9.2. We introduce the Cartan model in Sect. 9.3. We treat characteristic classes of bundles over classifying spaces in Sect. 9.4. We consider these characteristic classes in terms of the Cartan model in Sect. 9.5. We treat the equivariant first Chern class of a prequantum line bundle in Sect. 9.6. We treat Euler classes and equivariant Euler classes in Sect. 9.7 We treat the localization formula for torus actions in Sect. 9.8. Finally, we treat the abelian localization theorem of Atiyah-Bott and Berline-Vergne in Sect. 9.10. References for this chapter are Audin [1], Sect. 5 and Berline-Getzler-Vergne [2], Sect. 7. 9.2 Homotopy Quotients In this section, we first define classifying spaces, and then use them to define homotopy quotients.
On the classification ofG-spheres I: equivariant transversality
Acta Mathematica, 1988
This paper is the first in a series of three. Stated in geometric terms the papers examine locally linear group actions on spheres for odd order groups G. In essentially equivalent homotopy theoretic terms the papers study the homotopy types of the spaces PLc (V) and Tope(V) of equivariant PL-homeomorphisms and homeomorphisms of a linear representation V. In fact, it is the homogeneous spaces F6(V)/PLG(V) and Top6(V)/PL6(V) we study where Fc(V) is the space of proper equivariant homotopy equivalences. Our results generalize theorems of Haefliger, Kirby-Siebenmann, Sullivan and Wall, and others.
A geometric description of equivariant K-homology for proper actions
2009
Let G be a discrete group and let X be a G-finite, proper G-CW-complex. We prove that Kasparov's equivariant K-homology groups KK G * (C 0 (X), C) are isomorphic to the geometric equivariant K-homology groups of X that are obtained by making the geometric K-homology theory of Baum and Douglas equivariant in the natural way. This reconciles the original and current formulations of the Baum-Connes conjecture for discrete groups.
Equivariant Euler characteristics and K-homology Euler classes for proper cocompact G-manifolds
2003
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 K-homology of M. The universal equivariant Euler characteristic of M, which lives in a group U^G(M), counts the equivariant cells of M, taking the component structure of the various fixed point sets into account. We construct a natural homomorphism from U^G(M) to the equivariant KO-homology of M. The main result of this paper says that this map sends the universal equivariant Euler characteristic to the equivariant Euler class. In particular this shows that there are no `higher' equivariant Euler characteristics. We show that, rationally, the equivariant Euler class carries the same information as the collection of the orbifold Euler characteristics of the components of the L-fixed point sets M^L, where L runs through the finite cyclic subgroups of G. However, we g...
On the equivariant 2-type of a G-space
Journal of Pure and Applied Algebra, 1998
A classical theorem of Mac Lane and Whitehead states that the homotopy type of a topological space with trivial homotopy at dimensions 3 and greater can be reconstructed from its 711 and 712, and a cohomology class ks ~H~(rri, 7~). More recently, Moerdijk and Svensson suggested the possibility of using Bredon cohomology to extend this result to the equivariant case, that is, for spaces X equipped with an action by a fixed group G. In this paper we carry out this suggestion and prove an analogue of the classical result in the equivariant case.
Equivariant homology and cohomology of groups
Topology and its Applications, 2005
We provide and study an equivariant theory of group (co)homology of a group G with coefficients in a Γ-equivariant G-module A, when a separate group Γ acts on G and A, generalizing the classical Eilenberg-MacLane (co)homology theory of groups. Relationship with equivariant cohomology of topological spaces is established and application to algebraic K-theory is given.
On the localization formula in equivariant cohomology
We give a generalization of the Atiyah–Bott–Berline–Vergne localization theorem for the equivariant cohomology of a torus action. We replace the manifold having a torus action by an equivariant map of manifolds having a compact connected Lie group action. This provides a systematic method for calculating the Gysin homomorphism in ordinary cohomology of an equivariant map. As an example, we recover a formula of Akyildiz–Carrell for the Gysin homomorphism of flag manifolds.
Localization of Equivariant Cohomology for Compact and Non-Compact Group Actions
Journal of Dynamical Systems and Geometric Theories, 2005
We give a brief introduction to the Berline-Vergne localization formula for the finite-dimensional setting and indicate how the Duistermaat-Heckman formula is derived from it. We consider applications of the localization formula when it is specialized to a maximal dimensional co-adjoint orbit. In particular, the case when the co-adjoint orbit is a quotient G/T of a connected Lie group G modulo a maximal torus T is analyzed in detail. We describe also a generalization of the localization formula to non-compact group actions.