A Note on Equivariant Fixed Point Theory

2002

Suppose one is given a discrete group G, a cocompact proper G-manifold M, and a G-self-map f of 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 that these two classes agree. As a special case, we prove an equivariant Poincare-Hopf Theorem, computing the universal equivariant Euler characteristic in terms of the zeros of an equivariant vector field, and also obtain an orbifold Lefschetz fixed point theorem. Finally, we prove a realization theorem for universal equivariant Euler characteristics.

Topological Methods in Nonlinear Analysis, 2009

For any compact Lie group G, we give a decomposition of the group {X, Y } k G of (unpointed) stable G-homotopy classes as a direct sum of subgroups of fixed orbit types. This is done by interpreting the G-homotopy classes in terms of the generalized fixed-point transfer and making use of conormal maps.

Manuscripta Mathematica, 1987

Proceedings of the American Mathematical Society, 1983

Let G be a finite group. In this note we study the question of realizing a collection of graded commutative algebras over Q as the cohomology algebras with rational coefficients of the fixed point sets X" ( H < G) of a G-space X. Let tí be a graded commutative algebra over Q. The question of realizing & as the cohomology with rational coefficients of a space X is answered by Quillen [4] and more directly by Sullivan [5], In particular, Sullivan constructs a space X of finite type, i.e. ir¡(X) is a finitely generated abelian group for every /, which realizes 68. Now let G be a finite group which acts on tf from the left by algebra isomorphisms. Because of the functoriality of the constructions in and one can construct a G-space X such that H*(X;Q) = tf, where the isomorphism is G-equivariant. The space X in both cases is a rational space. In this note we consider a more general question. Let 6C be the category of canonical orbits of a finite group G [1], The objects of 0C are the quotient spaces G/H, where H is a subgroup of G (H < G), and the morphisms are the G-maps between them, where G acts on G/H by left multiplication. Definition 1. A system of graded commutative algebras (GA's) for G is a covariant functor from (3C into the category of graded commutative connected algebras over Q. We recall that a GA tí is said to be connected if 68° = Q and is said to be of finite type if tf " is a finite-dimensional vector space over Q. Let A be a G-space such that each fixed point set XH, H < G, is nonempty and connected. Given A, a system of GA's H*( X) is defined by H*(X)(G/H) =H*(X";Q) on objects of (P0. If/: G/H -G/K is a G-map, then there exists an element g E G such that g~lHg < K, and the map /is determined by H t-> gK. The map/induces a map /: XK -» X" by x i-> gx and therefore a unique map H*(X)(f)=f*: H*(X";Q) -H*(XK;Q). The main result of this paper is the following Theorem 2. Given a system H of connected GA's of finite type, there exists a G-CW-complex X of finite type such that H*(X) = H.

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.

arXiv (Cornell University), 2020

Algebraic &amp; Geometric Topology, 2018

We investigate certain adjunctions in derived categories of equivariant spectra, including a right adjoint to fixed points, a right adjoint to pullback by an isometry of universes, and a chain of two right adjoints to geometric fixed points. This leads to a variety of interesting other adjunctions, including a chain of 6 (sometimes 7) adjoints involving the restriction functor to a subgroup of a finite group on equivariant spectra indexed over the trivial universe.

Proceedings of the National Academy of Sciences of the United States of America, 1959

Fundamenta Mathematicae

In the theory of transformation groups, it is important to know what kind of isotropy subgroups of G do occur at points of the space upon which the given group G acts. In this article, for a finite group G, we prove the Equivariant Bundle Subtraction Theorem (Theorem 2.2) which allows us to construct smooth G-manifolds with prescribed isotropy subgroups around the G-fixed point sets. In Theorem 0.1, we restate Oliver's result about manifolds M and G-vector bundles over M that occur, respectively, as the G-fixed point sets and their equivariant normal bundles for smooth G-actions on disks. In Theorems 0.2 and 0.3, we prove the corresponding results for smooth G-actions on disks with prescribed isotropy subgroups around M. In Theorems 0.4 and 0.5, for large classes of finite groups G, we explicitly describe manifolds M that occur as the G-fixed point sets for such actions on disks. These actions are expected to be useful for answering the question of which manifolds occur as the G-fixed points sets for smooth G-actions on spheres. 0. Introduction. In the theory of transformation groups, it often happens that a solution of a particular problem depends on the family of the isotropy subgroups that we allow to occur at points in the space upon which a given group G acts. In [O4], for a finite group G not of prime power order, Oliver describes necessary and sufficient conditions under which a smooth manifold M occurs as the G-fixed point set and a smooth G-vector ν over M stably occurs as the equivariant normal bundle of M in D (resp., E