G-Spaces with Prescribed Equivariant Cohomology

Mathematische Zeitschrift, 1983

w 1. Introduction It is well known that the rationalization X o of an H-space X is homotopy equivalent to a product of Eilenberg-Mac Lane spaces, X o ~ @K(rc.(X)| n) n ([4]). The object of this paper is to study the question of whether there is an equivariant splitting of X o in the case that X has a finite group action compatible with the H-structure. Let G be a finite group. Definition 1.1. A Hop/" G-space is a Hopf space (= H-space) on which G acts in such a way that the multiplication m: X x X~ X is an equivariant map, the unit e~X is a fixed point, and the composite X v X_~X x X-+X is G-homotopic to the folding map V: X v X ~ X. Examples. Let X be a topological group and let G be a finite subgroup of the group of inner automorphisms of X. Then X is obviously a Hopf G-space (the multiplication is a G-map). Another example of a Hopf G-space is the loop space ~2(Y, y) of a G-space Y, where G leaves y fixed. Also Eilenberg-Mac Lane G-spaces are Hopf G-spaces. We recall that an Eilenberg-Mac Lane G-space of dimension n is a G-space K such that the fixed point set K" is an Eilenberg-Mac Lane space of dimension n for every subgroup H of G. Eilenberg-Mac Lane G-spaces are introduced in [13 and are constructed in a way similar to the non-equivariant case. They play the same role in equivariant Bredon cohomology as Eilenberg-Mac Lane spaces in ordinary cohomology. In particular, H"~(X; M)= IX, K(M, n)]G, where M is a coefficient system for G (definition in w K(M, n) is the corresponding Eilenberg-Mac Lane G-space of dimension n, and [, ]~ means G-homotopy classes of G-maps. The equivariant cohomology in turn is the right cohomology in which to express equivariant obstructions for extending or lifting G-maps.

1998

arXiv (Cornell University), 2020

2020

Smith theory says that the fixed point of a semi-free action of a group G on a contractible space is Z p-acyclic for any prime factor p of G. Jones proved the converse of Smith theory for the case G is a cyclic group acting on finite CW-complexes. We extend the theory to semi-free group action on finite CW-complexes of given homotopy type, in various settings. In particular, the converse of Smith theory holds if and only if certain K-theoretical obstruction vanishes. We also give some examples that show the effects of different types of the K-theoretical obstruction.

Inventiones Mathematicae, 1988

Topology and its Applications, 2003

The purpose of the present article is threefold. First of all, we rebuild the whole theory of cosimplicial models of mapping spaces by using systematically Kan adjunction techniques. Secondly, given two topological spaces X and Y , we construct a cochain algebra which is quasi-isomorphic (as an algebra) to the singular cochain algebra of the mapping space Y X . Here, X has to be homotopy equivalent to the geometric realization a finite simplicial set and of dimension less or equal to the connectivity of Y . At last, we apply these results to the study of finite group actions on mapping spaces that are induced by an action on the source.

Topology and its Applications, 2017

Let G be a matrix Lie group. We prove that a proper Gspace X of finite structure, which is metrizable by a G-invariant metric, is a G-ANR (resp., a G-AR) iff for any compact subgroup H ⊂ G the Hfixed point set X H is an ANR (resp., an AR). An equivariant embedding result for proper G-spaces of finite structure is also obtained. Problem 1.1 (Jaworowski's Problem). Let G be a compact Lie group and X a metrizable G-space that has a finitely many G-orbit types (e.g., when G is a finite group). Assume that for each compact subgroup H ⊂ G, the H-fixed point set X H = {x ∈ X | hx = x, ∀h ∈ H} is an ANR (resp., an AR). Is then X a G-ANR (resp., a G-AR)? This problem still remains open. Moreover, it is open even in a very special case when G = Z 2 , X = Q = [−1, 1] ∞ , the Hilbert cube, and X Z 2 = { * }, a sigleton. In this case the problem is equivalent to the following old problem of R.D. Anderson posed in the mid-sixties (see [39, p. 292], see also [40, p. 656] and [41, Problem 930]): Problem 1.2 (Anderson's Problem). Let α : Q → Q be a based-free involution, i.e., α 2 = 1 Q and α has a unique fixed point. Is then α conjugate

Canadian Journal of Mathematics, 2002

In this article we state and prove precise theorems on the homotopy classification of graded categorical groups and their homomorphisms. The results use equivariant group cohomology, and they are applied to show a treatment of the general equivariant group extension problem.

K-Theory, 1998

Colloquium Mathematicum, 1989

Commentarii Mathematici Helvetici, 1993

A well-known elementary application of algebraic topology to group theory is the theorem that any group which acts freely on a (simplicial) tree is a free group, and its corollary that any subgroup of a free group is free. A deep generalization of this is the theorem of Karrass Pietrowski-Solitar, Cohen and Scott which classifies groups with a free subgroup of finite index (free-by-finite groups) as those groups which act on trees with finite stabilizers of bounded order (see, e.g.

2015

Let G be a group and let W be an algebra over a field K. We will say that W is a G-graded twisted algebra if W can be written as W = ⊕g∈GWg with WaW b ⊂ W ab , and where each Wg is a one dimensional K-vector space. It is also assumed that W has no monomial which is a zero divisor which means that for each pair of nonzero elements wa ∈ Wa, w b ∈ W b , wa • w b = 0. We also demand that W has a multiplicative identity element. We focus in the case where G is a finite abelian group and K = C or K = R. In this article, using methods of group cohomology, we classify all associative G-graded twisted algebras in the case G is a finite abelian group. On the other hand, by generalizing some of the arguments developed in [1] we present a classification of all G-graded twisted algebras that satisfy certain symmetry condition.

2015

2010

Algebraic Topology - Old and New, 2009

We prove that if G is a locally compact Hausdorff group then every proper G-ANR space has the G-homotopy type of a G-CW complex. This is applied to extend the James-Segal G-homotopy equivalence theorem to the case of arbitrary locally compact proper group actions.