On extending actions of groups

Abstract

Problems of dense and closed extension of actions of compact transformation groups are solved. The method developed in the paper is applied to problems of extension of equivariant maps and of construction of equivariant compactifications

Изовариантные Экстензоры И Характеризация Эквивариантных Гомотопических Эквивалентностей

“…Пространство X называется Isov-порождающим, если для любого метри-ческого G-пространства Z существует изовариантное отображение η : Z → X. Если через Con T обозначить метрический конус над дискретным объедине-нием T всех однородных пространств G/H ∈ G-ANE, то счетная степень J (Con T) ω является Isov-порождающим [18]. Поскольку J является также Equiv-AE-пространством и J ∼ = J ω , из теоремы 1.9 следует, что J ∈ Isov-AE.…”

On Murayamaʼs theorem on extensor properties of G-spaces of given orbit types

“…Proof. By Lemma 5.2 V γ admits a nontrivial slice map ϕ : V γ → G/K, (K) ∈ C. The Slice theorem (≡ G/K ∈ G-ANE) implies that (7) the partial G-map P ′ γ 1 ←֓ V γ ϕ → G/K can be extended up to a slice map ψ :…”

“…We conjecture that the further generalization of Theorem 1.3 for compact group action on a metric space is also valid, provided that C ⊂ Orb G is a saturated family in which the intersection C ∩ E with the family E of extensor orbit types is cofinal in E 1 . The following theorem, proved in [7,Theorem 9], asserts in favor of this conjecture: if X is a metric G-ANE-space, then each G-subspace Y ⊂ X containing the bundle X E of extensor orbit types is a G-ANE.…”

“…The consequences of this theorem and another results of the theory of isovariant absolute extensors will be presented in the subsequent publications of the first author. The proofs of Theorems 1.3 and 1.4 will be based on the problem of extending the action of groups which was first posed by Shchepin in view of its connection with the problem of extending equivariant maps (see [7]). The diagram D = {X p → X i ֒→ Y }, in which the G-space X has the orbit type C ⊂ Orb G , p : X → X is an orbit projection and i is a closed topological embedding of the orbit space X into a space Y , will be called C-admissible.…”

“…We say that the problem of extending the action (denoted briefly by PEA) is solvable for the class F of spaces, if for each admissible diagram D in which X, X and Y belong to F there exists a solution of the PEA for which Y ∈ F. For compact group G, the PEA is solvable for the class of stratifiable spaces (see [7]). Here for the class of metric spaces we supplement this result with an information on the G-orbit type of a solution of the PEA.…”