CLASSIFICATION OFG-SPACES

Изовариантные Экстензоры И Характеризация Эквивариантных Гомотопических Эквивалентностей

“…Однако до сих пор даже существование Isov-AE-пространств было известно лишь при некото-рых ограничениях, налагаемых на размерность и орбитный тип [14], [16], [17]. Следующая теорема окончательно исправляет ситуацию.…”

On extending actions of groups

“…If G is a compact non-Lie group, then by (7) G ∈ ANE. It is known that there exists a free G-space X ∈ ANE [4]. Hence each its invariant open subset is not homeomorphic to a product G × U and therefore in this case the slice theorem fails [22].…”

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

“…We cannot omit the saturation condition from Theorem 1.3 (since there exists a 2-dimensional compact counterexample), but we do have a pleasant (and important) exception for (Σ, d)-universal (in the sense of Palais [18, p.59]) G-spaces. Until recently the solution of Palais problem on existence of universal G-spaces was known only for finite collection Σ ⊂ Orb G of orbit types and finite dimension d < ∞ [18, 2.6]; for finite dimension d [3]. The final solution of Palais problem (without any restrictions on dimension d and collection Σ) was obtained in [6]: the equivariant Hilbert space L 2 is an (Orb G , ∞)-universal G-space.…”