Wyniki wyszukiwania

1

G-ANR's with homotopy trivial fixed point sets

100%

Antonyan S.

Fundamenta Mathematicae

|

2007

|

tom 197

|

nr 1

1-16

EN

Let G be a compact group and X a G-ANR. Then X is a G-AR iff the H-fixed point set $X^H$ is homotopy trivial for each closed subgroup H ⊂ G.

2

The equivariant homotopy type of G-ANR's for proper actions of locally compact groups

63%

Antonyan S., Elfving E.

Banach Center Publications

|

2009

|

tom 85

|

nr 1

155-178

EN

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.

3

Affine group acting on hyperspaces of compact convex subsets of ℝⁿ

63%

Antonyan S., Jonard-Pérez N.

Fundamenta Mathematicae

|

2013

|

tom 223

|

nr 2

99-136

EN

For every n ≥ 2, let cc(ℝⁿ) denote the hyperspace of all nonempty compact convex subsets of the Euclidean space ℝⁿ endowed with the Hausdorff metric topology. Let cb(ℝⁿ) be the subset of cc(ℝⁿ) consisting of all compact convex bodies. In this paper we discover several fundamental properties of the natural action of the affine group Aff(n) on cb(ℝⁿ). We prove that the space E(n) of all n-dimensional ellipsoids is an Aff(n)-equivariant retract of cb(ℝⁿ). This is applied to show that cb(ℝⁿ) is homeomorphic to the product $Q × ℝ^{n(n+3)/2}$, where Q stands for the Hilbert cube. Furthermore, we investigate the action of the orthogonal group O(n) on cc(ℝⁿ). In particular, we show that if K ⊂ O(n) is a closed subgroup that acts nontransitively on the unit sphere $𝕊^{n-1}$, then the orbit space cc(ℝⁿ)/K is homeomorphic to the Hilbert cube with a point removed, while cb(ℝⁿ)/K is a contractible Q-manifold homeomorphic to the product (E(n)/K) × Q. The orbit space cb(ℝⁿ)/Aff(n) is homeomorphic to the Banach-Mazur compactum BM(n), while cc(ℝⁿ)/O(n) is homeomorphic to the open cone over BM(n).

Ograniczanie wyników

2 Fundamenta Mathematicae

1 Banach Center Publications

3 Antonyan S.

1 Elfving E.

1 Jonard-Pérez N.

1 2013

1 2009

1 2007

G-ANR's with homotopy trivial fixed point sets

The equivariant homotopy type of G-ANR's for proper actions of locally compact groups

Affine group acting on hyperspaces of compact convex subsets of ℝⁿ