Hyperspaces of Keller compacta and their orbit spaces Sa ul Ju - PowerPoint PPT Presentation

Keller compacta G -spaces Motivation Miscellaneous Result Hyperspaces of Keller compacta and their orbit spaces Sa´ ul Ju´ arez-Ord´ o˜ nez National University of Mexico (UNAM) 28th Summer Conference on Topology and its Applications, North Bay, Canada, July 22nd-26th

Keller compacta G -spaces Motivation Miscellaneous Result Joint work with Sergey Antonyan and Natalia Jonard-P´ erez. National University of Mexico (UNAM)

Keller compacta G -spaces Motivation Miscellaneous Result Index 1. Keller compacta 2. G -spaces 3. The problem and the result 4. Important notions 5. Sketch of the proof

Keller compacta G -spaces Motivation Miscellaneous Result Keller compacta An infinite-dimensional compact convex subset K of a topological linear space is called a Keller compactum , if it is affinely embeddable in the Hilbert space ℓ 2 : ∞ � x 2 K ֒ − − → ℓ 2 = { ( x n ) | x n ∈ R , n < ∞} . n = 1

Keller compacta G -spaces Motivation Miscellaneous Result Let K and V be convex subsets of linear spaces. A map f : K → V is called affine , if for every x 1 ,..., x n ∈ K and t 1 ,..., t n ∈ [ 0 , 1 ] such that � n i = 1 t i = 1 � n � n � � f t i x i = t i f ( x i ) . i = 1 i = 1 x 1 x n x 2 x 4 x 3

Keller compacta G -spaces Motivation Miscellaneous Result Let K and V be convex subsets of linear spaces. A map f : K → V is called affine , if for every x 1 ,..., x n ∈ K and t 1 ,..., t n ∈ [ 0 , 1 ] such that � n i = 1 t i = 1 � n � n � � f t i x i = t i f ( x i ) . i = 1 i = 1 f f ( x 3 ) x 1 f ( x 4 ) x n x 2 f ( x 2 ) f ( x n ) x 4 x 3 f ( x 1 )

Keller compacta G -spaces Motivation Miscellaneous Result Proposition Every infinite-dimensional metrizable compact convex subset of a locally convex linear space is a Keller compactum.

Keller compacta G -spaces Motivation Miscellaneous Result Proposition Every infinite-dimensional metrizable compact convex subset of a locally convex linear space is a Keller compactum. The Hilbert cube � ∞ [ − 1 , 1 ] n ⊂ R ∞ Q = n = 1 is affinely homeomorphic to { x ∈ ℓ 2 | | x n | ≤ 1 / n } ⊂ ℓ 2 .

Keller compacta G -spaces Motivation Miscellaneous Result Proposition Every infinite-dimensional metrizable compact convex subset of a locally convex linear space is a Keller compactum. The Hilbert cube � ∞ [ − 1 , 1 ] n ⊂ R ∞ Q = n = 1 is affinely homeomorphic to { x ∈ ℓ 2 | | x n | ≤ 1 / n } ⊂ ℓ 2 . The space P ( X ) of probability measures of an infinite compact metric space X endowed with the topology of weak convergence in measures: � f d µ n ❀ � f d µ µ n ❀ µ ⇐ ⇒ ∀ f ∈ C ( X ) .

Keller compacta G -spaces Motivation Miscellaneous Result Theorem (O. H. Keller) Every infinite-dimensional compact convex subset of the Hilbert space ℓ 2 is homeomorphic to the Hilbert cube Q.

Keller compacta G -spaces Motivation Miscellaneous Result Theorem (O. H. Keller) Every infinite-dimensional compact convex subset of the Hilbert space ℓ 2 is homeomorphic to the Hilbert cube Q. However, not all Keller compacta are affinely homeomorphic to each other.

Keller compacta G -spaces Motivation Miscellaneous Result Theorem (O. H. Keller) Every infinite-dimensional compact convex subset of the Hilbert space ℓ 2 is homeomorphic to the Hilbert cube Q. However, not all Keller compacta are affinely homeomorphic to each other. We consider Keller compacta together with its affine-topological structure.

Keller compacta G -spaces Motivation Miscellaneous Result G -spaces Let G be a topological group. A G -space is a topological space X together with a fixed continuous action of G : G × X − → X , ( g , x ) �− → gx .

� � Keller compacta G -spaces Motivation Miscellaneous Result G -spaces Let G be a topological group. A G -space is a topological space X together with a fixed continuous action of G : G × X − → X , ( g , x ) �− → gx . A map f : X → Y between G -spaces is called equivariant if for every x ∈ X and g ∈ G , � X f ( gx ) = gf ( x ) G × X 1 × f f � Y G × Y

Keller compacta G -spaces Motivation Miscellaneous Result Let X be a G -space. A subset A ⊂ X is called invariant if A = { ga | g ∈ G , a ∈ A } .

Keller compacta G -spaces Motivation Miscellaneous Result Let X be a G -space. A subset A ⊂ X is called invariant if A = { ga | g ∈ G , a ∈ A } . The orbit of x ∈ X is the smallest invariant subset containing x : Gx = { gx | g ∈ G } . gx x

Keller compacta G -spaces Motivation Miscellaneous Result The orbit space of X is the set X / G = { Gx | x ∈ X } endowed with the quotient topology given by the orbit map X − → X / G , x �− → Gx .

Keller compacta G -spaces Motivation Miscellaneous Result Let ( X , d ) be a metric space. The hyperspace of X : 2 X = { A ⊂ X | ∅ � = A compact } endowed with the topology induced by the Hausdorff metric: � � A , B ∈ 2 X . d H ( A , B ) = max sup d ( b , A ) , sup d ( a , B ) , b ∈ B a ∈ A

Keller compacta G -spaces Motivation Miscellaneous Result Let ( X , d ) be a metric space. The hyperspace of X : 2 X = { A ⊂ X | ∅ � = A compact } endowed with the topology induced by the Hausdorff metric: � � A , B ∈ 2 X . d H ( A , B ) = max sup d ( b , A ) , sup d ( a , B ) , b ∈ B a ∈ A Let X be a subset of a topological linear space. The cc-hyperspace of X : cc ( X ) = { A ∈ 2 X | A convex } .

Keller compacta G -spaces Motivation Miscellaneous Result If X is a metrizable G -space, then 2 X becomes a G -space with the induced action: G × 2 X − → 2 X , ( g , A ) �− → gA = { ga | a ∈ A } . In case X is a subset of a topological linear space and every g ∈ G preserves convexity, cc ( X ) is an invariant subspace of 2 X under this action. a ga A gA

Keller compacta G -spaces Motivation Miscellaneous Result Motivation Question (J. West, 1976) Let G be a compact connected Lie group. Is the orbit space 2 G / G an AR ? If it is, is it homeomorphic to the Hilbert cube Q?

Keller compacta G -spaces Motivation Miscellaneous Result Motivation Question (J. West, 1976) Let G be a compact connected Lie group. Is the orbit space 2 G / G an AR ? If it is, is it homeomorphic to the Hilbert cube Q? nczyk and West proved that 2 S 1 / S 1 ∈ AR not homeomorphic Toru´ to Q .

Keller compacta G -spaces Motivation Miscellaneous Result Motivation Question (J. West, 1976) Let G be a compact connected Lie group. Is the orbit space 2 G / G an AR ? If it is, is it homeomorphic to the Hilbert cube Q? nczyk and West proved that 2 S 1 / S 1 ∈ AR not homeomorphic Toru´ to Q . Antonyan proved that 2 S 1 / O ( 2 ) ∼ = BM ( 2 ) , which is an AR but not homeomorphic to Q.

Keller compacta G -spaces Motivation Miscellaneous Result Theorem (S. Antonyan) For n ≥ 2 , the orbit space 2 B n / O ( n ) is homeomorphic to the Hilbert cube Q.

Keller compacta G -spaces Motivation Miscellaneous Result Theorem (S. Antonyan) For n ≥ 2 , the orbit space 2 B n / O ( n ) is homeomorphic to the Hilbert cube Q. Theorem (S. Antonyan and N. Jonard-P´ erez) For n ≥ 2 , the orbit space cc ( B n ) / O ( n ) is homeomorphic to � BM ( n ) � . cone

Keller compacta G -spaces Motivation Miscellaneous Result Theorem (S. Antonyan) For n ≥ 2 , the orbit space 2 B n / O ( n ) is homeomorphic to the Hilbert cube Q. Theorem (S. Antonyan and N. Jonard-P´ erez) For n ≥ 2 , the orbit space cc ( B n ) / O ( n ) is homeomorphic to � BM ( n ) � . cone The Hilbert cube Q is a natural infinite-dimensional analog of B n . An analog for O ( n ) is the group O ( Q ) of affine isometries of Q , which leave the origin fixed.

Keller compacta G -spaces Motivation Miscellaneous Result Theorem (S. Antonyan) For n ≥ 2 , the orbit space 2 B n / O ( n ) is homeomorphic to the Hilbert cube Q. Theorem (S. Antonyan and N. Jonard-P´ erez) For n ≥ 2 , the orbit space cc ( B n ) / O ( n ) is homeomorphic to � BM ( n ) � . cone The Hilbert cube Q is a natural infinite-dimensional analog of B n . An analog for O ( n ) is the group O ( Q ) of affine isometries of Q , which leave the origin fixed. The purpose of this talk is to show that 2 Q / O ( Q ) ∼ cc ( Q ) / O ( Q ) ∼ = Q = Q .

Keller compacta G -spaces Motivation Miscellaneous Result Centrally symmetric Keller compacta are infinite-dimensional analogs of B n .

Keller compacta G -spaces Motivation Miscellaneous Result Centrally symmetric Keller compacta are infinite-dimensional analogs of B n . In analogy to the action of O ( n ) in B n , we consider actions of compact groups on centrally symmetric Keller compacta that respect their affine-topological structure.

Keller compacta G -spaces Motivation Miscellaneous Result Centrally symmetric Keller compacta are infinite-dimensional analogs of B n . In analogy to the action of O ( n ) in B n , we consider actions of compact groups on centrally symmetric Keller compacta that respect their affine-topological structure. We say that a group G acts affinely on a Keller compactum K if for every g ∈ G , x 1 ,..., x n ∈ K and t 1 ,..., t n ∈ [ 0 , 1 ] such that � n i = 1 t i = 1 � n � n � � g t i x i = t i gx i . i = 1 i = 1