Finite subset spaces of the circle Christopher Tuffley Institute of - PowerPoint PPT Presentation

Finite subset spaces The circle Finite subset spaces of the circle Christopher Tuffley Institute of Fundamental Sciences Massey University, Palmerston North Devonport Topology Festival, 2008 Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Outline Finite subset spaces 1 Definition Properties The circle 2 Known results New results Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces Definition The circle Properties Finite subset spaces X a topological space k a positive integer Definition (Borsuk and Ulam, 1931) The kth finite subset space of X is exp k X = { nonempty subsets of X of size ≤ k } , with the quotient topology from X k → exp k X ( x 1 , . . . , x k ) �→ { x 1 } ∪ · · · ∪ { x k } . Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces Definition The circle Properties The symmetric product Borsuk and Ulam called exp k X the k th symmetric product, but this has come to mean Sym k X = X k / S k . Comparing: exp 1 X ∼ = Sym 1 X ∼ = X exp 2 X ∼ = Sym 2 X exp k X is a proper quotient of Sym k X for k ≥ 3: in Sym 3 X , but ( a , a , b ) �∼ ( a , b , b ) ( a , a , b ) ∼ ( a , b , b ) �→ { a , b } in exp 3 X . Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces Definition The circle Properties Inclusions and inclusion-like maps If j ≤ k there is a natural inclusion exp j X ֒ → exp k X ; for X Hausdorff this is a homeo onto its image. Given x 0 ∈ X , have ∪{ x 0 } : exp k X → exp k + 1 X S ⊆ X �→ S ∪ { x 0 } . Image is exp k + 1 ( X , x 0 ) = { S ∈ exp k + 1 X : x 0 ∈ S } ; generically one-to-one, but not globally: S and S ∪ { x 0 } have the same image if | S | < k and x 0 �∈ S . Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces Definition The circle Properties Inclusions and inclusion-like maps If j ≤ k there is a natural inclusion exp j X ֒ → exp k X ; for X Hausdorff this is a homeo onto its image. Given x 0 ∈ X , have ∪{ x 0 } : exp k X → exp k + 1 X S ⊆ X �→ S ∪ { x 0 } . Image is exp k + 1 ( X , x 0 ) = { S ∈ exp k + 1 X : x 0 ∈ S } ; generically one-to-one, but not globally: S and S ∪ { x 0 } have the same image if | S | < k and x 0 �∈ S . Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces Definition The circle Properties Functoriality exp k is a homotopy functor: f : X → Y induces a map exp k f : exp k X → exp k Y S ⊆ X �→ f ( S ) ⊆ Y if { h t } is a homotopy between f and g then { exp k h t } is a homotopy between exp k f and exp k g Example exp k R n is contractible, because R n ≃ {∗} and � � exp k {∗} = {∗} for all k . Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces Definition The circle Properties Functoriality exp k is a homotopy functor: f : X → Y induces a map exp k f : exp k X → exp k Y S ⊆ X �→ f ( S ) ⊆ Y if { h t } is a homotopy between f and g then { exp k h t } is a homotopy between exp k f and exp k g Example exp k R n is contractible, because R n ≃ {∗} and � � exp k {∗} = {∗} for all k . Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces Known results The circle New results The second finite subset space Theorem exp 2 S 1 is a Möbius strip, with boundary exp 1 S 1 . Proof via cut-and-paste topology. (y,x) (x,y) (a) (b) (c) (d) Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces Known results The circle New results The inclusions Note: exp 2 S 1 ≃ exp 1 S 1 = S 1 , and exp 1 S 1 ֒ → exp 2 S 1 is degree two; exp 2 ( S 1 , ∗ ) ∼ = exp 1 S 1 = S 1 , and exp 1 S 1 ∪{∗} → exp 2 ( S 1 , ∗ ) ֒ → exp 2 S 1 − is a homotopy equivalence. Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces Known results The circle New results A second proof Proof via bundles. Map { x , y } to the “bisector” in RP 1 ∼ = S 1 . 1 y x Fibres may be identified with [ 0 , 2 π ] via arclength. 2 The bundle is twisted, because ℓ ∈ [ 0 , 2 π ] is equivalent to 3 2 π − ℓ . Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces Known results The circle New results The third finite subset space Theorem (Bott, 1952) The space exp 3 S 1 is homeomorphic to the three-sphere. So exp 1 S 1 ֒ → exp 3 S 1 is a map S 1 ֒ → S 3 , hence a knot, and: Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces Known results The circle New results The third finite subset space Theorem (Bott, 1952) The space exp 3 S 1 is homeomorphic to the three-sphere. So exp 1 S 1 ֒ → exp 3 S 1 is a map S 1 ֒ → S 3 , hence a knot, and: Theorem (Shchepin, unpublished) exp 1 S 1 ⊆ exp 3 S 1 is a trefoil knot. Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces Known results The circle New results Sketch proof of Bott’s result Reduce to the simplex 0 ≤ x ≤ y ≤ z ≤ 1, with face 1 gluings ( 0 , y , z ) ∼ ( y , z , 1 ) and ( x , x , y ) ∼ ( x , y , y ) . z a a y b a a a x (a) (b) Result is a 3-manifold, because χ = 1 − 2 + 2 − 1 = 0. 2 π 1 = � a , b | a 2 = a = b � ∼ = { 1 } , so result is S 3 . 3 Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces Known results The circle New results Sketch proof of Bott’s result Reduce to the simplex 0 ≤ x ≤ y ≤ z ≤ 1, with face 1 gluings ( 0 , y , z ) ∼ ( y , z , 1 ) and ( x , x , y ) ∼ ( x , y , y ) . z a a y b a a a x (a) (b) Result is a 3-manifold, because χ = 1 − 2 + 2 − 1 = 0. 2 π 1 = � a , b | a 2 = a = b � ∼ = { 1 } , so result is S 3 . 3 Or: Use a Heegard splitting to show the manifold is S 3 , and 3’ that exp 1 S 1 is a trefoil. Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces Known results The circle New results Bott and Shchepin via Seifert fibred spaces S 1 acts on its finite subset spaces by rotation. 1 This gives exp 3 S 1 the structure of a Seifert fibred space (a three-manifold that is a union of circles, the orbits). There are two exceptional fibres, of indices 2 and 3: 2 .S. is S 3 , with the S 1 The only such simply connected S.F 3 action λ · ( z , w ) = ( λ 2 z , λ 3 w ) . Generic orbits are ( 2 , 3 ) -torus knots, i.e. trefoils. Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces Known results The circle New results Bott and Shchepin via Seifert fibred spaces S 1 acts on its finite subset spaces by rotation. 1 This gives exp 3 S 1 the structure of a Seifert fibred space (a three-manifold that is a union of circles, the orbits). There are two exceptional fibres, of indices 2 and 3: 2 index 2 orbit generic orbit index 3 orbit .S. is S 3 , with the S 1 The only such simply connected S.F 3 action λ · ( z , w ) = ( λ 2 z , λ 3 w ) . Generic orbits are ( 2 , 3 ) -torus knots, i.e. trefoils. Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces Known results The circle New results Bott and Shchepin via Seifert fibred spaces S 1 acts on its finite subset spaces by rotation. 1 This gives exp 3 S 1 the structure of a Seifert fibred space (a three-manifold that is a union of circles, the orbits). There are two exceptional fibres, of indices 2 and 3: 2 index 2 orbit generic orbit index 3 orbit .S. is S 3 , with the S 1 The only such simply connected S.F 3 action λ · ( z , w ) = ( λ 2 z , λ 3 w ) . Generic orbits are ( 2 , 3 ) -torus knots, i.e. trefoils. Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces Known results The circle New results The general case Theorem (Tuffley, 2002) exp k S 1 has the homotopy type of an odd-dimensional 1 sphere, of dimension k or k − 1 (so exp 2 k − 1 S 1 ≃ exp 2 k S 1 ≃ S 2 k − 1 ). exp k ( S 1 , ∗ ) has the homotopy type of a point if k is odd, 2 and a ( k − 1 ) -sphere if k is even. exp 2 k − 1 S 1 ֒ → exp 2 k S 1 has degree two, while 3 exp 2 k − 1 S 1 ∪{∗} → exp 2 k S 1 has degree one. → exp 2 k ( S 1 , ∗ ) ֒ − The complement of exp k − 2 S 1 in exp k S 1 has the homotopy 4 type of a ( k − 1 , k ) –torus knot complement. Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces Known results The circle New results The general case Theorem (Tuffley, 2002) exp k S 1 has the homotopy type of an odd-dimensional 1 sphere, of dimension k or k − 1 (so exp 2 k − 1 S 1 ≃ exp 2 k S 1 ≃ S 2 k − 1 ). exp k ( S 1 , ∗ ) has the homotopy type of a point if k is odd, 2 and a ( k − 1 ) -sphere if k is even. exp 2 k − 1 S 1 ֒ → exp 2 k S 1 has degree two, while 3 exp 2 k − 1 S 1 ∪{∗} → exp 2 k S 1 has degree one. → exp 2 k ( S 1 , ∗ ) ֒ − The complement of exp k − 2 S 1 in exp k S 1 has the homotopy 4 type of a ( k − 1 , k ) –torus knot complement. Christopher Tuffley Finite subset spaces of the circle