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

1

Finite subset spaces Definition Properties

2

The circle Known results New results

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Definition 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 expkX = {nonempty subsets of X of size ≤ k}, with the quotient topology from X k → expkX (x1, . . . , xk) → {x1} ∪ · · · ∪ {xk}.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Definition Properties

The symmetric product

Borsuk and Ulam called expkX the kth symmetric product, but this has come to mean SymkX = X k/Sk. Comparing: exp1X ∼ = Sym1X ∼ = X exp2X ∼ = Sym2X expkX is a proper quotient of SymkX for k ≥ 3: (a, a, b) ∼ (a, b, b) in Sym3X, but (a, a, b) ∼ (a, b, b) → {a, b} in exp3X.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Definition Properties

Inclusions and inclusion-like maps

If j ≤ k there is a natural inclusion expjX ֒ → expkX; for X Hausdorff this is a homeo onto its image. Given x0 ∈ X, have ∪{x0} : expkX → expk+1X S ⊆ X → S ∪ {x0}.

Image is expk+1(X, x0) = {S ∈ expk+1X : x0 ∈ S}; generically one-to-one, but not globally: S and S ∪ {x0} have the same image if |S| < k and x0 ∈ S.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Definition Properties

Inclusions and inclusion-like maps

If j ≤ k there is a natural inclusion expjX ֒ → expkX; for X Hausdorff this is a homeo onto its image. Given x0 ∈ X, have ∪{x0} : expkX → expk+1X S ⊆ X → S ∪ {x0}.

Image is expk+1(X, x0) = {S ∈ expk+1X : x0 ∈ S}; generically one-to-one, but not globally: S and S ∪ {x0} have the same image if |S| < k and x0 ∈ S.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Definition Properties

Functoriality

expk is a homotopy functor: f : X → Y induces a map expkf : expkX → expkY S ⊆ X → f(S) ⊆ Y if {ht} is a homotopy between f and g then {expkht} is a homotopy between expkf and expkg Example expkRn is contractible, because Rn ≃ {∗} and expk{∗} =

• {∗}
• for all k.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Definition Properties

Functoriality

expk is a homotopy functor: f : X → Y induces a map expkf : expkX → expkY S ⊆ X → f(S) ⊆ Y if {ht} is a homotopy between f and g then {expkht} is a homotopy between expkf and expkg Example expkRn is contractible, because Rn ≃ {∗} and expk{∗} =

• {∗}
• for all k.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Known results New results

The second finite subset space

Theorem exp2S1 is a Möbius strip, with boundary exp1S1. 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 The circle Known results New results

The inclusions

Note: exp2S1 ≃ exp1S1 = S1, and exp1S1 ֒ → exp2S1 is degree two; exp2(S1, ∗) ∼ = exp1S1 = S1, and exp1S1 ∪{∗} − → exp2(S1, ∗) ֒ → exp2S1 is a homotopy equivalence.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Known results New results

A second proof

Proof via bundles.

1

Map {x, y} to the “bisector” in RP1 ∼ = S1.

x y

2

Fibres may be identified with [0, 2π] via arclength.

3

The bundle is twisted, because ℓ ∈ [0, 2π] is equivalent to 2π − ℓ.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Known results New results

The third finite subset space

Theorem (Bott, 1952) The space exp3S1 is homeomorphic to the three-sphere. So exp1S1 ֒ → exp3S1 is a map S1 ֒ → S3, hence a knot, and:

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Known results New results

The third finite subset space

Theorem (Bott, 1952) The space exp3S1 is homeomorphic to the three-sphere. So exp1S1 ֒ → exp3S1 is a map S1 ֒ → S3, hence a knot, and: Theorem (Shchepin, unpublished) exp1S1 ⊆ exp3S1 is a trefoil knot.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Known results New results

Sketch proof of Bott’s result

1

Reduce to the simplex 0 ≤ x ≤ y ≤ z ≤ 1, with face gluings (0, y, z) ∼ (y, z, 1) and (x, x, y) ∼ (x, y, y).

(a) z y x (b) a a a b a a

2

Result is a 3-manifold, because χ = 1 − 2 + 2 − 1 = 0.

3

π1 = a, b|a2 = a = b ∼ = {1}, so result is S3.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Known results New results

Sketch proof of Bott’s result

1

Reduce to the simplex 0 ≤ x ≤ y ≤ z ≤ 1, with face gluings (0, y, z) ∼ (y, z, 1) and (x, x, y) ∼ (x, y, y).

(a) z y x (b) a a a b a a

2

Result is a 3-manifold, because χ = 1 − 2 + 2 − 1 = 0.

3

π1 = a, b|a2 = a = b ∼ = {1}, so result is S3. Or:

3’

Use a Heegard splitting to show the manifold is S3, and that exp1S1 is a trefoil.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Known results New results

Bott and Shchepin via Seifert fibred spaces

1

S1 acts on its finite subset spaces by rotation. This gives exp3S1 the structure of a Seifert fibred space (a three-manifold that is a union of circles, the orbits).

2

There are two exceptional fibres, of indices 2 and 3:

3

The only such simply connected S.F .S. is S3, with the S1 action λ · (z, w) = (λ2z, λ3w). Generic orbits are (2, 3)-torus knots, i.e. trefoils.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Known results New results

Bott and Shchepin via Seifert fibred spaces

1

S1 acts on its finite subset spaces by rotation. This gives exp3S1 the structure of a Seifert fibred space (a three-manifold that is a union of circles, the orbits).

2

There are two exceptional fibres, of indices 2 and 3:

generic orbit index 2 orbit index 3 orbit

3

The only such simply connected S.F .S. is S3, with the S1 action λ · (z, w) = (λ2z, λ3w). Generic orbits are (2, 3)-torus knots, i.e. trefoils.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Known results New results

Bott and Shchepin via Seifert fibred spaces

1

S1 acts on its finite subset spaces by rotation. This gives exp3S1 the structure of a Seifert fibred space (a three-manifold that is a union of circles, the orbits).

2

There are two exceptional fibres, of indices 2 and 3:

generic orbit index 2 orbit index 3 orbit

3

The only such simply connected S.F .S. is S3, with the S1 action λ · (z, w) = (λ2z, λ3w). Generic orbits are (2, 3)-torus knots, i.e. trefoils.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Known results New results

The general case

Theorem (Tuffley, 2002)

1

expkS1 has the homotopy type of an odd-dimensional sphere, of dimension k or k − 1 (so exp2k−1S1 ≃ exp2kS1 ≃ S2k−1).

2

expk(S1, ∗) has the homotopy type of a point if k is odd, and a (k − 1)-sphere if k is even.

3

exp2k−1S1 ֒ → exp2kS1 has degree two, while exp2k−1S1 ∪{∗} − → exp2k(S1, ∗) ֒ → exp2kS1 has degree one.

4

The complement of expk−2S1 in expkS1 has the homotopy type of a (k − 1, k)–torus knot complement.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Known results New results

The general case

Theorem (Tuffley, 2002)

1

expkS1 has the homotopy type of an odd-dimensional sphere, of dimension k or k − 1 (so exp2k−1S1 ≃ exp2kS1 ≃ S2k−1).

2

expk(S1, ∗) has the homotopy type of a point if k is odd, and a (k − 1)-sphere if k is even.

3

exp2k−1S1 ֒ → exp2kS1 has degree two, while exp2k−1S1 ∪{∗} − → exp2k(S1, ∗) ֒ → exp2kS1 has degree one.

4

The complement of expk−2S1 in expkS1 has the homotopy type of a (k − 1, k)–torus knot complement.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Known results New results

The general case

Theorem (Tuffley, 2002)

1

expkS1 has the homotopy type of an odd-dimensional sphere, of dimension k or k − 1 (so exp2k−1S1 ≃ exp2kS1 ≃ S2k−1).

2

expk(S1, ∗) has the homotopy type of a point if k is odd, and a (k − 1)-sphere if k is even.

3

exp2k−1S1 ֒ → exp2kS1 has degree two, while exp2k−1S1 ∪{∗} − → exp2k(S1, ∗) ֒ → exp2kS1 has degree one.

4

The complement of expk−2S1 in expkS1 has the homotopy type of a (k − 1, k)–torus knot complement.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Known results New results

The general case

Theorem (Tuffley, 2002)

1

expkS1 has the homotopy type of an odd-dimensional sphere, of dimension k or k − 1 (so exp2k−1S1 ≃ exp2kS1 ≃ S2k−1).

2

expk(S1, ∗) has the homotopy type of a point if k is odd, and a (k − 1)-sphere if k is even.

3

exp2k−1S1 ֒ → exp2kS1 has degree two, while exp2k−1S1 ∪{∗} − → exp2k(S1, ∗) ֒ → exp2kS1 has degree one.

4

The complement of expk−2S1 in expkS1 has the homotopy type of a (k − 1, k)–torus knot complement.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Known results New results

Sketch proof — part 1 only

Let d = k if k is odd, and k − 1 if k is even.

1

Reduce to the simplex 0 ≤ x1 ≤ x2 ≤ · · · ≤ xk ≤ 1. Resulting cell structure for expkS1 has 1 0-cell, 2 i-cells, 1 ≤ i ≤ k − 1, 1 k-cell.

2

expkS1 has the homology of Sd, and is simply connected for k ≥ 3.

3

Hurewicz: the first non-vanishing homotopy group is πd = Hd = Z.

4

Whitehead: the generator of πd is a homotopy equivalence Sd → expkS1.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Known results New results

Sketch proof — part 1 only

Let d = k if k is odd, and k − 1 if k is even.

1

Reduce to the simplex 0 ≤ x1 ≤ x2 ≤ · · · ≤ xk ≤ 1. Resulting cell structure for expkS1 has 1 0-cell, 2 i-cells, 1 ≤ i ≤ k − 1, 1 k-cell.

2

expkS1 has the homology of Sd, and is simply connected for k ≥ 3.

3

Hurewicz: the first non-vanishing homotopy group is πd = Hd = Z.

4

Whitehead: the generator of πd is a homotopy equivalence Sd → expkS1.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Known results New results

Sketch proof — part 1 only

Let d = k if k is odd, and k − 1 if k is even.

1

Reduce to the simplex 0 ≤ x1 ≤ x2 ≤ · · · ≤ xk ≤ 1. Resulting cell structure for expkS1 has 1 0-cell, 2 i-cells, 1 ≤ i ≤ k − 1, 1 k-cell.

2

expkS1 has the homology of Sd, and is simply connected for k ≥ 3.

3

Hurewicz: the first non-vanishing homotopy group is πd = Hd = Z.

4

Whitehead: the generator of πd is a homotopy equivalence Sd → expkS1.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Known results New results

Sketch proof — part 1 only

Let d = k if k is odd, and k − 1 if k is even.

1

Reduce to the simplex 0 ≤ x1 ≤ x2 ≤ · · · ≤ xk ≤ 1. Resulting cell structure for expkS1 has 1 0-cell, 2 i-cells, 1 ≤ i ≤ k − 1, 1 k-cell.

2

expkS1 has the homology of Sd, and is simply connected for k ≥ 3.

3

Hurewicz: the first non-vanishing homotopy group is πd = Hd = Z.

4

Whitehead: the generator of πd is a homotopy equivalence Sd → expkS1.

Christopher Tuffley Finite subset spaces of the circle

Finite subset spaces The circle Known results New results

A last word

A similar approach may be used to study the finite subset spaces of a graph G. Find for example that expk(G, ∗) ≃ a wedge of k − 1 spheres. Results for graphs form the basis of an induction to prove that the kth finite subset space of a connected cell complex is (k − 2)-connected.

Christopher Tuffley Finite subset spaces of the circle