Incomputability of Simply Connected Planar Continua Takayuki Kihara - - PowerPoint PPT Presentation

Introduction Main Theorem

Incomputability of Simply Connected Planar Continua

Takayuki Kihara

Mathematical Institute, Tohoku University Computability in Europe 2011

June 29, 2011

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Introduction

Every nonempty Σ

∼ 1 set in Rn contains a computable point.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Introduction

Every nonempty Σ

∼ 1 set in Rn contains a computable point.

Not every nonempty Π0

1 set in Rn contains a computable point.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Introduction

Every nonempty Σ

∼ 1 set in Rn contains a computable point.

Not every nonempty Π0

1 set in Rn contains a computable point.

If a nonempty Π0

1 subset F ⊆ R1 contains no computable

points, then F must be disconnected.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Introduction

Every nonempty Σ

∼ 1 set in Rn contains a computable point.

Not every nonempty Π0

1 set in Rn contains a computable point.

If a nonempty Π0

1 subset F ⊆ R1 contains no computable

points, then F must be disconnected. Does there exist a nonempty (simply) connected Π0

1 set in Rn

without computable points?

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Introduction

Every nonempty Σ

∼ 1 set in Rn contains a computable point.

Not every nonempty Π0

1 set in Rn contains a computable point.

If a nonempty Π0

1 subset F ⊆ R1 contains no computable

points, then F must be disconnected. Does there exist a nonempty (simply) connected Π0

1 set in Rn

without computable points? The main theme of this talk is Computability Theory for Connected Spaces.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Definition

{Be}e∈N: an effective enumeration of all rational open balls.

1

x ∈ Rn is computable if {e ∈ N : x ∈ Be} is c.e. Equivalently, x = (x1, . . . , xn) ∈ Rn is computable iff xi is computable for each i ≤ n.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Definition

{Be}e∈N: an effective enumeration of all rational open balls.

1

x ∈ Rn is computable if {e ∈ N : x ∈ Be} is c.e. Equivalently, x = (x1, . . . , xn) ∈ Rn is computable iff xi is computable for each i ≤ n.

2

F ⊆ Rn is Π0

1 (or co-c.e. closed) if F = Rn \ ∪ e∈W Be for a

c.e. set W.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Definition

{Be}e∈N: an effective enumeration of all rational open balls.

1

x ∈ Rn is computable if {e ∈ N : x ∈ Be} is c.e. Equivalently, x = (x1, . . . , xn) ∈ Rn is computable iff xi is computable for each i ≤ n.

2

F ⊆ Rn is Π0

1 (or co-c.e. closed) if F = Rn \ ∪ e∈W Be for a

c.e. set W.

1

Not every nonempty Π0

1 set in R1 contains a computable point.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Definition

{Be}e∈N: an effective enumeration of all rational open balls.

1

x ∈ Rn is computable if {e ∈ N : x ∈ Be} is c.e. Equivalently, x = (x1, . . . , xn) ∈ Rn is computable iff xi is computable for each i ≤ n.

2

F ⊆ Rn is Π0

1 (or co-c.e. closed) if F = Rn \ ∪ e∈W Be for a

c.e. set W.

1

Not every nonempty Π0

1 set in R1 contains a computable point.

2

(Category) Every nonempty co-meager Π0

1 set in Rn contains

a computable point.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Definition

{Be}e∈N: an effective enumeration of all rational open balls.

1

x ∈ Rn is computable if {e ∈ N : x ∈ Be} is c.e. Equivalently, x = (x1, . . . , xn) ∈ Rn is computable iff xi is computable for each i ≤ n.

2

F ⊆ Rn is Π0

1 (or co-c.e. closed) if F = Rn \ ∪ e∈W Be for a

c.e. set W.

1

Not every nonempty Π0

1 set in R1 contains a computable point.

2

(Category) Every nonempty co-meager Π0

1 set in Rn contains

a computable point.

3

(Measure) Not every nonempty positive measure Π0

1 set in R1

contains a computable point.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Definition

{Be}e∈N: an effective enumeration of all rational open balls.

1

x ∈ Rn is computable if {e ∈ N : x ∈ Be} is c.e. Equivalently, x = (x1, . . . , xn) ∈ Rn is computable iff xi is computable for each i ≤ n.

2

F ⊆ Rn is Π0

1 (or co-c.e. closed) if F = Rn \ ∪ e∈W Be for a

c.e. set W.

1

Not every nonempty Π0

1 set in R1 contains a computable point.

2

(Category) Every nonempty co-meager Π0

1 set in Rn contains

a computable point.

3

(Measure) Not every nonempty positive measure Π0

1 set in R1

contains a computable point.

4

(Connectedness) What about connected, simply connected,

• r contractible Π0

1 sets?

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Connected Π0

1 Sets

Observation

1

Every nonempty connected Π0

1 subset P ⊆ R1 contains a

computable point.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Connected Π0

1 Sets

Observation

1

Every nonempty connected Π0

1 subset P ⊆ R1 contains a

computable point. Fact

1

There exists a nonempty connected Π0

1 subset P(2) ⊆ R2

without computable points.

2

There exists a nonempty simply connected Π0

1 subset

P(3) ⊆ R3 without computable points.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Example ✓ ✏ P P(2) ✒ ✑ P(n) = ∪

k<n([0, 1]k × P × [0, 1]n−k−1) for P ⊆ [0, 1].

． ．

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Example ✓ ✏ P P(2) ✒ ✑ P(n) = ∪

k<n([0, 1]k × P × [0, 1]n−k−1) for P ⊆ [0, 1].

Let P ⊆ [0, 1] be a Π0

1 set without computable points.

． ．

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Example ✓ ✏ P P(2) ✒ ✑ P(n) = ∪

k<n([0, 1]k × P × [0, 1]n−k−1) for P ⊆ [0, 1].

Let P ⊆ [0, 1] be a Π0

1 set without computable points.

P(2) ⊆ [0, 1]2 is a connected Π0

1 set without computable

points． ．

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Example ✓ ✏ P P(2) ✒ ✑ P(n) = ∪

k<n([0, 1]k × P × [0, 1]n−k−1) for P ⊆ [0, 1].

Let P ⊆ [0, 1] be a Π0

1 set without computable points.

P(2) ⊆ [0, 1]2 is a connected Π0

1 set without computable

points． P(3) ⊆ [0, 1]3 is a simply connected Π0

1 set without

computable points．

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

X is n-connected ⇐

⇒ the first n + 1 homotopy groups

vanish identically. X is path-connected ⇐

⇒ X is 0-connected.

X is simply connected ⇐

⇒ X is 1-connected.

X is contractible ⇐

⇒ the identity map on X is null-homotopic.

X is contractible =

⇒ X is n-connected for any n.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

X is n-connected ⇐

⇒ the first n + 1 homotopy groups

vanish identically. X is path-connected ⇐

⇒ X is 0-connected.

X is simply connected ⇐

⇒ X is 1-connected.

X is contractible ⇐

⇒ the identity map on X is null-homotopic.

X is contractible =

⇒ X is n-connected for any n.

Observation Let P ⊆ [0, 1] be a Π0

1 set without computable points.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

X is n-connected ⇐

⇒ the first n + 1 homotopy groups

vanish identically. X is path-connected ⇐

⇒ X is 0-connected.

X is simply connected ⇐

⇒ X is 1-connected.

X is contractible ⇐

⇒ the identity map on X is null-homotopic.

X is contractible =

⇒ X is n-connected for any n.

Observation Let P ⊆ [0, 1] be a Π0

1 set without computable points.

P(n+2) ⊆ [0, 1]n+2 is n-connected, but not n + 1-connected.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

X is n-connected ⇐

⇒ the first n + 1 homotopy groups

vanish identically. X is path-connected ⇐

⇒ X is 0-connected.

X is simply connected ⇐

⇒ X is 1-connected.

X is contractible ⇐

⇒ the identity map on X is null-homotopic.

X is contractible =

⇒ X is n-connected for any n.

Observation Let P ⊆ [0, 1] be a Π0

1 set without computable points.

P(n+2) ⊆ [0, 1]n+2 is n-connected, but not n + 1-connected. P(n) is not contractible for any n.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Observation (Restated) Not every nonempty n-connected Π0

1 set in Rn+2 contains a

computable point, for any n ∈ N.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Observation (Restated) Not every nonempty n-connected Π0

1 set in Rn+2 contains a

computable point, for any n ∈ N. Every nonempty n-connected Π0

1 set in Rn+1 contains a

computable point, for n = 0.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Observation (Restated) Not every nonempty n-connected Π0

1 set in Rn+2 contains a

computable point, for any n ∈ N. Every nonempty n-connected Π0

1 set in Rn+1 contains a

computable point, for n = 0. Question

1

(Le Roux-Ziegler) Does every simply connected planar Π0

1 set

contain a computable point?

2

Does every contractible Euclidean Π0

1 set contain a

computable point?

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Observation (Restated) Not every nonempty n-connected Π0

1 set in Rn+2 contains a

computable point, for any n ∈ N. Every nonempty n-connected Π0

1 set in Rn+1 contains a

computable point, for n = 0. Question

1

(Le Roux-Ziegler) Does every simply connected planar Π0

1 set

contain a computable point?

2

Does every contractible Euclidean Π0

1 set contain a

computable point? Main Theorem Not every nonempty contractible planar Π0

1 set contains a

computable point.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Main Theorem

Main Theorem Not every nonempty contractible planar Π0

1 set contains a

computable point.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Main Theorem

Main Theorem Not every nonempty contractible planar Π0

1 set contains a

computable point. Proof Idea Let P ⊆ R be a Π0

1 set without computable points.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Main Theorem

Main Theorem Not every nonempty contractible planar Π0

1 set contains a

computable point. Proof Idea Let P ⊆ R be a Π0

1 set without computable points.

Stretch [0, 1] × P along a stray snake A whose destination is a fixed incomputable point (i.e., A is Miller’s computable arc whose end-point is an incomputable left-c.e. real).

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Main Theorem

Main Theorem Not every nonempty contractible planar Π0

1 set contains a

computable point. Proof Idea Let P ⊆ R be a Π0

1 set without computable points.

Stretch [0, 1] × P along a stray snake A whose destination is a fixed incomputable point (i.e., A is Miller’s computable arc whose end-point is an incomputable left-c.e. real). All path-component of [0, 1] × P will be bundled at the destination end-point.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Main Theorem

Main Theorem Not every nonempty contractible planar Π0

1 set contains a

computable point. Proof Idea Let P ⊆ R be a Π0

1 set without computable points.

Stretch [0, 1] × P along a stray snake A whose destination is a fixed incomputable point (i.e., A is Miller’s computable arc whose end-point is an incomputable left-c.e. real). All path-component of [0, 1] × P will be bundled at the destination end-point. Thus, the desired Π0

1 set will be homeomorphic to the Cantor

fan.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Proof Idea Stretch [0, 1] × P along a stray snake A. ✓ ✏ Destination of the snake (incomputable point) A snake A (Cantor fan) ([0, 1] × P) ✒ ✑ The desired Π0

1 set D will be homeomorphic to the Cantor fan.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

A fat approximation of Cantor set: ✓ ✏ A construction of Cantor set Fat approx. of Cantor set ✒ ✑ P: a Π0

1 subset of Cantor set.

Ps: a fat approximation of P at stage s. ls, rs: the leftmost and rightmost of Ps.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

A fat approximation of Cantor set: ✓ ✏ A construction of Cantor set Fat approx. of Cantor set ✒ ✑ P: a Π0

1 subset of Cantor set.

Ps: a fat approximation of P at stage s. ls, rs: the leftmost and rightmost of Ps.

[ls, ls+1] ∩ Ps, [rs+1, rs] ∩ Ps contains intervals Il

s, Ir s.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

A fat approximation of Cantor set: ✓ ✏ A construction of Cantor set Fat approx. of Cantor set ✒ ✑ P: a Π0

1 subset of Cantor set.

Ps: a fat approximation of P at stage s. ls, rs: the leftmost and rightmost of Ps.

[ls, ls+1] ∩ Ps, [rs+1, rs] ∩ Ps contains intervals Il

s, Ir s.

We call these intervals Il

s, Ir s ⊆ Ps \ Ps+1 free blocks.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Prepare a stretched Π0

1 class D− 0 = P × [0, 1].

✓ ✏ Body Free block Free block Stretched ✒ ✑ P ⊆ R1: a Π0

1 set without computable points.

Ps: a fat approximation of P (Note that P = ∩

s Ps).

D−

0 = [0, 1] × P0.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

D0 is the following connected closed set. ✓ ✏ Body Free block Free block Stretched ✒ ✑ The desired Π0

1 set D will be obtained by carving D0.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Destination

α ∈ R: an incomputable left-c.e. real.

There is a computable sequence {Js} of rational open intervals s.t.

min Js → α as s → ∞. diam(Js) → 0 as s → ∞. Either Js+1 ⊂ Js or max Js < min Js+1, for each s.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Our construction starts with D0. ✓ ✏ Body Free block Free block Stretched ✒ ✑

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

By carving free blocks, stretch P0 toward max J0. ✓ ✏ max J0 ✒ ✑

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

By carving free blocks, stretch P0 toward min J0. ✓ ✏ min J0 ✒ ✑

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Proceed one step with a fat approximation of P. ✓ ✏ min J0 max J0 ✒ ✑

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

D1 is defined by this, ✓ ✏ Zoom ✒ ✑

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

D1 is defined by this, ✓ ✏ Zoom ✒ ✑ If J1 ⊂ J0, then the construction of D2 is similar as that of D1. i.e., on the top block, stretch toward max J1 and back to min J1, by caving free blocks.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

D1 is defined by this, ✓ ✏ Zoom ✒ ✑ If J1 ⊂ J0, then the construction of D2 is similar as that of D1. i.e., on the top block, stretch toward max J1 and back to min J1, by caving free blocks.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

D1 is defined by this, ✓ ✏ Zoom ✒ ✑ If J1 ⊂ J0, then the construction of D2 is similar as that of D1. i.e., on the top block, stretch toward max J1 and back to min J1, by caving free blocks. In general, similar for Js+1 ⊂ Js.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

D1 is defined by this, ✓ ✏ Zoom ✒ ✑ If J1 ⊂ J0, then the construction of D2 is similar as that of D1. i.e., on the top block, stretch toward max J1 and back to min J1, by caving free blocks. In general, similar for Js+1 ⊂ Js. Only the problem is the case of Js+1 Js!

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

In the case of Js+1 Js: ✓ ✏ Jp Js Js+1 Overview of Ds (above Dp) ✒ ✑ Pick the greatest p ≤ s such that Js+1 ⊂ Jp.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

In the case of Js+1 Js: ✓ ✏ Jp Js+1 Overview of Ds (above Dp) ✒ ✑ Go back to Dp by caving free blocks into the shape of P.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

✓ ✏ Jp Js+1 Overview of Ds (above Dp) ✒ ✑ By caving free blocks on Dp into the shape of P, stretch toward max Js+1 and back to min Js+1.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Main Theorem (Restated) Not every nonempty contractible planar Π0

1 set contains a

computable point. D = ∩

s Ds is Π0 1.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Main Theorem (Restated) Not every nonempty contractible planar Π0

1 set contains a

computable point. D = ∩

s Ds is Π0 1.

D is obtained by bundling [0, 1] × P at (α, y) ∈ R2 for some y.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Main Theorem (Restated) Not every nonempty contractible planar Π0

1 set contains a

computable point. D = ∩

s Ds is Π0 1.

D is obtained by bundling [0, 1] × P at (α, y) ∈ R2 for some y. D is path-connected by the property of an approximation {Js}

• f the incomputable left-c.e. real α.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Main Theorem (Restated) Not every nonempty contractible planar Π0

1 set contains a

computable point. D = ∩

s Ds is Π0 1.

D is obtained by bundling [0, 1] × P at (α, y) ∈ R2 for some y. D is path-connected by the property of an approximation {Js}

• f the incomputable left-c.e. real α.

Therefore, D is homeomorphic to Cantor fan, and contractible.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Main Theorem (Restated) Not every nonempty contractible planar Π0

1 set contains a

computable point. D = ∩

s Ds is Π0 1.

D is obtained by bundling [0, 1] × P at (α, y) ∈ R2 for some y. D is path-connected by the property of an approximation {Js}

• f the incomputable left-c.e. real α.

Therefore, D is homeomorphic to Cantor fan, and contractible. Stretching [0, 1] × P cannot introduce new computable points.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Main Theorem (Restated) Not every nonempty contractible planar Π0

1 set contains a

computable point. D = ∩

s Ds is Π0 1.

D is obtained by bundling [0, 1] × P at (α, y) ∈ R2 for some y. D is path-connected by the property of an approximation {Js}

• f the incomputable left-c.e. real α.

Therefore, D is homeomorphic to Cantor fan, and contractible. Stretching [0, 1] × P cannot introduce new computable points. Of course, (α, y) is also incomputable.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Main Theorem (Restated) Not every nonempty contractible planar Π0

1 set contains a

computable point. D = ∩

s Ds is Π0 1.

D is obtained by bundling [0, 1] × P at (α, y) ∈ R2 for some y. D is path-connected by the property of an approximation {Js}

• f the incomputable left-c.e. real α.

Therefore, D is homeomorphic to Cantor fan, and contractible. Stretching [0, 1] × P cannot introduce new computable points. Of course, (α, y) is also incomputable. Hence, D has no computable points.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Corollary For every Π0

1 class P, there is a contractible planar Π0 1 set D such

that D is Turing-degree-isomorphic to P.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Corollary For every Π0

1 class P, there is a contractible planar Π0 1 set D such

that D is Turing-degree-isomorphic to P. Definition (RCA0) A sequence (Bi)i∈N of open rational balls is disk-like if ∪

i<n Bi is

homeomorphic to (0, 1)2 for any n ∈ N. Corollary The following are equivalent over RCA0: WKL0: Every infinite tree has a path; Heine-Borel: Every covering of [0, 1] has a finite subcovering. Heine-Borel(Disk): Every disk-like covering of [0, 1]2 has a finite subcovering.

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Question What about Medvedev degrees of contractible planar Π0

1 sets?

Does every nonempty locally connected planar Π0

1 set contain

a computable point?

Takayuki Kihara Incomputability of Simply Connected Planar Continua

Introduction Main Theorem

Thank you!

Takayuki Kihara Incomputability of Simply Connected Planar Continua