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 1 set in R n contains a computable point. 0 Every nonempty Σ ∼ Takayuki Kihara Incomputability of Simply Connected Planar Continua
Introduction Main Theorem Introduction 1 set in R n contains a computable point. 0 Every nonempty Σ ∼ 1 set in R n contains a computable point. Not every nonempty Π 0 Takayuki Kihara Incomputability of Simply Connected Planar Continua
Introduction Main Theorem Introduction 1 set in R n contains a computable point. 0 Every nonempty Σ ∼ 1 set in R n contains a computable point. Not every nonempty Π 0 1 subset F ⊆ R 1 contains no computable If a nonempty Π 0 points, then F must be disconnected . Takayuki Kihara Incomputability of Simply Connected Planar Continua
Introduction Main Theorem Introduction 1 set in R n contains a computable point. 0 Every nonempty Σ ∼ 1 set in R n contains a computable point. Not every nonempty Π 0 1 subset F ⊆ R 1 contains no computable If a nonempty Π 0 points, then F must be disconnected . Does there exist a nonempty (simply) connected Π 0 1 set in R n without computable points? Takayuki Kihara Incomputability of Simply Connected Planar Continua
Introduction Main Theorem Introduction 1 set in R n contains a computable point. 0 Every nonempty Σ ∼ 1 set in R n contains a computable point. Not every nonempty Π 0 1 subset F ⊆ R 1 contains no computable If a nonempty Π 0 points, then F must be disconnected . Does there exist a nonempty (simply) connected Π 0 1 set in R n 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 { B e } e ∈ N : an effective enumeration of all rational open balls. x ∈ R n is computable if { e ∈ N : x ∈ B e } is c.e. 1 Equivalently, x = ( x 1 , . . . , x n ) ∈ R n is computable iff x i is computable for each i ≤ n . Takayuki Kihara Incomputability of Simply Connected Planar Continua
Introduction Main Theorem Definition { B e } e ∈ N : an effective enumeration of all rational open balls. x ∈ R n is computable if { e ∈ N : x ∈ B e } is c.e. 1 Equivalently, x = ( x 1 , . . . , x n ) ∈ R n is computable iff x i is computable for each i ≤ n . F ⊆ R n is Π 0 1 (or co-c.e. closed ) if F = R n \ ∪ e ∈ W B e for a 2 c.e. set W . Takayuki Kihara Incomputability of Simply Connected Planar Continua
Introduction Main Theorem Definition { B e } e ∈ N : an effective enumeration of all rational open balls. x ∈ R n is computable if { e ∈ N : x ∈ B e } is c.e. 1 Equivalently, x = ( x 1 , . . . , x n ) ∈ R n is computable iff x i is computable for each i ≤ n . F ⊆ R n is Π 0 1 (or co-c.e. closed ) if F = R n \ ∪ e ∈ W B e for a 2 c.e. set W . 1 set in R 1 contains a computable point. Not every nonempty Π 0 1 Takayuki Kihara Incomputability of Simply Connected Planar Continua
Introduction Main Theorem Definition { B e } e ∈ N : an effective enumeration of all rational open balls. x ∈ R n is computable if { e ∈ N : x ∈ B e } is c.e. 1 Equivalently, x = ( x 1 , . . . , x n ) ∈ R n is computable iff x i is computable for each i ≤ n . F ⊆ R n is Π 0 1 (or co-c.e. closed ) if F = R n \ ∪ e ∈ W B e for a 2 c.e. set W . 1 set in R 1 contains a computable point. Not every nonempty Π 0 1 1 set in R n contains ( Category ) Every nonempty co-meager Π 0 2 a computable point. Takayuki Kihara Incomputability of Simply Connected Planar Continua
Introduction Main Theorem Definition { B e } e ∈ N : an effective enumeration of all rational open balls. x ∈ R n is computable if { e ∈ N : x ∈ B e } is c.e. 1 Equivalently, x = ( x 1 , . . . , x n ) ∈ R n is computable iff x i is computable for each i ≤ n . F ⊆ R n is Π 0 1 (or co-c.e. closed ) if F = R n \ ∪ e ∈ W B e for a 2 c.e. set W . 1 set in R 1 contains a computable point. Not every nonempty Π 0 1 1 set in R n contains ( Category ) Every nonempty co-meager Π 0 2 a computable point. ( Measure ) Not every nonempty positive measure Π 0 1 set in R 1 3 contains a computable point. Takayuki Kihara Incomputability of Simply Connected Planar Continua
Introduction Main Theorem Definition { B e } e ∈ N : an effective enumeration of all rational open balls. x ∈ R n is computable if { e ∈ N : x ∈ B e } is c.e. 1 Equivalently, x = ( x 1 , . . . , x n ) ∈ R n is computable iff x i is computable for each i ≤ n . F ⊆ R n is Π 0 1 (or co-c.e. closed ) if F = R n \ ∪ e ∈ W B e for a 2 c.e. set W . 1 set in R 1 contains a computable point. Not every nonempty Π 0 1 1 set in R n contains ( Category ) Every nonempty co-meager Π 0 2 a computable point. ( Measure ) Not every nonempty positive measure Π 0 1 set in R 1 3 contains a computable point. ( Connectedness ) What about connected , simply connected , 4 or contractible Π 0 1 sets? Takayuki Kihara Incomputability of Simply Connected Planar Continua
Introduction Main Theorem Connected Π 0 1 Sets Observation 1 subset P ⊆ R 1 contains a Every nonempty connected Π 0 1 computable point. Takayuki Kihara Incomputability of Simply Connected Planar Continua
Introduction Main Theorem Connected Π 0 1 Sets Observation 1 subset P ⊆ R 1 contains a Every nonempty connected Π 0 1 computable point. Fact 1 subset P ( 2 ) ⊆ R 2 There exists a nonempty connected Π 0 1 without computable points. There exists a nonempty simply connected Π 0 1 subset 2 P ( 3 ) ⊆ R 3 without computable points. Takayuki Kihara Incomputability of Simply Connected Planar Continua
． ． Introduction Main Theorem Example ✓ ✏ P ( 2 ) P ✒ ✑ 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 ( 2 ) P ✒ ✑ P ( n ) = ∪ k < n ([ 0 , 1 ] k × P × [ 0 , 1 ] n − k − 1 ) for P ⊆ [ 0 , 1 ] . Let P ⊆ [ 0 , 1 ] be a Π 0 1 set without computable points. Takayuki Kihara Incomputability of Simply Connected Planar Continua
． Introduction Main Theorem Example ✓ ✏ P ( 2 ) P ✒ ✑ P ( n ) = ∪ k < n ([ 0 , 1 ] k × P × [ 0 , 1 ] n − k − 1 ) for P ⊆ [ 0 , 1 ] . Let P ⊆ [ 0 , 1 ] be a Π 0 1 set without computable points. P ( 2 ) ⊆ [ 0 , 1 ] 2 is a connected Π 0 1 set without computable points ． Takayuki Kihara Incomputability of Simply Connected Planar Continua
Introduction Main Theorem Example ✓ ✏ P ( 2 ) P ✒ ✑ P ( n ) = ∪ k < n ([ 0 , 1 ] k × P × [ 0 , 1 ] n − k − 1 ) for P ⊆ [ 0 , 1 ] . Let P ⊆ [ 0 , 1 ] be a Π 0 1 set without computable points. P ( 2 ) ⊆ [ 0 , 1 ] 2 is a connected Π 0 1 set without computable points ． P ( 3 ) ⊆ [ 0 , 1 ] 3 is a simply connected Π 0 1 set without computable points ． Takayuki Kihara Incomputability of Simply Connected Planar Continua
Introduction Main Theorem ⇒ the first n + 1 homotopy groups X is n -connected ⇐ 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 ⇒ the first n + 1 homotopy groups X is n -connected ⇐ 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 Π 0 1 set without computable points. Takayuki Kihara Incomputability of Simply Connected Planar Continua
Introduction Main Theorem ⇒ the first n + 1 homotopy groups X is n -connected ⇐ 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 Π 0 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 ⇒ the first n + 1 homotopy groups X is n -connected ⇐ 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 Π 0 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) 1 set in R n + 2 contains a Not every nonempty n -connected Π 0 computable point, for any n ∈ N . Takayuki Kihara Incomputability of Simply Connected Planar Continua
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