The space of short ropes and the classifying space of the space of - PowerPoint PPT Presentation

The space of short ropes and the classifying space of the space of long knots Shunji Moriya 1 and Keiichi Sakai 2 1 Osaka Prefecture University moriyasy@gmail.com a 2 Shinshu University ksakai@math.shinshu-u.ac.jp Mathematics of knots, IX Nihon University, December 22, 2016

The space of long knots D 2 : the unit open disk Long knots : embeddings f : R 1 ֒ → R 1 × D 2 (or their images) satisfying x � [0 , 1] = ⇒ f ( t ) = ( t , 0 , 0) . K : = { long knots } with C ∞ -topology Fact . π 0 ( K ) = K / isotopy � { knots in S 3 } / isotopy. K is a topological monoid via concatenation (connected-sum). Thus ▶ π 0 ( K ) is a monoid, and ▶ the classifying space B K can be defined (later). Main Theorem . B K is weakly equivalent to the space R of short ropes . Corollary (J. Mostovoy, 2002). π 1 ( R ) � � π 0 ( K ) (the group completion). Classification of loops on R classification of knots ←→ up to homotopy up to isotopy

The space of short ropes • • Ropes : embeddings r : [0 , 1] ֒ → R 1 × D 2 a short rope satisfying r (0) = (0 , 0 , 0) and r (1) = (1 , 0 , 0) . Short ropes : ropes of length < 3 . • • R : = { short ropes } with C ∞ -topology a non-short rope π 0 ( R ) = { r 0 } , r 0 ( t ) : = ( t , 0 , 0) (the tight rope) • • no classification problem of ropes r 0 Generators of π 1 ( R ) � � π 0 ( K ) (Mostovoy). For f ∈ K , (1) tie f around (0 , 0 , 0) ; r 0 • • → • • → • • f → • • → • • (2) unknot f around (1 , 0 , 0) in a “reversed way”

Classifying spaces of (topological) categories For a (topological) category C , → c k ) ; composable k morphisms } ⊂ Mor × k f 1 f 2 f k N k C : = { ( c 0 − → c 1 − → · · · − C N ∗ C : = { N k C } k ≥ 0 (the nerve of C ) is a simplicial space via compositions / insertion of identities. ⊔ ( N k C × ∆ k ) Definition . B C : = | N ∗ C | = / ∼ : the classifying space of C k ≥ 0 ∆ k = { 0 ≤ t 1 ≤ · · · ≤ t k ≤ 1 } ) ∈ N k C × ∆ k gives an element of B C ▶ ( ( f i ) k i = 1 ; ( t i ) k i = 1 ) = ( . . . , f i + 1 ◦ f i , . . . ; . . . , t i , t i + 2 , . . . ) ∈ B C ▶ t i = t i + 1 = ⇒ ( ( f i ) i ; ( t i ) i ) = ( . . . , f i − 1 , f i + 1 , . . . ; . . . , t i , t i + 2 , . . . ) ∈ B C ▶ f i = id = ⇒ ( ( f i ) i ; ( t i ) i In the following ▶ C = K : the category of long knots, f 1 f 2 f k • • • ▶ f i ⇐⇒ long knots, t 1 t 2 t k ▶ composition ⇐⇒ connected-sum “Connected-sum of long knots”

The space ψ s of “long” 1 -manifolds M A : = M ∩ ( A × D 2 ) for A ⊂ R 1 and a manifold M ⊂ R 1 × D 2 Definition (S. Galatius, O. Randal-Williams, 2010). ψ s : = { M 1 ⊂ R 1 × D 2 w/o boundary | ▶ M T is compact for ∀ T ∈ R 1 , ▶ ∀ connected component of M is “long” in at least one direction of R 1 , ▶ exactly one comp. L ⊂ M is “long” in both directions; L T � ∅ for ∀ T ∈ R 1 , ▶ ∃ at least one T ∈ R 1 s.t. M T is a one point set } M T ∈ ψ s • L T Topologize ψ s so that “ M is close to N if they are close in a compact set.” Example . M ( T ) T →∞ ∈ ψ s − − − − → T T + 1

The space of long knots as a topological category The category K ; Ob( K ) = D 2 , Mor K ( p , q ) = { ( T , M ) ∈ R 1 ≥ 0 × ψ s | M connected, ∃ ϵ > 0 s.t. M ( −∞ ,ϵ ] = { p } × ( −∞ , ϵ ] , M [ T − ϵ, ∞ ) = { q } × [ T − ϵ, ∞ ) } × × p q 0 T ▶ Mor K ( p , q ) ≃ { long knots } , ▶ N k K = {( 0 ≤ T 1 ≤ · · · ≤ T k ; f ) | f T i are one point sets } . ( f = f [0 , T 1 ] # f [ T 1 , T 2 ] # · · · # f [ T k − 1 , T k ] ) Want to know B K = | N ∗ K| .

The space of long knots as a topological category The partially ordered sets (posets) D and D ⊥ ; D : = { ( T , M ) ∈ R × ψ s | M T is a one point set } , D ⊥ : = { ( T , M ) ∈ D | ∃ ϵ > 0 s.t. M ( T − ϵ, T + ϵ ) = M T × ( T − ϵ, T + ϵ ) } ⊂ D , def ⇐⇒ M = M ′ and T ≤ T ′ ( T , M ) ≤ ( T ′ , M ′ ) ∈ D ⊥ ∈ D T T Posets are categories;  x ≤ y ∈ D ( ⊥ ) , {∗}  Ob( D ( ⊥ ) ) : = D ( ⊥ ) ,  Mor D ( ⊥ ) ( x , y ) : =   ∅ otherwise.   Mor D = { ( T 0 ≤ T 1 ; M ) | M T i are one point sets } , N k D = { ( T 0 ≤ · · · ≤ T k ; M ) | M T i are one point sets } Remark . All the “half-long” components ⊂ ( ( −∞ , T 0 ] ⊔ [ T k , ∞ ) ) × D 2 .

The classifying space of long knots Theorem (essentially due to S. Galatius and O. Randal-Williams). ≃ − N ∗ D ⊥ ≃ ∃ simplicial maps N ∗ D ← − → N ∗ K that are levelwise homotopy ≃ − B D ⊥ ≃ equivalences. Thus B D ← − → B K . Main point: N k D ⊥ → N k K → N k D ⊥ is given by “cut-off” ⇝ T 0 T k T k − T 0 0 This is homotopic to id by the definition of the topology of ψ s ; s →−∞ , t → + ∞ s t

� � � � � ψ s is the classifying space of long knots ) �→ M . N ∗ D × ∆ ∗ ∋ ( ( T i ) i ; M ) , ( t i ) i ∃ u : B D → ψ s , induced by Theorem (essentially due to Galatius and Randal-Williams). The map u is a weak equivalence. Thus B K ∼ ψ s . Outline of proof. Want to show π m ( ψ s , B D ) = 0 for ∀ m . ? Given the strict arrows = ⇒ ∃ the dotted g ? f ∂ D m B D � � g u f ψ s D m ∀ a ∈ R , U a : = { x ∈ D m | f ( x ) a is a one point set } = ⇒ { U a } a ∈ R is an open covering of D m . Pick a finitely many subcover U = { U a i } i and a partition of unity { λ i } i subordinate to U . ) ∈ N ∗ D × ∆ ∗ . Roughly g ( x ) : = ( (( a i ) i ; f ( x )) , ( λ i ( x )) i

ψ s is the space of short ropes R : = { short ropes } ∋ r = ⇒ length ( r ) < 3 Remark . R ֒ → { r : rope | r t is a one point set for ∃ t ∈ (0 , 1) } . r • • • • t a short rope a non-short rope Lemma . The above inclusion is a weak equivalence. Below R : = { r : rope | r t is a one point set for ∃ t ∈ (0 , 1) } . ≈ Fix f : (0 , 1) − → R . Theorem (Moriya-S). The “cut-off” map c : R → ψ s , c ( r ) : = ( f × id D 2 )( r (0 , 1) ) , is a weak equivalence. Thus B K ∼ R . • r • c 0 1

ψ s is the space of short ropes The posets (categories) E ⊥ ⊂ E ; E : = { ( t , r ) ∈ (0 , 1) × R | r t is a one point set } E ⊥ : = { ( t , r ) ∈ E | ∃ ϵ > 0 s.t. r ( t − ϵ, t + ϵ ) = r t × ( t − ϵ, t + ϵ ) } def ( t , r ) ≤ ( t ′ , r ′ ) ⇐⇒ r = r ′ , t ≤ t ′ • • • • ∈ E ⊥ ∈ E r r 0 t 1 0 t 1 Mor E ( ⊥ ) = { (0 < t 0 ≤ t 1 < 1; r ) | r t i are one point sets } N k E = { (0 < t 0 ≤ · · · ≤ t k < 1; r ) | r t i are one point sets } Theorem (essentially due to Galatius and Randal-Williams). ∃ (weak) equivalences B E ⊥ ≃ ∼ − → B E − → R . Proof. Similar to the proof of B D ⊥ ≃ ∼ − → B D − → ψ s .

� � � ψ s is the space of short ropes Theorem . A simplicial map Φ : N ∗ E ⊥ → N ∗ D ⊥ , Φ ( t 0 ≤ · · · ≤ t k ; r ) : = ( T 0 ≤ · · · ≤ T k ; c ( r )) where T i : = f ( t i ) is a levelwise homotopy equivalence. Thus B E ⊥ ≃ → B D ⊥ . − Main point: N k E ⊥ Φ → N k D ⊥ → N k E ⊥ “unknots r around the endpoints” − • • r • • t 0 t k t 0 t k 0 1 0 1 This is homotopic to id ; ∃ a canonical way to unknot r ( −∞ , t 0 ] and r [ t k , ∞ ) (Mostovoy). Conclusion . c ψ s R ∼ ∼ ⟳ ∼ Φ ∼ � B D ⊥ “cut-off” � B K B E ⊥ ≃