Double quandle coverings Franois Renaud Universit catholique de - PowerPoint PPT Presentation

Double quandle coverings François Renaud Université catholique de Louvain Institut de Recherche en Mathématique et Physique Funding: FRIA & Bourses de voyage de la Communauté française Edinburgh CT2019

? Double quandle coverings ? Previous work: D.E. Joyce (1979) (Supervised by Peter J. Freyd) An algebraic approach to symmetry and applications in knot theory M. Eisermann (2007) Quandle coverings and their Galois correspondence V. Even (2014) A Galois-Theoretic Approach to the Covering Theory of Quandles A motivation: Illustrate, in algebra, an instance of Galois theory with geometrical intuition – display homotopical information... also in higher dimensions A first step: "What are double central extensions for quandles ?" [Same question for groups in 1991: R.Brown asks G.Janelidze]

Categorical Galois theory [G.Janelidze 1990]

� � � � � � �� �� � � � � � �� � Higher categorical Galois theory I 1 • Start: Galois structure in dimension 1 � ⇐ F 1 α 1 A 1 B 1 � f B consider the category of extensions Ext: α ↔ f A f A �� f B � B 0 A 0 α 0 I 2 Get: Galois structure in dimension 2 CExt Ext ⇐ F 2 α 1 A 1 B 1 A 0 × B 0 B 1 good notion of double extensions: f A f B � � � B 0 A 0 α 0 Question: "What are double central extensions ?"

� � � context – Racks [Conway and Wraith 1959] Definition: A set X equipped with: symmetries/inner-automorphisms two inverse and self distributive assigned to each point binary operations S ⊳ S − 1 � X X X × X ⊳ − 1 � X , X S y ( x ) ⇔ x ⊳ y (R1) ( x ⊳ y ) ⊳ − 1 y = x = ( x ⊳ − 1 y ) ⊳ y (R2) ( x ⊳ y ) ⊳ z = ( x ⊳ z ) ⊳ ( y ⊳ z )

� � � context – Racks [Conway and Wraith 1959] Definition: A set X equipped with: symmetries/inner-automorphisms two inverse and self distributive assigned to each point binary operations S ⊳ X × X ⊳ − 1 � X , X S − 1 � Aut( X ) S y ( x ) ⇔ x ⊳ y (R1) ( x ⊳ y ) ⊳ − 1 y = x = ( x ⊳ − 1 y ) ⊳ y (R2) ( x ⊳ y ) ⊳ z = ( x ⊳ z ) ⊳ ( y ⊳ z )

� � � context – Racks [Conway and Wraith 1959] Definition: A set X equipped with: symmetries/inner-automorphisms two inverse and self distributive assigned to each point binary operations S ⊳ X × X ⊳ − 1 � X , X S − 1 � Aut( X ) S y ( x ) ⇔ x ⊳ y (R1) x ⊳ y ⊳ − 1 y = x = x ⊳ − 1 y ⊳ y (R2’) x ⊳ ( y ⊳ z ) = x ⊳ − 1 z ⊳ y ⊳ z

Examples – Quandles A rack X is a quandle if moreover (idempotency) (Q1) x ⊳ x = x For instance: 1 Sets y ⊳ x = x I : Set → Qnd → Rac 2 Groups Conj: Grp → Qnd → Rac ( G , · , e ) �→ ( G , ⊳, ⊳ − 1 ) . = y − 1 xy x ⊳ y . 3 Knot quandles 4 Symmetric spaces [O. Loos 1969]

� � Connected components adjunction I Set Rac ⇐ π 0 Define: Elements x and y in a rack X are connected ( x ∼ X y ) y = x ⊳ δ 1 a 1 · · · ⊳ δ n a n if there is a primitive path from x to y : a 1 δ 1 ... a n δ n x y > ��� Send X to π 0 ( X ) . . = X / ∼ X it’s set of connected components

Primitive paths – Observations – Invert / concatenate primitive paths: • • • • • > > > < > > – A lot of different prim. paths from x to y x y . . . > – Prim. paths which could be equivalent ? Using axiom (R1) x ⊳ δ 1 a 1 · · · ⊳ δ n a n = x ⊳ δ 1 a 1 · · · ⊳ δ n a n ⊳ − 1 z ⊳ z Using axiom (R2), say a i = y ⊳ z x ⊳ δ 1 a 1 · · · ⊳ δ i ( y ⊳ z ) · · · ⊳ δ n a n = x ⊳ δ 1 a 1 · · · ⊳ − δ i z ⊳ y ⊳ δ i z · · · ⊳ δ n a n

The group of paths – homotopy equivalent prim. paths Define the functor Pth � Grp Pth( A ) . . = F g ( A ) / � ( x ⊳ a ) − 1 a − 1 xa | a , x ∈ A � Rac Representatives of the symmetries: pth A : a ∈ A �→ a ∈ Pth( A ) Action by inner-automorphisms: given g = a 1 δ 1 . . . a n δ n in Pth( A ) x . g = x . ( a 1 δ 1 . . . a n δ n ) = x ⊳ δ 1 a 1 · · · ⊳ δ n a n g x x . g > The group of paths is left adjoint to Conj: Grp → Rac

� � � � � � � Commutative square of adjunctions I Rac Set ⇒ π 0 Conj F ab ⇓ Pth ⇓ U I Grp � Ab . ⇒ ab

The free rack [R. Fenn and C. Rourke 1991] Given a set A the free rack is F r ( A ) . . = A ⋊ F g ( A ) g elements are pairs ( a , g ) a << a . g >> > A path acts on another « with its codomain » ( a , g ) ⊳ ( b , h ) = ( a , gh − 1 bh ) Unit: A → F r ( A ): a �→ ( a , e ) ⋆ The group of paths Pth(F r ( A )) = F g ( A ) acts freely on F r ( A ) : g = g 1 δ 1 · · · g n δ n ∈ F g ( A ) ( a , h ) . g = ( a , h ) ⊳ δ 1 ( g 1 , e ) · · · ⊳ δ n ( g n , e ) = ( a , hg 1 δ 1 · · · g n δ n ) = ( a , hg )

� ✤ � � � � � � Trivial extensions Definition: η A A π 0 ( A ) t π 0 ( t ) � π 0 ( B ) . B η B Characterization: A path sent to a loop was already a loop Pth[ t ]( g ) g g t ( a a . g ) t ( a ) = t ( a . g ) ⇒ a = a . g

� � � � � Characterization of central extensions ? Objective: condition on extention c s.t. there is p such that ¯ c is trivial p ¯ E × B A A c c ¯ � B . E ∃ p Condition [Eisermann]: c is a covering if c ( a ) = c ( b ) ⇒ x ⊳ a = x ⊳ b Geometric interpretation: x x c a 1 a 2 a 3 � b 1 b 2 b 3 � b 1 b 2 b 3 a 1 a 2 a 3 ⇒ c c x . ( a 1 a 2 a 3 ) x . ( b 1 b 2 b 3 ) x . ( a 1 a 2 a 3 ) = x . ( b 1 b 2 b 3 )

� �� � � �� � � � � � �� � Characterization of central extensions [V.Even 2014] – new proof Objective: Test if t is trivial: if t sends a path l to a loop ˜ t ( l ) c a covering ⇒ t trivial l P A ( x x . ( l )) ❴ s t t c ① � B . F r (U B ) ǫ B t ( x ) = t ( x ) . (˜ t ( l )) 1 Downstairs: paths act freely ⇒ loops are trivial ⇒ ˜ t ( l ) = e 2 Send trivial loop back up via splitting s : s ˜ ˜ t ( l ) = e 3 Upstairs: path l and loop ˜ s ˜ t ( l ) act the same because t is a covering. x x t ( st ( a 1 ) st ( a 2 ) st ( a 3 )) a 1 a 2 a 3 � e l ⇒ t t s ˜ x . ( l ) x . (˜ t ( l )) x . ( l ) = x

� � � � � � Towards higher dimensions g X C D × Y C Double extension f � Y D Condition for double covering ? 1-dimensional covering : act on x ∈ X with 1-dimensional data a b f 2-dimensional covering : act on x ∈ X with 2-dimensional data a b f g g a ′ b ′ f

Double covering

Commutator condition Given: quandle X congruences R and S Define: [ R , S ] the congruence generated by the pairs ( x ⊳ a ⊳ − 1 b , x ⊳ c ⊳ − 1 d ) for any x , a , b , c and d in X such that a b R S S d . c R 1 [ R , S ] = [ S , R ] ⊂ R ∩ S 2 [ X × X , X × X ] = ∼ X i.e. connectedness 3 1-dimensional centrality ⇔ ( [Eq( f ) , X × X ] = ∆ X ) 4 2-dimensional centrality ⇔ ( [Eq( f ) , Eq( g )] = ∆ X )

� � � � � � � � Double trivial coverings g X C D × Y C A double extension is trivial iff in X : f � Y D x x g g a 1 a 2 a 3 � b 1 b 2 b 3 a 3 a 2 a 1 � b 1 b 2 b 3 g ⇒ g g g y x . ( a 1 a 2 a 3 ) x . ( b 1 b 2 b 3 ) f y = x . ( a 1 a 2 a 3 ) = x . ( b 1 b 2 b 3 )

� �� � � � � � �� � � Characterization of double central extensions Objective: Test if t is trivial: ✤ t � closed membr. c a double covering ⇒ t trivial ? if open membr. P A x a 1 a 2 a 3 b 1 b 2 b 3 s t c ⑤ � B . x . ( a 1 a 2 a 3 ) x . ( b 1 b 2 b 3 ) t F q • 1 closed membrane = trivial loop in the domain of F • 2 obtain trivial loop in P via splitting s 3 closed membrane above the open membrane, fitting into a cone...

References M. Eisermann. Quandle coverings and their Galois correspondence. Fund. Math. , 225(1):103–168, 2014. V. Even. A Galois-theoretic approach to the covering theory of quandles. Appl. Categ. Structures , 22(5–6):817–831, 2014. R. Fenn and C. Rourke. Racks and links in codimension two. J. Knot Theory Ramifications , 1(4):343–406, 1992. G. Janelidze. Pure Galois Theory in Categories. J. Algebra , 132:270–286, 1990. G. Janelidze, What is a double central extension? (The question was asked by Ronald Brown), Cah. Top. Géom. Diff. Catég. XXXII (1991), no. 3, 191–201. D. Joyce. An Algebraic Approach to Symmetry With Applications to Knot Theory. PhD thesis, University of Pennsylvania , 1979.

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