Classifying spaces of quandles and low dimensional topology - PowerPoint PPT Presentation

Classifying spaces of quandles and low dimensional topology 野坂 武史 Takefumi Nosaka Kyoto univ. RIMS 京都大学 数理解析研究所

Introduction X : a quandle BX : a rack space [95, Fenn-Rourke-Sanderson] [FRS] studied BX , particularly, π ∗ ( BX ) and a “ link bordism ” .

Introduction X : a quandle BX : a rack space [95, Fenn-Rourke-Sanderson] [FRS] studied BX , particularly, π ∗ ( BX ) and a “ link bordism ” . Quandle cocycle invariant (Carter-Jelsovsky-Kamada -Langford-Saito, 99)   L : S 1 ֒ → S 3 or Σ g ֒  → S 4    Quandle cocycle invariant X : finite quandle  � Φ ϕ ( L ) ∈ Z [ A ]    ϕ ∈ H 3 ( BX ; A ) with a condition

Introduction X : a quandle BX : a rack space [95, Fenn-Rourke-Sanderson] [FRS] studied BX , particularly, π ∗ ( BX ) and a “ link bordism ” . Quandle cocycle invariant (Carter-Jelsovsky-Kamada -Langford-Saito, 99)   L : S 1 ֒ → S 3 or Σ g ֒  → S 4    Quandle cocycle invariant X : finite quandle  � Φ ϕ ( L ) ∈ Z [ A ]    ϕ ∈ H 3 ( BX ; A ) with a condition Questions • What does the space BX classify? • How about more applications to low-dim. topology?

The content of this talk § 1 Definition of quandles and examples § 2 Review of classifying spaces BX § 3 X -colorings and their homotopy groups π 2 ( BX ). § 4 Some applications to low-dimensional topology

{ X : a set A quandle ( X, ∗ ) is a pair satisfying ∗ : X × X − → X • ∀ x ∈ X, x ∗ x = x • ∀ y ∈ X , • ∗ y : X − → X is a bijection. • ∀ x, y, z ∈ X, ( x ∗ y ) ∗ z = ( x ∗ z ) ∗ ( y ∗ z ) Ex. Conjugacy quandle x ∗ y def = y − 1 xy ∀ x, y ∈ X X = G grp. Ex. Alexander quandle on a finite field F q : ( F q , ∗ ω ) ω ∈ F q \ { 0 , 1 } ωa x ∗ ω y def = y + ω ( x − y ) y x ∗ a x y ( • ∗ ω y ) = ω multiple centered at y

Ex. The fundamental quandle Q ( M, N ) N ⊂ M : an oriented manifold pair of codimension 2. = { * Q ( M, N ) def → ( M, N ) } / homotopy ∞ ∞ * def� Fact (Joyce, Matveev) K 1 , K 2 Knots S 1 ֒ → S 3 emb. ⇒ ∃ quand. isom. Q ( S 3 , K 1 ) ∼ = Q ( S 3 , K 2 ) K 1 ≃ K 2 isotopic ⇐ cf. ∃ K, K ′ ⊂ S 3 s.t. K ̸≃ K ′ & π 1 ( S 3 \ K ) ∼ = π 1 ( S 3 \ K ′ ).

X : a quandle over-arc D : an oriented link diagram ⊂ S 2 An X -coloring of D is a map C : { over-arcs } → X satisfying α β C ( α ) ∗ C ( β ) = C ( γ ) γ D : a digram of a link L ⊂ S 3 Properties • { X -coloring of D } 1:1 → Hom Qnd ( Q ( S 3 , L ) , X ) − β β γ α α γ

∪ Rack space (Fenn-Rourke-Sanderson) BX def = ( d -skeleton) 1-skeleton 2-skeleton = (( a, b )-cells) ∪ 1-skeleton a . . . . . X ∋ b . . . b b . . X ∋ a . . . a ∗ b 3-skeleton=(( a, b, c )-cells) ∪ 2-skeleton a b a ∗ b b c c c a ∗ c b ∗ c ( a ∗ b ) ∗ c = ( a ∗ c ) ∗ ( b ∗ c )

∪ Rack space (Fenn-Rourke-Sanderson) BX def = ( d -skeleton) 1-skeleton 2-skeleton = (( a, b )-cells) ∪ 1-skeleton a . . . . . X ∋ b . . . b b . . X ∋ a . . . a ∗ b 3-skeleton=(( a, b, c )-cells) 4-skeleton=(( a, b, c, d )-cells) ∪ 2-skeleton ∪ 3-skeleton a b a ∗ b b c c c a ∗ c b ∗ c ( a ∗ b ) ∗ c = ( a ∗ c ) ∗ ( b ∗ c )

∪ Rack space (Fenn-Rourke-Sanderson) BX def = ( d -skeleton) 1-skeleton 2-skeleton = (( a, b )-cells) ∪ 1-skeleton a . . . . . X ∋ b . . . b b . . X ∋ a . . . a ∗ b 3-skeleton=(( a, b, c )-cells) 4-skeleton=(( a, b, c, d )-cells) ∪ 2-skeleton ∪ 3-skeleton a b a ∗ b b c c c a ∗ c b ∗ c ( a ∗ b ) ∗ c = ( a ∗ c ) ∗ ( b ∗ c ) Rem. BX was defined by a fat realization of a “ cubical set ” .

Known results on the rack homology H ∗ ( BX ) = H R ∗ ( X ) ℓ : # of “ connected components ” of X • (03, Etingof-Gra˜ na) = Q ℓ n ⇒ H n ( BX ; Q ) ∼ | X | < ∞ =

Known results on the rack homology H ∗ ( BX ) = H R ∗ ( X ) ℓ : # of “ connected components ” of X • (03, Etingof-Gra˜ na) = Q ℓ n ⇒ H n ( BX ; Q ) ∼ | X | < ∞ = • (03, T. Mochizuki) X = F q , ω ∈ F q , x ∗ y = ωx +(1 − ω ) y He determined H 2 ⊕ H 3 ( BX ; F q ) with their base. • (09, N.) Let q = p . ( ) He determined the quandle homologies H Q ∗ ( X ; Z ) ⊂ H ∗ ( BX ; Z ) • (10, Clauwens) Let q = p and ω = − 1 He determined the rack homology H ∗ ( BX ; Z ).

Known results on the rack homology H ∗ ( BX ) = H R ∗ ( X ) ℓ : # of “ connected components ” of X • (03, Etingof-Gra˜ na) = Q ℓ n ⇒ H n ( BX ; Q ) ∼ | X | < ∞ = • (03, T. Mochizuki) X = F q , ω ∈ F q , x ∗ y = ωx +(1 − ω ) y He determined H 2 ⊕ H 3 ( BX ; F q ) with their base. • (09, N.) Let q = p . ( ) He determined the quandle homologies H Q ∗ ( X ; Z ) ⊂ H ∗ ( BX ; Z ) • (10, Clauwens) Let q = p and ω = − 1 He determined the rack homology H ∗ ( BX ; Z ). Next, we discuss π ∗ ( BX ) by low-dim. topology.

{ } We can have ( C, D ) : X -coloring C of D C,D → π 2 ( BX ) { } Π 2 ( X ) def = ( C, D ) C,D / R-II, III moves, concordance rel. a a a � a FACT (Fenn-Rourke-Sanderson) ( cf. Thom’s fund. theorem) ∀ X quandle. There exists an isom. Π 2 ( X ) ∼ = π 2 ( BX ). Rem Π n ( X ) → π n ( BX ) is known. But whether it is an isom. or not is unknown for n > 2. Rem (What is the quandle cocycle invariant [CJKLS] ？ ) ⟨ ϕ, • ⟩ H ϕ ∈ H 2 ( BX ; A ) π 2 ( BX ) − → H 2 ( BX ; A ) − → A

How to compute the homotopy grp, π 2 ( BX ) & π 3 ( BX ) Top. monoid str. on the universal cov. of BX by Clauwens π 1 ( BX ) ∼ = Adj( X ) := ⟨ x ∈ X | x · y = y · ( x ∗ y ) ⟩ ∪ ( Adj( X ) × ([0 , 1] × X ) n ) � BX ≃ / ∼ n ≥ 0 µ : ( G × [0 , 1] n × X n ) × ( G × [0 , 1] m × X m ) → G × [0 , 1] n + m × X n + m , µ ([ g ; t 1 , . . . , t n , x 1 , . . . , x n ] , [ h ; t ′ 1 , . . . , t ′ m , x ′ 1 , . . . , x ′ m ]) := [ gh ; t 1 , . . . , t n , t ′ 1 , . . . , t ′ m , x 1 ∗ h, . . . , x n ∗ h, x ′ 1 , . . . , x ′ m ] , Rem. π 1 ( BX ) is non-comm. grp. So BX admits no t.p.l monoid str. Classical Fact The 2-nd Postnikov inv. of connected t.p.l monoid is annihilated by 2.

Thm. (10, N.) X = F q with p > 2. ω ∈ F q , x ∗ y = ωx + (1 − ω ) y = ⇒ ∃ a splitting exact sequence → Λ 2 ( ) 0 − → π 2 ( BX ) − → H 3 ( BX ; Z ) − H 2 ( BX ; Z ) − → 0 Exa. (N.) If q = p h and ω = − 1, ( ) = h 2 ( h 2 +11) + 1. then dim π 2 ( BX ) ⊗ Z p 12

Thm. (10, N.) X = F q with p > 2. ω ∈ F q , x ∗ y = ωx + (1 − ω ) y = ⇒ ∃ a splitting exact sequence → Λ 2 ( ) 0 − → π 2 ( BX ) − → H 3 ( BX ; Z ) − H 2 ( BX ; Z ) − → 0 Exa. (N.) If q = p h and ω = − 1, ( ) = h 2 ( h 2 +11) + 1. then dim π 2 ( BX ) ⊗ Z p 12 → S 4 ) Thm (10, N.) (On π 3 ( BX ) vs. knotted surfaces Σ g ֒ Further, if X satisfies the vanishing H Q 2 ( X ; Z ) ∼ = 0, = ⇒ π 3 ( BX ) ∼ = Z 2 ⊕ H 4 ( BX ; Z ) . Cor. (N.) If q = p and ω = − 1, then π 3 ( BX ) ∼ = Z 2 ⊕ ( Z p ) 2 .

Some Applications to low-dimensional topology (I) Closed 3-mfds via branched covering spaces M → S 3 Fact ∀ M , ∃ L ⊂ S 3 s.t. M = 4-fold bran. cov. along L . Prop. (09, E. Hatakenaka) ∀ G grp, ∃ a quandle � G s.t. → { Q ( S 3 , L ) f 1:1 → � Hom( π 1 ( M ) , G ) × G ← G | · · · } N. constructed a 3-mfd inv. ∈ Z [ π 2 ( B � G ) / ∼ ] • (10, Hatakenaka-N.) G ) → H gr ∀ a grp G , we constructed an epi. π 2 ( B � 3 ( G ; Z ). We further related the link invariant to the Dijkgraaf- Witten inv. of 3-mfds.

(II) Lefshetz fibrations over the 2-sphere. Dehn quandle D g := { simple closed curves γ ⊂ Σ g } / isotopy x ∗ y := Dehn twist of x along y . Lem. (Y. Matsumoto, D. Yetter) ρ { LF over S 2 } { Q ( S 2 , n) → D g qnd. hom. | ρ ( c 1 ) · · · ρ ( c n ) = 1 M g } 1:1 ← → “ conjugacy actions ” Q ( S 2 , n) � B n ( S 2 ) , isom. D g � M g

(II) Lefshetz fibrations over the 2-sphere. Dehn quandle D g := { simple closed curves γ ⊂ Σ g } / isotopy x ∗ y := Dehn twist of x along y . Lem. (Y. Matsumoto, D. Yetter) ρ { LF over S 2 } { Q ( S 2 , n) → D g qnd. hom. | ρ ( c 1 ) · · · ρ ( c n ) = 1 M g } 1:1 ← → “ conjugacy actions ” Q ( S 2 , n) � B n ( S 2 ) , isom. D g � M g

T n,n (II) Lefshetz fibrations over the 2-sphere. Dehn quandle D g := { simple closed curves γ ⊂ Σ g } / isotopy x ∗ y := Dehn twist of x along y . Lem. (Y. Matsumoto, D. Yetter) ρ { LF over S 2 } { Q ( S 2 , n) → D g qnd. hom. | ρ ( c 1 ) · · · ρ ( c n ) = 1 M g } 1:1 ← → “ conjugacy actions ” Q ( S 2 , n) � B n ( S 2 ) , isom. D g � M g ρ { Q ( S 3 , T n,n ) → D g qnd. hom. | ρ ( c 1 ) · · · ρ ( c n ) = 1 M g } 1:1 Lem. (N.) ← → “ conjugacy actions ” Q ( S 3 , T n,n ) � B n ( S 2 ) , D g � M g ∃ ϕ ∈ H 2 ( B D g ; G ) s.t. Thm. (11, N.) ∀ E : LF over S 2 , ⟨ ϕ, [ Q ( S 3 , T n,n )] ⟩ = Sign( E ) − n .

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