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

Classifying spaces of quandles and low dimensional topology Takefumi Nosaka

野坂 武史

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 : S1 ֒ → S3 or Σg ֒ → S4 X : finite quandle ϕ ∈ H3(BX; A) with a condition      Φϕ(L) ∈ Z[A] Quandle cocycle invariant

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 : S1 ֒ → S3 or Σg ֒ → S4 X : finite quandle ϕ ∈ H3(BX; A) with a condition      Φϕ(L) ∈ Z[A] Quandle cocycle invariant 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 π2(BX). §4 Some applications to low-dimensional topology

A quandle (X, ∗) is a pair { X : a set ∗ : X × X − → X satisfying

• ∀ 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 = G grp. x ∗ y def = y−1xy ∀ x, y ∈ X Ex. Alexander quandle on a finite field Fq : ( Fq, ∗ω ) ω ∈ Fq \ {0, 1} x ∗ω y def = y + ω(x − y) y ∗ x y x a ωa (• ∗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) K1, K2 Knots S1 ֒ → S3 emb. K1 ≃ K2 isotopic⇐ ⇒∃quand. isom.Q(S3, K1) ∼ = Q(S3, K2)

• cf. ∃ K, K′ ⊂ S3 s.t. K ̸≃K′ & π1(S3\K) ∼

= π1(S3\K′).

X: a quandle D: an oriented link diagram ⊂ S2

• ver-arc

An X-coloring of D is a map C : { over-arcs } → X satisfying α β γ C(α) ∗ C(β) = C(γ) Properties D: a digram of a link L ⊂ S3

• {X-coloring of D} 1:1

− → HomQnd(Q(S3, L), X) β α γ α β γ

Rack space (Fenn-Rourke-Sanderson) BX def = ∪ (d-skeleton) 1-skeleton 2-skeleton = ((a, b)-cells) ∪ 1-skeleton

. . . . . . . . . . . . .

X ∋ a X ∋ b a b b a ∗ b

3-skeleton=((a, b, c)-cells) ∪ 2-skeleton

a b b a ∗ b c c c b ∗ c (a ∗ b) ∗ c = (a ∗ c) ∗ (b ∗ c) a ∗ c

Rack space (Fenn-Rourke-Sanderson) BX def = ∪ (d-skeleton) 1-skeleton 2-skeleton = ((a, b)-cells) ∪ 1-skeleton

. . . . . . . . . . . . .

X ∋ a X ∋ b a b b a ∗ b

3-skeleton=((a, b, c)-cells) 4-skeleton=((a, b, c, d)-cells) ∪ 2-skeleton ∪ 3-skeleton

a b b a ∗ b c c c b ∗ c (a ∗ b) ∗ c = (a ∗ c) ∗ (b ∗ c) a ∗ c

Rack space (Fenn-Rourke-Sanderson) BX def = ∪ (d-skeleton) 1-skeleton 2-skeleton = ((a, b)-cells) ∪ 1-skeleton

. . . . . . . . . . . . .

X ∋ a X ∋ b a b b a ∗ b

3-skeleton=((a, b, c)-cells) 4-skeleton=((a, b, c, d)-cells) ∪ 2-skeleton ∪ 3-skeleton

a b b a ∗ b c c c b ∗ c (a ∗ b) ∗ c = (a ∗ c) ∗ (b ∗ c) a ∗ c

• Rem. BX was defined by a fat realization of a“cubical set”

.

Known results on the rack homology H∗(BX) = HR

∗ (X)

• (03, Etingof-Gra˜

na)

ℓ : # of“connected components”of X

|X| < ∞ = ⇒ Hn(BX; Q) ∼ = Qℓn

Known results on the rack homology H∗(BX) = HR

∗ (X)

• (03, Etingof-Gra˜

na)

ℓ : # of“connected components”of X

|X| < ∞ = ⇒ Hn(BX; Q) ∼ = Qℓn

• (03, T. Mochizuki) X = Fq, ω ∈ Fq, x∗y = ωx+(1−ω)y

He determined H2 ⊕ H3(BX; Fq) with their base.

• (09, N.) Let q = p.

He determined the quandle homologies HQ

∗ (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) = HR

∗ (X)

• (03, Etingof-Gra˜

na)

ℓ : # of“connected components”of X

|X| < ∞ = ⇒ Hn(BX; Q) ∼ = Qℓn

• (03, T. Mochizuki) X = Fq, ω ∈ Fq, x∗y = ωx+(1−ω)y

He determined H2 ⊕ H3(BX; Fq) with their base.

• (09, N.) Let q = p.

He determined the quandle homologies HQ

∗ (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 → π2(BX)

Π2(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. Π2(X) ∼

= π2(BX). Rem Πn(X) → πn(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 ∈ H2(BX; A) π2(BX)

H

− → H2(BX; A)

⟨ ϕ,• ⟩

− → A

How to compute the homotopy grp, π2(BX) & π3(BX)

• Top. monoid str. on the universal cov. of BX by Clauwens

π1(BX) ∼ = Adj(X) := ⟨x ∈ X | x · y = y · (x ∗ y)⟩

• BX ≃

∪

n≥0

( Adj(X) × ([0, 1] × X)n) / ∼

µ : (G × [0, 1]n × Xn) × (G × [0, 1]m × Xm) → G × [0, 1]n+m × Xn+m, µ([g; t1, . . . , tn, x1, . . . , xn], [h; t′

1, . . . , t′ m, x′ 1, . . . , x′ m]) :=

[gh; t1, . . . , tn, t′

1, . . . , t′ m, x1 ∗ h, . . . , xn ∗ h, x′ 1, . . . , x′ m],

• Rem. π1(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 = Fq with p > 2. ω ∈ Fq, x ∗ y = ωx + (1 − ω)y = ⇒

∃ a splitting exact sequence

0 − → π2(BX) − → H3(BX; Z) − → Λ2( H2(BX; Z) ) − → 0

• Exa. (N.) If q = ph and ω = −1,

then dim ( π2(BX) ⊗ Zp ) = h2(h2+11)

12

+ 1.

• Thm. (10, N.)

X = Fq with p > 2. ω ∈ Fq, x ∗ y = ωx + (1 − ω)y = ⇒

∃ a splitting exact sequence

0 − → π2(BX) − → H3(BX; Z) − → Λ2( H2(BX; Z) ) − → 0

• Exa. (N.) If q = ph and ω = −1,

then dim ( π2(BX) ⊗ Zp ) = h2(h2+11)

12

+ 1. Thm (10, N.) (On π3(BX) vs. knotted surfaces Σg ֒ → S4) Further, if X satisfies the vanishing HQ

2 (X; Z) ∼

= 0, = ⇒ π3(BX) ∼ = Z2 ⊕H4(BX; Z).

• Cor. (N.) If q = p and ω = −1, then π3(BX) ∼

= Z2 ⊕(Zp)2.

Some Applications to low-dimensional topology (I) Closed 3-mfds via branched covering spaces M → S3 Fact ∀ M, ∃L ⊂ S3 s.t. M= 4-fold bran. cov. along L.

• Prop. (09, E. Hatakenaka) ∀ G grp, ∃ a quandle

G s.t. Hom(π1(M), G) × G

1:1

← → {Q(S3, L) f → G | · · · }

• N. constructed a 3-mfd inv. ∈ Z[π2(B

G)/ ∼]

• (10, Hatakenaka-N.)

∀ a grp G, we constructed an epi. π2(B

G) → Hgr

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

Dg := { simple closed curves γ ⊂ Σg}/ isotopy x ∗ y := Dehn twist of x along y.

• Lem. (Y. Matsumoto, D. Yetter)

{LF over S2} isom.

1:1

← → {Q(S2, n)

ρ

→ Dg qnd. hom. | ρ(c1) · · · ρ(cn) = 1Mg} Q(S2, n) Bn(S2), Dg Mg

“conjugacy actions”

(II) Lefshetz fibrations over the 2-sphere.

Dehn quandle

Dg := { simple closed curves γ ⊂ Σg}/ isotopy x ∗ y := Dehn twist of x along y.

• Lem. (Y. Matsumoto, D. Yetter)

{LF over S2} isom.

1:1

← → {Q(S2, n)

ρ

→ Dg qnd. hom. | ρ(c1) · · · ρ(cn) = 1Mg} Q(S2, n) Bn(S2), Dg Mg

“conjugacy actions”

(II) Lefshetz fibrations over the 2-sphere.

Dehn quandle

Dg := { simple closed curves γ ⊂ Σg}/ isotopy x ∗ y := Dehn twist of x along y. Tn,n

• Lem. (Y. Matsumoto, D. Yetter)

{LF over S2} isom.

1:1

← → {Q(S2, n)

ρ

→ Dg qnd. hom. | ρ(c1) · · · ρ(cn) = 1Mg} Q(S2, n) Bn(S2), Dg Mg

“conjugacy actions”

• Lem. (N.)

1:1

← → {Q(S3, Tn,n)

ρ

→ Dg qnd. hom. | ρ(c1) · · · ρ(cn) = 1Mg} Q(S3, Tn,n) Bn(S2), Dg Mg

“conjugacy actions”

• Thm. (11, N.)

∃ϕ ∈ H2(BDg; G) s.t. ∀E: LF over S2, ⟨ϕ, [Q(S3, Tn,n)]⟩ = Sign(E) − n.

Thank you