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The space of short ropes and the classifying space of the space of long knots

Shunji Moriya1 and Keiichi Sakai2

1Osaka Prefecture University

moriyasy@gmail.com a

2Shinshu University

ksakai@math.shinshu-u.ac.jp

Mathematics of knots, IX Nihon University, December 22, 2016

The space of long knots

D2 : the unit open disk Long knots : embeddings f : R1 ֒→ R1 × D2 (or their images) satisfying x [0, 1] =⇒ f(t) = (t, 0, 0). K := {long knots} with C∞-topology

• Fact. π0(K) = K/isotopy {knots in S 3}/isotopy.

K is a topological monoid via concatenation (connected-sum). Thus

▶ π0(K) is a monoid, and ▶ the classifying space BK can be defined (later).

Main Theorem. BK is weakly equivalent to the space R of short ropes. Corollary (J. Mostovoy, 2002). π1(R) π0(K) (the group completion). Classification of loops on R up to homotopy ←→ classification of knots up to isotopy

The space of short ropes

Ropes : embeddings r : [0, 1] ֒→ R1 × D2 satisfying r(0) = (0, 0, 0) and r(1) = (1, 0, 0). Short ropes : ropes of length < 3. R := {short ropes} with C∞-topology π0(R) = {r0}, r0(t) := (t, 0, 0) (the tight rope) no classification problem of ropes

• a short rope
• a non-short rope
• r0

Generators of π1(R) π0(K) (Mostovoy). For f ∈ K, (1) tie f around (0, 0, 0);

• → •
• →
• →

→

• r0

f (2) unknot f around (1, 0, 0) in a “reversed way”

Classifying spaces of (topological) categories

For a (topological) category C , NkC := {(c0

f1

− → c1

f2

− → · · ·

fk

− → ck) ; composable k morphisms} ⊂ Mor×k

C

N∗C := {NkC }k≥0 (the nerve of C ) is a simplicial space via compositions / insertion of identities.

• Definition. BC := |N∗C | =

⊔

k≥0

( NkC × ∆k) /∼ : the classifying space of C ∆k = {0 ≤ t1 ≤ · · · ≤ tk ≤ 1}

▶ ((fi)k i=1; (ti)k i=1

) ∈ NkC × ∆k gives an element of BC

▶ ti = ti+1 =⇒ (( fi)i; (ti)i

) = (. . . , fi+1 ◦ fi, . . . ; . . . , ti, ti+2, . . . ) ∈ BC

▶ fi = id =⇒ ((fi)i; (ti)i

) = (. . . , fi−1, fi+1, . . . ; . . . , ti, ti+2, . . . ) ∈ BC In the following

▶ C = K : the category of long knots, ▶ fi ⇐⇒ long knots, ▶ composition ⇐⇒ connected-sum

• t1

t2 tk f1 f2 fk

“Connected-sum of long knots”

The space ψs of “long” 1-manifolds

MA := M ∩ (A × D2) for A ⊂ R1 and a manifold M ⊂ R1 × D2 Definition (S. Galatius, O. Randal-Williams, 2010). ψs := {M1 ⊂ R1 × D2 w/o boundary |

▶ MT is compact for ∀T ∈ R1, ▶ ∀ connected component of M is “long” in at least one direction of R1, ▶ exactly one comp. L ⊂ M is “long” in both directions; LT ∅ for

∀T ∈ R1,

▶ ∃ at least one T ∈ R1 s.t. MT is a one point set } T

∈ ψs L

MT

• Topologize ψs so that “M is close to N if they are close in a compact set.”

Example.

T T+1 T→∞

− − − − → ∈ ψs

M(T)

The space of long knots as a topological category

The category K; Ob(K) = D2, MorK(p, q) = {(T, M) ∈ R1

≥0 × ψs | M connected,

∃ϵ > 0 s.t. M(−∞,ϵ] = {p} × (−∞, ϵ], M[T−ϵ,∞) = {q} × [T − ϵ, ∞)} T × p × q

▶ MorK(p, q) ≃ {long knots}, ▶ NkK = {(0 ≤ T1 ≤ · · · ≤ Tk; f) | fTi are one point sets}.

( f = f[0,T1]# f[T1,T2]# · · · # f[Tk−1,Tk]) Want to know BK = |N∗K|.

The space of long knots as a topological category

The partially ordered sets (posets) D and D⊥; D := {(T, M) ∈ R × ψs | MT is a one point set}, D⊥ := {(T, M) ∈ D | ∃ϵ > 0 s.t. M(T−ϵ,T+ϵ) = MT × (T − ϵ, T + ϵ)} ⊂ D, (T, M) ≤ (T ′, M′)

def

⇐⇒ M = M′ and T ≤ T ′ T T ∈ D ∈ D⊥ Posets are categories; Ob(D(⊥)) := D(⊥), MorD(⊥)(x, y) :=        {∗} x ≤ y ∈ D(⊥), ∅

• therwise.

MorD = {(T0 ≤ T1; M) | MTi are one point sets}, NkD = {(T0 ≤ · · · ≤ Tk; M) | MTi are one point sets}

• Remark. All the “half-long” components ⊂ ((−∞, T0] ⊔ [Tk, ∞)) × D2.

The classifying space of long knots

Theorem (essentially due to S. Galatius and O. Randal-Williams). ∃ simplicial maps N∗D

≃

← − N∗D⊥ ≃ − → N∗K that are levelwise homotopy

• equivalences. Thus BD

≃

← − BD⊥ ≃ − → BK. Main point: NkD⊥ → NkK → NkD⊥ is given by “cut-off” T0 Tk Tk − T0 ⇝ This is homotopic to id by the definition of the topology of ψs; s t

s→−∞, t→+∞

ψs is the classifying space of long knots

∃u : BD → ψs, induced by N∗D × ∆∗ ∋ ((Ti)i; M), (ti)i ) → M. Theorem (essentially due to Galatius and Randal-Williams). The map u is a weak equivalence. Thus BK ∼ ψs. Outline of proof. Want to show πm(ψs, BD) = 0 for ∀m. Given the strict arrows

?

=⇒ ∃ the dotted g ? ∂Dm

f

• BD

u

• Dm

f

• g
• ψs

∀a ∈ R, Ua := {x ∈ Dm | f(x)a is a one point set} =⇒ {Ua}a∈R is an open covering of Dm. Pick a finitely many subcover U = {Uai}i and a partition of unity {λi}i subordinate to U. Roughly g(x) := (((ai)i; f(x)), (λi(x))i ) ∈ N∗D × ∆∗.

ψs is the space of short ropes

R := {short ropes} ∋ r =⇒ length(r) < 3

• Remark. R ֒→ {r : rope | rt is a one point set for ∃t ∈ (0, 1)}.
• r
• a short rope

t

• a non-short rope
• Lemma. The above inclusion is a weak equivalence.

Below R := {r : rope | rt is a one point set for ∃t ∈ (0, 1)}. Fix f : (0, 1)

≈

− → R. Theorem (Moriya-S). The “cut-off” map c : R → ψs, c(r) := (f × idD2)(r(0,1)), is a weak equivalence. Thus BK ∼ R. 1

• c

r

ψs is the space of short ropes

The posets (categories) E⊥ ⊂ E ; E := {(t, r) ∈ (0, 1) × R | rt is a one point set} E⊥ := {(t, r) ∈ E | ∃ϵ > 0 s.t. r(t−ϵ,t+ϵ) = rt × (t − ϵ, t + ϵ)} (t, r) ≤ (t′, r′)

def

⇐⇒ r = r′, t ≤ t′

• 1

t

∈ E

• ∈ E⊥

1 t

r r MorE(⊥) = {(0 < t0 ≤ t1 < 1; r) | rti are one point sets} NkE = {(0 < t0 ≤ · · · ≤ tk < 1; r) | rti are one point sets} Theorem (essentially due to Galatius and Randal-Williams). ∃(weak) equivalences BE⊥ ≃ − → BE

∼

− → R.

• Proof. Similar to the proof of BD⊥ ≃

− → BD

∼

− → ψs.

ψs is the space of short ropes

• Theorem. A simplicial map Φ : N∗E⊥ → N∗D⊥,

Φ(t0 ≤ · · · ≤ tk; r) := (T0 ≤ · · · ≤ Tk; c(r)) where Ti := f(ti) is a levelwise homotopy equivalence. Thus BE⊥ ≃ − → BD⊥. Main point: NkE⊥ Φ − → NkD⊥ → NkE⊥ “unknots r around the endpoints”

1

• t0

tk

• r

t0 tk 1

This is homotopic to id; ∃ a canonical way to unknot r(−∞,t0] and r[tk,∞) (Mostovoy). Conclusion. R

c ∼

• ⟳

ψs BE⊥

∼

• Φ

≃

BD⊥

∼

• ∼

“cut-off” BK