A spectral sequence for cohomology of knot space Syunji Moriya - - PowerPoint PPT Presentation

A spectral sequence for cohomology of knot space

Syunji Moriya

Osaka Prefecture University moriyasy@gmail.com a

Syunji Moriya (O.P .U.) Knot space 1 / 45

Notations

M : closed smooth manifold of dimension d ≥ 4. Emb(S1, M) : The space of smooth embeddings S1 → M with C∞-topology, which we call the space of knots in M (without any base point condition). k : a fixed commutative ring (which is a PID). We do not restrict to a field of characteristic H∗ (H∗) : singular (co)homology with coefficients in k.

Syunji Moriya (O.P .U.) Knot space 2 / 45

Motivation

Recently, Emb(S1, M) is studied by Arone-Szymik, Budney-Gabai, and Kupers using Goodwillie-Weiss embedding calculus Motivation : construction of a computable spectral sequence (s.s.) converging to H∗(Emb(S1, M); k) for a simply connected M

Syunji Moriya (O.P .U.) Knot space 3 / 45

Main results

Main results

Syunji Moriya (O.P .U.) Knot space 4 / 45

Main results

Our spectral sequence, which we call ˇ Cech spectral sequence and denote by ˇ

Ep q

r

, has an algebraic presentation of E2-page when H∗(M) is a free k-module, and the Euler number χ(M) = 0 ∈ k or χ(M) is invertible in k (χ(M) ∈ k via the ring hom Z → k) We state main results separately into the cases of χ(M) = 0 or invertible

Syunji Moriya (O.P .U.) Knot space 5 / 45

Main results

Poincar´ e algebra

Definition 1 A Poincar´ e algebra H∗ of dimension d is a pair of a graded commutative algebra H∗ and a linear isomorphism ϵ : Hd → k s. t.

H∗ ⊗ H∗ multiplication −→ H∗

ϵ

→ k

induces a linear isomorphism H∗ (Hd−∗)∨. Let {ai}i be a linear basis of H∗ and

(bij)ij denote the inverse of the matrix (ϵ(ai · aj))ij. ∆H : the diagonal class for H∗ given by ∆H =

• i,j

(−1)|aj|bji ai ⊗ aj .

Syunji Moriya (O.P .U.) Knot space 6 / 45

Main results

Poincar´ e algebra

If M is oriented, and H∗(M) is a free k-module, fixing an orientation on M, H∗(M) is Poincar´ e algebra by ϵ : fund.class → 1 ∈ k.

Syunji Moriya (O.P .U.) Knot space 7 / 45

Main results

simplicial dg-algebra A⋆ ∗

• (H)

H∗ : 1-connected (i.e. H1 = 0) Poincar´

e algebra of dim. d. ei : H∗ → (H∗)⊗n+1 : a → 1 ⊗ · · · ⊗ a ⊗ · · · ⊗ 1, insertion to i-th factor. A⋆ ∗

n (H) := (H∗)⊗ n+1 ⊗

• yi, gi j | 0 ≤ i, j ≤ n
• /I

with deg yi = (0, d − 1),

deg gi j = (−1, d).

The ideal I is generated by y2

i = g2 i j = 0,

gi i = 0,

(eia − eja)gi j = 0 (a ∈ H∗),

gi j = (−1)dgj i, gi jgj k + gj kgk i + gk igi j = 0

(3-term relation)

The differential is given by ∂(a) = 0 for a ∈ H⊗n+1 and ∂(gij) = fij∆H, where fij : H ⊗ H → H⊗n+1 is insertion to i-th and j-th factors.

Syunji Moriya (O.P .U.) Knot space 8 / 45

Main results

simplicial dg-algebra A⋆ ∗

• (H)

The face di : A⋆ ∗

n (H) → A⋆ ∗ n−1(H) (0 ≤ i ≤ n) : is given by

di(a0 ⊗ · · · ⊗ an) =

      

a0 ⊗ · · · ⊗ aiai+1 ⊗ · · · an

(0 ≤ i ≤ n − 1) ±ana0 ⊗ · · · ⊗ an−1 (i = n)

and di(gj,k) = gj′,k ′ where j′ =

      

j

(j ≤ i)

j − 1

(j > i)

, similarly for k ′. the degeneracy si : A⋆ ∗

n (H) → A⋆ ∗ n+1(H) : insertion of 1 to i-th factor and skip the index

i + 1.

Syunji Moriya (O.P .U.) Knot space 9 / 45

Main results

Main theorem : the case of χ(M) = 0

A⋆ ∗

• (H) −→ NA⋆ ∗
• (H) (normalization)

−→ H(NA⋆ ∗

• (H)) (homology of total complex)

Theorem 2 M : 1-connected manifold. Set H∗ = H∗(M) and suppose that H∗ is a free k-module and χ(M) = 0 ∈ k

∃ a spec. seq. : ˇ

Ep q

2

H(NA⋆ ∗

• (H)) ⇒ Hp+q(Emb(S1, M)),

where bidegree is given by p = ∗, q = ⋆ − •

Syunji Moriya (O.P .U.) Knot space 10 / 45

Main results

Remark 3

ˇ Ep q

2

has a graded commutative ring structure but its relation to the ring H∗(Emb(S1, M) an whether it induces ring structure on pages after E2 is unclear for the speaker. It may be related to comparison of filtered ring objects in spectra and complexes

Syunji Moriya (O.P .U.) Knot space 11 / 45

Main results

simplicial dg-algebra B⋆ ∗

• (H)

H∗ : 1-connected Poincar´

e algebra of dimension d. Define a Poincar´ e algebra SH∗ of dimension 2d − 1 as follows: SH∗ = H≤d−2 ⊕ H≥2[d − 1] a · ¯ b = a · b for a ∈ H≤d−2, ¯ b ∈ H≥2[d − 1] corresponding to b ∈ H≥2

Syunji Moriya (O.P .U.) Knot space 12 / 45

Main results

simplicial dg-algebra B⋆ ∗

• (H)

Set B⋆ ∗

n (H) := (SH∗)⊗ n+1 ⊗

• hi j, gi j | 0 ≤ i, j ≤ n
• /J

with deg gi j = (−1, d),

deg hi j = (−1, 2d − 1). The ideal J is generated by

g2

i j = h2 i j = 0,

hi i = gi i = 0, gi j = gj i hi j = −hj i

(eia − eja)gi j = 0, (eia − eja)hi j = 0 (a ∈ SH∗),

3-term relations for gi j and for hi j ,

(hi j + hk i)gj k = (hi j + hj k)gi j

The differential is given by ∂a = 0 for a ∈ SH⊗n+1 and

∂(gi j) = fi j∆H, ∂(hij) = fi j∆SH.

The face and degeneracy is similar to A⋆ ∗

• (H).

Syunji Moriya (O.P .U.) Knot space 13 / 45

Main results

Main theorem : the case χ(M) is invertible

Theorem 4 M: 1-connected manifold. Set H∗ = H∗(M) and suppose that H∗ is a free k-module and χ(M) is invertible in k

∃ a spec. seq. : ˇ

Ep q

2

H(NB⋆ ∗

• (H)) ⇒ Hp+q(Emb(S1, M)),

where bidegree is given by p = ∗, q = ⋆ − • We call the above spectral sequences the ˇ Cech spectral sequences.

Syunji Moriya (O.P .U.) Knot space 14 / 45

Main results

Remark 5

ˇ Ep q

2

has a graded commutative ring structure but its relation to the ring H∗(Emb(S1, M) an whether it induces ring structure on pages after E2 is unclear for the speaker. It may be related to comparison of filtered ring objects in spectra and complexes

Syunji Moriya (O.P .U.) Knot space 15 / 45

Main results

Other spectral sequences

Vassiliev (1997) defined a s.s. converging to H∗(LM, Emb(S1, M)) by discriminant method.

It is applicable to arbitrary manifold (including non-orientable one). Its E2-page has an interesting description but somewhat complicated for the speaker.

Sinha (2009) defined a cosimplicial model for a variant of Emb(S1, M), which induces a Bousfield-Kan cohomology s.s.

A version of this s.s. for long knots in Rd leads to the collapse of Vassiliev s.s. by Lambrechts-Turchin-Voli´ c (2010) in ch(k) = 0 and vanish of some differentials by de Brito-Horel (2020) in ch(k) > 0. E2-page is described by cohomology of ordered configuration spaces of points in M with a tangent vector, which is difficult to compute for general M.

Syunji Moriya (O.P .U.) Knot space 16 / 45

Main results

Computation for M = Sk × Sl, (odd)×(even)

Corollary 6 k : Z or Fp with p prime. k : an odd number, l : an even number with k + 5 ≤ l ≤ 2k − 3 and |3k − 2l| ≥ 2, or l + 5 ≤ k ≤ 2l − 3 and |3l − 2k| ≥ 2. H∗ := H∗(Emb(S1, Sk × Sl)).

1

We have isomorphisms Hi = k

(i = k − 1, k, 2k − 2, 2k − 1, k + l).

2

If k = Fp with p 2, we have isomorphisms Hi = k2 (i = k + l − 2, k + l − 1, 2k + l − 3, 2k + l − 2, 2k + l − 1). The inequalities ensure that differentials vanish by degree reason.

Syunji Moriya (O.P .U.) Knot space 17 / 45

Main results

Computation for M = Sk × Sl, (even)×(even)

Corollary 7 Suppose 2 ∈ k×. k, l : two even numbers with k + 2 ≤ l ≤ 2k − 2 and |3k − 2l| ≥ 2. H∗ := H∗(Emb(S1, Sk × Sl)). We have isomorphisms Hi = k

(i = k − 1, k, l − 1, l, k + l − 3, k + l − 2, k + l − 1, 3k).

For any other degree i ≤ 2k + l, Hi = 0.

□

The inequalities ensure that differentials vanish by degree reason.

Syunji Moriya (O.P .U.) Knot space 18 / 45

Main results

π1(Emb(S1, M)) for 4-dimensional M

Imm(S1, M) : the space of immersions S1 → M Question by Arone-Szymik : Is there a simp. conn. 4-dim M s.t. the inclusion iM : Emb(S1, M) → Imm(S1, M) has a non-trivial kernel on π1. (This map is always surjective.)

Syunji Moriya (O.P .U.) Knot space 19 / 45

Main results

π1(Emb(S1, M)) for 4-dimensional M

Corollary 8 M : simply connected, d = 4, H2(M; Z) 0, and the intersection form on H2(M; F2) is represented by a matrix of which the inverse has at least

• ne non-zero diagonal component.

Then, the inclusion iM induces an isomorphism on π1. In particular,

π1(Emb(S1, M)) H2(M; Z).

For example, M = CP2#CP2 satisfies the assumption while M = S2 × S2 does not. For the case H2(M) = 0, by Arone-Szymik, Emb(S1, M) is simply connected. The case of all of the diagonal components of the matrix being zero is unclear for the speaker.

Syunji Moriya (O.P .U.) Knot space 20 / 45

Construction of spectral sequence

Construction of ˇ Cech s.s.

Syunji Moriya (O.P .U.) Knot space 21 / 45

Construction of spectral sequence

Sinha’s cosimplicial model

Goodwillie-Weiss embedding calculus is a framework which relates embedding spaces and configuration spaces of points in manifolds. Based on this, Turchin (2013) and de Brito-Weiss (2013) prove a beautiful theorem which states that that Emb(N, M) is weak htpy equiv. to a space of derived maps of right modules of (framed) configuration spaces of points in N or M. For knot spaces, another beautiful model which fits with Bousfield-Kan s.s. is Sinha’s cosimplicial model. This is also based on the calculus.

Syunji Moriya (O.P .U.) Knot space 22 / 45

Construction of spectral sequence

(co)module over an operad

A (non-symmetric) operad is a (non-symmetric) sequence {O(n)}n≥1 with a partial composition (− ◦i −) : O(m) ⊗ O(n) → O(m + n − 1) satisfying some axioms. ( ⊗ : the monoidal product of the underlying monoidal category) A (right) O-module is a symmetric sequence X = {X(n)}n≥1 with a partial composition

(− ◦i −) : X(n) ⊗ O(m) → X(m + n − 1).

A (left) O-comodule is a symmetric sequence X = {X(n)}n≥1 with a partial composition

(− ◦i −) : O(m) ⊗ X(m + n − 1) → X(n).

Syunji Moriya (O.P .U.) Knot space 23 / 45

Construction of spectral sequence

little interval operad D1

D1 : the little interval operads

An element of D1(n) is the n-tuple c = (c1, . . . , cn) of closed intervals ci ⊂

• − 1

2, 1 2

• s. t.

ci ∩ cj = ∅ for i j, and the labeling of 1, . . . , n is consistent with order of the interval

[−1/2, 1/2] d c e = d ◦2 c

d1 d2 d3 c1 c2 e2 e3 e1 e4

D1(3) D1(2) D1(4) × → (− ◦2 −) :

Figure: partial composition of D1

Syunji Moriya (O.P .U.) Knot space 24 / 45

Construction of spectral sequence

A D1-module FM

Fix a Riemanniann metric on M,

• M : the tangent sphere bundle of M

δ : a number s.t. 0 < δ <the injectivity radius of M

Balln(M) := {(D1, . . . , Dn) | Di is a closed geodesic ball of radius < δ, Di ∩ Dj = ∅ if i j}, topologized as a subspace of Mn × Rn via (center, radius)-inclusion Define FM(n) as the following pullback FM(n)

• M×n

projection×n

• Balln(M)center M×n

M

Syunji Moriya (O.P .U.) Knot space 25 / 45

Construction of spectral sequence

partial composition (− ◦i −) : FM(n) × D1(m) → FM(m + n − 1)

The partial composition is a ”perturbed diagonal map” FM(n)

D1(m)

xi

• i

is defined by (2) (3) xi (1)

Syunji Moriya (O.P .U.) Knot space 26 / 45

Construction of spectral sequence

A A∞-comodule XA

A∞ : the associahedral chain operad

generators { µk ∈ A∞(k) }k≥2 ( |µk| = −k + 2 ) dµk =

• l, p, q

l + p = k − 1 ± µl ◦p+1 µq

For an A∞-algebra A, Define a A∞-comodule XA by

XA(n) := A⊗n µm ◦i (a1 ⊗ · · · am+n−1) := a0 ⊗ · · · ⊗ µm(ai, . . . , ai+m−1) ⊗ · · · ⊗ am+n−1 the action of Σn is the standard permutation of factors.

Syunji Moriya (O.P .U.) Knot space 27 / 45

Construction of spectral sequence

Hochschild complex of A∞-comodule

For an A∞-algebra A, Getzler-Jones defined a Hochschild complex C(A, A) as a natural generalization of that of an associative algebra. The following lemma is a straightforward extension of Getzler-Jones. Lemma 9 For a A∞-comodule, X, there is a functorial bigraded complex CH•X s.t. For X = XA, CH•XA is quasi-isom. to C(A, A).

CHnX = X(n + 1)

total degree is ∗ − •, where ∗ is the original cochain degree of X(n + 1)

Syunji Moriya (O.P .U.) Knot space 28 / 45

Construction of spectral sequence

from module to comodule

D1-module FM −→ C∗(D1)-module C∗(FM) −→ C∗(D1)-comodule C∗(FM)

((α ◦i f)(σ) = f(σ ◦i α) for α ∈ C∗(D1(m)), σ ∈ C∗(FM(n)), f ∈ C∗(FM(m + n − 1)))

−→ A∞-comodule C∗(FM).

(pulling back partial comp. by a fixed map A∞ → C∗(D1))

Syunji Moriya (O.P .U.) Knot space 29 / 45

Construction of spectral sequence

Sinha spectral sequence

Filtering CH•C∗(FM) by the grading •, we have a spectral sequence Ep q

r

Lemma 10

Ep q

r

is isom. to Bousfield-Kan cohomology s.s. associated to the (analogue of )Sinha’s cosimplicial model, (essentially, Sinha 2009) Ep q

r

converges to H∗(Emb(S1, M)) if M is simp. conn.

Ep q

1

Hq(FM(p + 1))

(Sinha considered manifolds with boundary and embeddings with some base point condition.)

Syunji Moriya (O.P .U.) Knot space 30 / 45

Construction of spectral sequence

Idea of construction of ˇ Cech spectral sequence

FM(n) is htpy equiv. to ⃗ Cn(M), the configuration spaces of points with tangent vector in M, the following pullback

⃗

Cn(M)

• Cn(M)
• M×n

M×n

,

Cn(M) = {(x1, . . . , xn) | xi xj if i j}

∆fat(M) := ∪pq∆p,q(M) ⊂ M×n, ∆p,q(M) = {xp = xq}, ⃗ ∆fat(M) : the space defined by the pullback ⃗ ∆fat(M)

• ∆fat(M)
• M×n

M×n

Syunji Moriya (O.P .U.) Knot space 31 / 45

Construction of spectral sequence

Idea of construction of ˇ Cech spectral sequence

Idea : replace configuration spaces with fat diagonals via Poincar´ e-Lefschetz duality C∗(⃗ Cn(M)) ≃ C∗( M×n, ⃗

∆fat(M))

coming from M×n − ⃗ Cn(M) = ⃗

∆fat(M) (we are loose on degree) and use ˇ

Cech resolution C∗( M×n, ⃗

∆fat(M)) ← ˇ

C0 n(M) ← ˇ C1 n(M) ← · · ·

ˇ

Ck,n(M) =

      

C∗( M×n)

(k = 0) ⊕IC∗(⃗ ∆IM) (k ≥ 1)

where I runs through set of pairs (p, q) with #I = k, and ∆I(M) = ∩(p,q)∈I∆p,q(M), following Bendersky-Gitler.

Syunji Moriya (O.P .U.) Knot space 32 / 45

Construction of spectral sequence

Idea of construction of ˇ Cech spectral sequence

We want to extend this to a resolution of the comodule. Soppose we could define partial composition compatible with the differential of ˇ Cech complex

C∗D1(m) ⊗ C∗FM(m + n − 1)

(−◦i−)

• C∗D1(m) ⊗ ˇ

C0 m+n−1(M)

(−◦i−)

• P.D.
• C∗D1(m) ⊗ ˇ

C1 m+n−1(M)

(−◦i−)

• · · ·
• C∗(FM(n))

ˇ C0 n(M)

P.D.

• ˇ

C1 n(M)

• · · ·
• Here, P.D means zigzag C∗FM(n)

≃

→ C∗(⃗

Cn(M))

≃

→ C∗(

M×n, ⃗

∆fat(M)) ← ˇ

C0,n(M) (In fact, construction of partial composition is main difficulty)

Syunji Moriya (O.P .U.) Knot space 33 / 45

Construction of spectral sequence

Idea of construction of ˇ Cech spectral sequence

So we would have C∗D1-comodule of ˇ CM

∗ ⋆ of double complexes by ˇ

CM

∗ ⋆(n) = ˇ

C⋆ n(M) (∗ : homological, ⋆ : ˇ Cech ).

−→ CH• ˇ

CM

∗ ⋆

By filtering by ⋆ + •, we would get ˇ Cech s.s. ˇ

E, and

By filtering by •, we get Sinha s.s. E Using this intermediate complex, we could prove convergence for simply connected M.

Syunji Moriya (O.P .U.) Knot space 34 / 45

Construction of spectral sequence

Difficulty in construction

It is difficult (for me) to define partial compositions compatible with ˇ Cech resolution on the chain level. This problem is analogous to construction of a chain-level intersection product which is associative, has some ”geometric description” , and makes the following diagram commutative C∗(M) ⊗ C∗(M)

P.D. ∪

• C∗(M) ⊗ C∗(M)

int.prod.

• C∗(M)

P.D.

C∗(M)

A nice solution is Atiyah duality and its refinement due to R. Cohen

Syunji Moriya (O.P .U.) Knot space 35 / 45

Construction of spectral sequence

Atiyah duality

Here we work in the classical homotopy category of spectra. (Though we need some model category of spectra to justify technical issue.) For an embedding e : M → RK, ν : a tubuler nbd of e(M) in RK. M−TM := Σ−NTh(ν). Different embeddings give equivalent spectra M−TM and equivalence can be chosen

• consistently. A multiplication on M−TM:

ν∆ : a tubuler neighborhood of image of M in R2K by the map

M

diagonal

−→

M × M

e×e

−→ RK × RK

taken so small that ν∆ ⊂ ν × ν multiplication M−TM ∧ M−TM → M−TM is induced by the composition

Σ−NTh(ν) ∧ Σ−NTh(ν) Σ−2NTh(ν × ν)

collapse

−→ Σ−2NTh(ν∆) M−TM

Syunji Moriya (O.P .U.) Knot space 36 / 45

Construction of spectral sequence

Atiyah duality

M∨ : Spanier-Whitehead dual of M with disjoint base point, i.e., M∨ = Map(M+, S) (S: sphere spectrum) M∨ has natural multiplication induced by pullback by ∆ : M → M × M. Theorem 11 (Atiyah) There is an equivalence of commutative ring spectrum M∨ M−TM

• R. Cohen gave a refinement of this in the category of symmetric spectra. We can justify our

idea using this refinement.

Syunji Moriya (O.P .U.) Knot space 37 / 45

Construction of spectral sequence

Remark 12 Using the refinement of the duality, Cohen-Jones (2002) proved there is an isomorphism of graded algebra

(H∗+d(LM), loop product) (HH∗(C∗(M); C∗(M)), cup product)

Syunji Moriya (O.P .U.) Knot space 38 / 45

Construction of spectral sequence

dual comodule

O: topological operad, X: O-module O can be considered as an operad in the category of spectra.

An O-comodule X∨ (in spectra) is defined as follows: X∨(n) = X(n)∨(= Map(X(n)+, S))

(a ◦i f)(x) = f(x ◦i a)

(a ∈ O(m), f ∈ X∨(n), x ∈ X(n))

Syunji Moriya (O.P .U.) Knot space 39 / 45

Construction of spectral sequence

Key theorem

Theorem 13 (M.) There exists a left D1-comodule T HM in symmetric spectra as follows.

1

There exists a zigzag of π∗-isomorphisms of left D1-comodules

(FM)∨ ≃ T HM .

2

T HM has a natural ˇ

Cech resolution. There is a suitable chain functor from spectra to complexes We can justify our idea of construction with these notions.

Syunji Moriya (O.P .U.) Knot space 40 / 45

Construction of spectral sequence

Outline of proof of Cor. 8

Corollary 14 (=Cor. 8) M : simply connected, d = 4, H2(M; Z) 0, and the intersection form on H2(M; F2) is represented by a matrix of which the inverse has at least

• ne non-zero diagonal component.

Then, the inclusion iM induces an isomorphism on π1. In particular,

π1(Emb(S1, M)) H2(M; Z).

Syunji Moriya (O.P .U.) Knot space 41 / 45

Construction of spectral sequence

Outline of proof of Cor. 8

Set H2 = H2(M; Z). By Smale-Hirsch theorem, Imm(S1, M) ≃ L M, so π1(Imm(S1, M)) H2.

π1(Emb(S1, M)) is finitely generated and nilpotent by a theorem for nilpotency of

homotopy limits by Farjoun (2003) and the Bousfield-Kan homotopy s.s. of Sinha’s model. It is enough to show the composition Emb(S1, M)

iM

→ Imm(S1, M)

cl

→ K(H2, 1)

induces isomorphism on H1(−; k) and monomorphism on H2(−; k) for any field k by a theorem of Stallings (1965). (cl is the classifying map.) iM is induced by a map of comodules so it induces map of s.s. ˇ

Er → Er (Er is a s.s. for

L M). Observing this map we have the claim on H1, H2.

Syunji Moriya (O.P .U.) Knot space 42 / 45

Construction of spectral sequence

Remark 15 If all of the diagonal components of the inverse of intersection matrix on H2(M; F2) is zero, the map ˇ

E∞ → E∞ is not a monomorphism for k = F2 but this does not necessarily imply the

• riginal (non-associated graded) map is not a monomorphism. So in this case, it is still unclear

whether iM is an isomorphism on π1.

Syunji Moriya (O.P .U.) Knot space 43 / 45

Construction of spectral sequence

question/speculation

Is there an essentially new element i.e. one not coming from Imm(S1, M) in H∗(Emb(S1, M) of degree higher than any given degree? related question : Are there any operations (e.g. multiplication) on ˇ

Ep q

r

. E2-page has a multiplication but it is unclear for Er>2. For the case of long knots modulo immersion Embc(R, Rd), an analogue of our construction present C∗(Embc(R, Rd) as a homotopy colimit of a diagram of desuspended sphere spectra ( d ≥ 4). This may lead to a new collapse result.

Syunji Moriya (O.P .U.) Knot space 44 / 45

Construction of spectral sequence

Thank you for attention!

Syunji Moriya (O.P .U.) Knot space 45 / 45