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 ( S 1 , M ) : The space of smooth embeddings S 1 → 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 0 H ∗ ( H ∗ ) : singular (co)homology with coefficients in k . Syunji Moriya (O.P .U.) Knot space 2 / 45

Motivation Recently, Emb ( S 1 , 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 ( S 1 , 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 E p q Cech spectral sequence and denote by ˇ , has an r algebraic presentation of E 2 -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 e algebra H ∗ of dimension d is A Poincar´ a pair of a graded commutative algebra H ∗ and a linear isomorphism ϵ : H d → k s. t. H ∗ ⊗ H ∗ multiplication ϵ H ∗ −→ → k induces a linear isomorphism H ∗ � ( H d −∗ ) ∨ . Let { a i } i be a linear basis of H ∗ and ( b ij ) ij denote the inverse of the matrix ( ϵ ( a i · a j )) ij . ∆ H : the diagonal class for H ∗ given by � ( − 1 ) | a j | b ji a i ⊗ a j . ∆ H = i , j 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. H 1 = 0) Poincar´ e algebra of dim. d . e i : H ∗ → ( H ∗ ) ⊗ n + 1 : a �→ 1 ⊗ · · · ⊗ a ⊗ · · · ⊗ 1, insertion to i -th factor. �� � n ( H ) := ( H ∗ ) ⊗ n + 1 ⊗ A ⋆ ∗ y i , g i j | 0 ≤ i , j ≤ n / I with deg y i = ( 0 , d − 1 ) , deg g i j = ( − 1 , d ) . The ideal I is generated by y 2 i = g 2 ( a ∈ H ∗ ) , i j = 0 , g i i = 0 , ( e i a − e j a ) g i j = 0 g i j = ( − 1 ) d g j i , g i j g j k + g j k g k i + g k i g i j = 0 ( 3-term relation ) The differential is given by ∂ ( a ) = 0 for a ∈ H ⊗ n + 1 and ∂ ( g ij ) = f ij ∆ H , where f ij : 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 d i : A ⋆ ∗ n ( H ) → A ⋆ ∗ n − 1 ( H ) ( 0 ≤ i ≤ n ) : is given by    a 0 ⊗ · · · ⊗ a i a i + 1 ⊗ · · · a n ( 0 ≤ i ≤ n − 1 )  d i ( a 0 ⊗ · · · ⊗ a n ) = and    ± a n a 0 ⊗ · · · ⊗ a n − 1 ( i = n )    ( j ≤ i ) j  d i ( g j , k ) = g j ′ , k ′ where j ′ = , similarly for k ′ .    ( j > i ) j − 1 the degeneracy s i : 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 E p q � H ( NA ⋆ ∗ • ( H )) ⇒ H p + q ( Emb ( S 1 , M )) , 2 where bidegree is given by p = ∗ , q = ⋆ − • Syunji Moriya (O.P .U.) Knot space 10 / 45

Main results Remark 3 E p q ˇ has a graded commutative ring structure but its relation to the ring H ∗ ( Emb ( S 1 , M ) an 2 whether it induces ring structure on pages after E 2 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 . e algebra S H ∗ of dimension 2 d − 1 as follows: Define a Poincar´ S H ∗ = 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 �� � n ( H ) := ( S H ∗ ) ⊗ n + 1 ⊗ B ⋆ ∗ h i j , g i j | 0 ≤ i , j ≤ n / J with deg g i j = ( − 1 , d ) , deg h i j = ( − 1 , 2 d − 1 ) . The ideal J is generated by g 2 i j = h 2 i j = 0 , h i i = g i i = 0 , g i j = g j i h i j = − h j i ( a ∈ S H ∗ ) , ( e i a − e j a ) g i j = 0 , ( e i a − e j a ) h i j = 0 3-term relations for g i j and for h i j , ( h i j + h k i ) g j k = ( h i j + h j k ) g i j The differential is given by ∂ a = 0 for a ∈ S H ⊗ n + 1 and ∂ ( g i j ) = f i j ∆ H , ∂ ( h ij ) = f i j ∆ S H . 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 E p q � H ( NB ⋆ ∗ • ( H )) ⇒ H p + q ( Emb ( S 1 , M )) , 2 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 E p q ˇ has a graded commutative ring structure but its relation to the ring H ∗ ( Emb ( S 1 , M ) an 2 whether it induces ring structure on pages after E 2 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 ( S 1 , M )) by discriminant method. It is applicable to arbitrary manifold (including non-orientable one). Its E 2 -page has an interesting description but somewhat complicated for the speaker. Sinha (2009) defined a cosimplicial model for a variant of Emb ( S 1 , M ) , which induces a Bousfield-Kan cohomology s.s. A version of this s.s. for long knots in R d 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. E 2 -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 = S k × S l , (odd) × (even) Corollary 6 k : Z or F p with p prime. k : an odd number, l : an even number with k + 5 ≤ l ≤ 2 k − 3 and | 3 k − 2 l | ≥ 2, or l + 5 ≤ k ≤ 2 l − 3 and | 3 l − 2 k | ≥ 2. H ∗ := H ∗ ( Emb ( S 1 , S k × S l )) . We have isomorphisms H i = k ( i = k − 1 , k , 2 k − 2 , 2 k − 1 , k + l ) . 1 If k = F p with p � 2, we have isomorphisms 2 H i = k 2 ( i = k + l − 2 , k + l − 1 , 2 k + l − 3 , 2 k + l − 2 , 2 k + 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 = S k × S l , (even) × (even) Corollary 7 Suppose 2 ∈ k × . k , l : two even numbers with k + 2 ≤ l ≤ 2 k − 2 and | 3 k − 2 l | ≥ 2. H ∗ := H ∗ ( Emb ( S 1 , S k × S l )) . We have isomorphisms H i = k ( i = k − 1 , k , l − 1 , l , k + l − 3 , k + l − 2 , k + l − 1 , 3 k ) . For any other degree i ≤ 2 k + l , H i = 0. □ The inequalities ensure that differentials vanish by degree reason. Syunji Moriya (O.P .U.) Knot space 18 / 45

Main results π 1 ( Emb ( S 1 , M )) for 4-dimensional M Imm ( S 1 , M ) : the space of immersions S 1 → M Question by Arone-Szymik : Is there a simp. conn. 4-dim M s.t. the inclusion i M : Emb ( S 1 , M ) → Imm ( S 1 , M ) has a non-trivial kernel on π 1 . (This map is always surjective.) Syunji Moriya (O.P .U.) Knot space 19 / 45

Main results π 1 ( Emb ( S 1 , M )) for 4-dimensional M Corollary 8 M : simply connected, d = 4, H 2 ( M ; Z ) � 0, and the intersection form on H 2 ( M ; F 2 ) is represented by a matrix of which the inverse has at least one non-zero diagonal component. Then, the inclusion i M induces an isomorphism on π 1 . In particular, π 1 ( Emb ( S 1 , M )) � H 2 ( M ; Z ) . For example, M = C P 2 # C P 2 satisfies the assumption while M = S 2 × S 2 does not. For the case H 2 ( M ) = 0, by Arone-Szymik, Emb ( S 1 , 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