Effective Computation of Generalized Spectral Sequences Andrea - PowerPoint PPT Presentation

Effective Computation of Generalized Spectral Sequences Andrea Guidolin 1 and Ana Romero 2 1 Basque Center for Applied Mathematics (Spain) 2 University of La Rioja (Spain) ISSAC, New York, July 2018 1/15

Chain complexes, homology, filtrations 2/15

Chain complexes, homology, filtrations Consider the chain complex d n +1 d n C ∗ : · · · ← − C n − 1 ← − C n ← − − C n +1 ← − · · · 2/15

Chain complexes, homology, filtrations Consider the chain complex d n +1 d n C ∗ : · · · ← − C n − 1 ← − C n ← − − C n +1 ← − · · · The n - homology group of C ∗ is defined as H n ( C ∗ ) := Ker d n Im d n +1 and its rank β n is called n -th Betti number . 2/15

Chain complexes, homology, filtrations Consider the chain complex d n +1 d n C ∗ : · · · ← − C n − 1 ← − C n ← − − C n +1 ← − · · · The n - homology group of C ∗ is defined as H n ( C ∗ ) := Ker d n Im d n +1 and its rank β n is called n -th Betti number . A filtration of the chain complex C ∗ is a sequence ( F p C ∗ ) p ∈ Z . . . ⊆ F p − 1 C ∗ ⊆ F p C ∗ ⊆ . . . ⊆ C ∗ 2/15

Spectral sequence of a filtered chain complex 3/15

Spectral sequence of a filtered chain complex Given a Z -filtration of a chain complex C ∗ = ( C n , d n ), a spectral sequence ( E r p , d r p ) is defined as follows: 3/15

Spectral sequence of a filtered chain complex Given a Z -filtration of a chain complex C ∗ = ( C n , d n ), a spectral sequence ( E r p , d r p ) is defined as follows: p , q := F p C p + q ∩ d − 1 ( F p − r C p + q − 1 ) + F p − 1 C p + q E r terms of the s.s. d ( F p + r − 1 C p + q +1 ) + F p − 1 C p + q 3/15

Spectral sequence of a filtered chain complex Given a Z -filtration of a chain complex C ∗ = ( C n , d n ), a spectral sequence ( E r p , d r p ) is defined as follows: p , q := F p C p + q ∩ d − 1 ( F p − r C p + q − 1 ) + F p − 1 C p + q E r terms of the s.s. d ( F p + r − 1 C p + q +1 ) + F p − 1 C p + q d r d r p + r − E r − E r p − E r · · · ← ← ← − p + r ← − · · · differentials induced by d p − r p 3/15

Spectral sequence of a filtered chain complex Given a Z -filtration of a chain complex C ∗ = ( C n , d n ), a spectral sequence ( E r p , d r p ) is defined as follows: p , q := F p C p + q ∩ d − 1 ( F p − r C p + q − 1 ) + F p − 1 C p + q E r terms of the s.s. d ( F p + r − 1 C p + q +1 ) + F p − 1 C p + q d r d r p + r − E r − E r p − E r · · · ← ← ← − p + r ← − · · · differentials induced by d p − r p It holds: ∼ E r +1 = Ker d r p / Im d r p + r p 3/15

� � � � � � � � � � � � � � � � � � � � � Spectral sequence of a filtered chain complex Given a Z -filtration of a chain complex C ∗ = ( C n , d n ), a spectral sequence ( E r p , d r p ) is defined as follows: p , q := F p C p + q ∩ d − 1 ( F p − r C p + q − 1 ) + F p − 1 C p + q E r terms of the s.s. d ( F p + r − 1 C p + q +1 ) + F p − 1 C p + q d r d r p + r − E r − E r p − E r · · · ← ← ← − p + r ← − · · · differentials induced by d p − r p It holds: ∼ E r +1 = Ker d r p / Im d r p + r p r=1 r=2 r=3 q q q • • • • • • • • • • • • • • • d 2 d 2 d 3 d 3 2 , 2 3 , 2 • • • • • • • • • • • • • • • 3 , 1 4 , 1 d 2 d 2 d 1 d 1 d 1 d 1 d 3 3 , 1 4 , 1 • • • • • 1 , 2 2 , 2 3 , 2 4 , 2 • • • • • • • • • • 3 , 0 d 1 d 1 d 1 d 1 p p p • • • • • 1 , 1 2 , 1 3 , 1 4 , 1 • • • • • • • • • • 3/15

Generalized filtrations and spectral systems The notion of spectral sequence of a filtered complex has been recently generalized by B. Matschke for a filtration indexed over a poset I , i.e. a collection of sub-chain complexes { F i C ∗ } i ∈ I with F i C ∗ ⊆ F j C ∗ if i ≤ j , as a set of groups, for all z ≤ s ≤ p ≤ b in I and for each degree n : S n [ z , s , p , b ] = F p C n ∩ d − 1 n ( F z C n − 1 ) + F s C n d n +1 ( F b C n +1 ) + F s C n 4/15

Generalized filtrations and spectral systems The notion of spectral sequence of a filtered complex has been recently generalized by B. Matschke for a filtration indexed over a poset I , i.e. a collection of sub-chain complexes { F i C ∗ } i ∈ I with F i C ∗ ⊆ F j C ∗ if i ≤ j , as a set of groups, for all z ≤ s ≤ p ≤ b in I and for each degree n : S n [ z , s , p , b ] = F p C n ∩ d − 1 n ( F z C n − 1 ) + F s C n d n +1 ( F b C n +1 ) + F s C n and differential maps d n : S n [ z 2 , s 2 , p 2 , b 2 ] → S n − 1 [ z 1 , s 1 , p 1 , b 1 ] . 4/15

Generalized filtrations and spectral systems The notion of spectral sequence of a filtered complex has been recently generalized by B. Matschke for a filtration indexed over a poset I , i.e. a collection of sub-chain complexes { F i C ∗ } i ∈ I with F i C ∗ ⊆ F j C ∗ if i ≤ j , as a set of groups, for all z ≤ s ≤ p ≤ b in I and for each degree n : S n [ z , s , p , b ] = F p C n ∩ d − 1 n ( F z C n − 1 ) + F s C n d n +1 ( F b C n +1 ) + F s C n and differential maps d n : S n [ z 2 , s 2 , p 2 , b 2 ] → S n − 1 [ z 1 , s 1 , p 1 , b 1 ] . Example: Z -filtration ( F p ) p ∈ Z , indices z ≤ s ≤ p ≤ b in Z : p − r p − 1 p p + r − 1 E r p z s p b S [ z, s, p, b ] 4/15

The posets Z m and D ( Z m ) 5/15

The posets Z m and D ( Z m ) Consider Z m , seen as the poset ( Z m , ≤ ) with the coordinate-wise order relation: P = ( p 1 , . . . , p m ) ≤ Q = ( q 1 , . . . , q m ) if and only if p i ≤ q i , for all 1 ≤ i ≤ m . 5/15

The posets Z m and D ( Z m ) Consider Z m , seen as the poset ( Z m , ≤ ) with the coordinate-wise order relation: P = ( p 1 , . . . , p m ) ≤ Q = ( q 1 , . . . , q m ) if and only if p i ≤ q i , for all 1 ≤ i ≤ m . A downset of Z m is a subset p ⊆ Z m such that if P ∈ p and Q ≤ P in Z m then Q ∈ p . 5/15

The posets Z m and D ( Z m ) Consider Z m , seen as the poset ( Z m , ≤ ) with the coordinate-wise order relation: P = ( p 1 , . . . , p m ) ≤ Q = ( q 1 , . . . , q m ) if and only if p i ≤ q i , for all 1 ≤ i ≤ m . A downset of Z m is a subset p ⊆ Z m such that if P ∈ p and Q ≤ P in Z m then Q ∈ p . We denote D ( Z m ) the collection of all downsets of Z m , which is a poset with respect to the inclusion ⊆ . 5/15

Motivating example 6/15

Motivating example Theorem (Serre, 1951) Let G ֒ − → E → B be a fibration and suppose the base B is 1-reduced. There is a spectral sequence converging to H ∗ ( E ) whose second page is given by E 2 p , q = H p ( B ; H q ( G )). 6/15

Motivating example Theorem (Serre, 1951) Let G ֒ − → E → B be a fibration and suppose the base B is 1-reduced. There is a spectral sequence converging to H ∗ ( E ) whose second page is given by E 2 p , q = H p ( B ; H q ( G )). Theorem (Matschke, 2013) Consider a tower of fibrations E N B G M and suppose the base B is 1-reduced. There exists a D ( Z 2 )-spectral system converging to H ∗ ( E ) whose second page is given by P = ( p 1 , p 2 ) ∈ Z 2 . S ∗ n ( P ; 2) = H p 2 ( B ; H p 1 ( M ; H n − p 1 − p 2 ( G ))) , 6/15

Multidimensional persistence and persistence of I -filtrations Multidimensional filtrations (or Z m -filtrations) of simplicial complexes: K 1 N ′ K 2 N ′ · · · K NN ′ · · · · · · · · · · · · K 12 K 22 K N 2 · · · K 11 K 21 K N 1 7/15

Multidimensional persistence and persistence of I -filtrations Multidimensional filtrations (or Z m -filtrations) of simplicial complexes: K 1 N ′ K 2 N ′ · · · K NN ′ · · · · · · · · · · · · K 12 K 22 K N 2 · · · K 11 K 21 K N 1 Associated invariant: rank invariant β P , Q P , Q ∈ Z m , := dim F Im( H n ( K P ) → H n ( K Q )) , P ≤ Q . n 7/15

Multidimensional persistence and persistence of I -filtrations Multidimensional filtrations (or Z m -filtrations) of simplicial complexes: K 1 N ′ K 2 N ′ · · · K NN ′ · · · · · · · · · · · · K 12 K 22 K N 2 · · · K 11 K 21 K N 1 Associated invariant: rank invariant β P , Q P , Q ∈ Z m , := dim F Im( H n ( K P ) → H n ( K Q )) , P ≤ Q . n Similarly, for an I -filtration ( F i ) i ∈ I , we define rank invariant the collection of integers β n ( v , w ) := dim F Im( H n ( F v ) → H n ( F w )) , v , w ∈ I , v ≤ w . 7/15

Algorithms 8/15

Algorithms We have developed a set of programs computing generalized spectral sequences implemented in the Computer Algebra System Kenzo. 8/15

Algorithms We have developed a set of programs computing generalized spectral sequences implemented in the Computer Algebra System Kenzo. If the I -filtered chain complex C ∗ is of finite type, the groups S n [ z , s , p , b ] can be determined by means of diagonalization operations on matrices. 8/15

Algorithms We have developed a set of programs computing generalized spectral sequences implemented in the Computer Algebra System Kenzo. If the I -filtered chain complex C ∗ is of finite type, the groups S n [ z , s , p , b ] can be determined by means of diagonalization operations on matrices. The result is a basis-divisors description of the group, that is: a list of combinations ( c 1 , . . . , c α + k ) a list of torsion coefficients ( b 1 , . . . , b k , 0 , α . . ., 0). 8/15