Computing Spectral Sequences

with the Kenzo system

A. Romero1, J. Rubio1, F. Sergeraert2

1Universidad de La Rioja (Spain)2Universite Joseph Fourier, Grenoble (France)

Madrid, August 2006

Spectral sequences

Spectral sequences

Definition. A Spectral Sequence E = Er, dr is a family of Z-bigraded

modules E1, E2,. . . , each provided with a differential dr = drp,q of bide-

gree (−r, r − 1) and with isomorphisms H(Er, dr) ∼= Er+1, r = 1, 2, . . .

Spectral sequences

Definition. A Spectral Sequence E = Er, dr is a family of Z-bigraded

modules E1, E2,. . . , each provided with a differential dr = drp,q of bide-

gree (−r, r − 1) and with isomorphisms H(Er, dr) ∼= Er+1, r = 1, 2, . . .

“If we think of a spectral sequence as a black box, then the input is a differential bi-

graded module, usually E1∗,∗ , and, with each turn of the handle, the machine computes

a successive homology according to a sequence of differentials. If some differential is

unknown, then some other (any other) principle is needed to proceed. [...] In the

nontrivial cases, it is often a deep geometric idea that is caught up in the knowledge

of a differential.”

John McCleary, User’s guide to spectral sequences (Publish or Perish, 1985)

Effective homology

Effective homology

X1, X2, . . . , Xn

ϕ

XEH1 , XEH

2 , . . . , XEHn

X XEH

Effective homology

X1, X2, . . . , Xn

ϕ

XEH1 , XEH

2 , . . . , XEHn

X XEH

Effective homology

X1, X2, . . . , Xn

ϕ

XEH1 , XEH

2 , . . . , XEHn

ϕEH

X XEH

Spectral sequences vs. effective homology

Spectral sequences vs. effective homology

X1, X2, . . . , Xn

ϕ

XEH1 , XEH

2 , . . . , XEHn H∗(X1), H∗(X2), . . . , H∗(Xn)

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

X XEH H∗(X)

Spectral sequences vs. effective homology

X1, X2, . . . , Xn

ϕ

XEH1 , XEH

2 , . . . , XEHn H∗(X1), H∗(X2), . . . , H∗(Xn)

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

X XEH ???H∗(X)???

Spectral sequences vs. effective homology

X1, X2, . . . , Xn

ϕ

XEH1 , XEH

2 , . . . , XEHn H∗(X1), H∗(X2), . . . , H∗(Xn)

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

X XEH ???H∗(X)???

Spectral sequences vs. effective homology

X1, X2, . . . , Xn

ϕ

XEH1 , XEH

2 , . . . , XEHn H∗(X1), H∗(X2), . . . , H∗(Xn)

SS

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

X XEH H∗(X)

Spectral sequences vs. effective homology

X1, X2, . . . , Xn

ϕ

XEH1 , XEH

2 , . . . , XEHn H∗(X1), H∗(X2), . . . , H∗(Xn)

OO

SS

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

X XEH H∗(X)

Spectral sequences vs. effective homology

X1, X2, . . . , Xn

ϕ

XEH1 , XEH

2 , . . . , XEHn

ϕEH

H∗(X1), H∗(X2), . . . , H∗(Xn)OO

SS

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

X XEH H∗(X)

Spectral sequences vs. effective homology

X1, X2, . . . , Xn

ϕ

XEH1 , XEH

2 , . . . , XEHn

ϕEH

H∗(X1), H∗(X2), . . . , H∗(Xn)OO

SS

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

X XEH // H∗(X)

Spectral sequences vs. effective homology

X1, X2, . . . , Xn

ϕ

XEH1 , XEH

2 , . . . , XEHn

ϕEH

H∗(X1), H∗(X2), . . . , H∗(Xn)OO

SS

................ //

X XEH // H∗(X)

Effective homology

Effective homology

Definition. A reduction ρ between two chain complexes A and B (de-

noted by ρ : A⇒⇒⇒B) is a triple ρ = (f, g, h)

A

h f

++B

gkk

satisfying the following relations:

......fg = idB; gf + dAh + hdA = idA;

......fh = 0; hg = 0; hh = 0.

Effective homology

Definition. A reduction ρ between two chain complexes A and B (de-

noted by ρ : A⇒⇒⇒B) is a triple ρ = (f, g, h)

A

h f

++B

gkk

satisfying the following relations:

......fg = idB; gf + dAh + hdA = idA;

......fh = 0; hg = 0; hh = 0.

Remark. If A⇒⇒⇒B, then A = B ⊕ C, with C acyclic, which implies

that H∗(A) ∼= H∗(B).

Effective homology

Definition. A (strong chain) equivalence between the complexes A

and B (A⇐⇐⇐⇒⇒⇒B) is a triple (D, ρ, ρ′) where D is a chain complex,

ρ : D⇒⇒⇒A and ρ′ : D⇒⇒⇒B.

Effective homology

Definition. A (strong chain) equivalence between the complexes A

and B (A⇐⇐⇐⇒⇒⇒B) is a triple (D, ρ, ρ′) where D is a chain complex,

ρ : D⇒⇒⇒A and ρ′ : D⇒⇒⇒B.

A⇐⇐⇐D⇒⇒⇒E14

10⇐⇐⇐ 42

30⇒⇒⇒ 21

15

Effective homology

Definition. A (strong chain) equivalence between the complexes A

and B (A⇐⇐⇐⇒⇒⇒B) is a triple (D, ρ, ρ′) where D is a chain complex,

ρ : D⇒⇒⇒A and ρ′ : D⇒⇒⇒B.

A⇐⇐⇐D⇒⇒⇒E14

10⇐⇐⇐ 42

30⇒⇒⇒ 21

15

Definition. An object with effective homology is a triple (X, EC, ε)

where EC is an effective chain complex and C(X)ε⇐⇐⇐⇒⇒⇒ EC.

Effective homology

Definition. A (strong chain) equivalence between the complexes A

and B (A⇐⇐⇐⇒⇒⇒B) is a triple (D, ρ, ρ′) where D is a chain complex,

ρ : D⇒⇒⇒A and ρ′ : D⇒⇒⇒B.

A⇐⇐⇐D⇒⇒⇒E14

10⇐⇐⇐ 42

30⇒⇒⇒ 21

15

Definition. An object with effective homology is a triple (X, EC, ε)

where EC is an effective chain complex and C(X)ε⇐⇐⇐⇒⇒⇒ EC.

Remark. This implies that H∗(X) ∼= H∗(EC).

The Kenzo system

The Kenzo system

Kenzo is a Lisp 16,000 lines program devoted to Symbolic Computation in

Algebraic Topology.

The Kenzo system

Kenzo is a Lisp 16,000 lines program devoted to Symbolic Computation in

Algebraic Topology.

It works with complex algebraic structures: chain complexes, differential

graded algebras, simplicial sets, simplicial groups, morphisms between these

objects, etc.

The Kenzo system

Kenzo is a Lisp 16,000 lines program devoted to Symbolic Computation in

Algebraic Topology.

It works with complex algebraic structures: chain complexes, differential

graded algebras, simplicial sets, simplicial groups, morphisms between these

objects, etc.

In particular, it implements the effective homology method and can be used

to compute homology groups of complicated spaces.

The Kenzo system

The Kenzo system

How does Kenzo compute the homology groups of a (complicated) space?

The Kenzo system

How does Kenzo compute the homology groups of a (complicated) space?

• If the complex is effective, then its homology groups can be determined

by means of elementary operations with integer matrices.

The Kenzo system

How does Kenzo compute the homology groups of a (complicated) space?

• If the complex is effective, then its homology groups can be determined

by means of elementary operations with integer matrices.

• Otherwise, the program uses the effective homology, because the ho-

mology groups of the initial space are isomorphic to those of its effective

complex.

Computing spectral sequences

Computing spectral sequences

For the spectral sequence associated with a filtered chain complex, there

exists a formal expression for the groups Erp,q, but they can only be deter-

mined in very simple situations.

Computing spectral sequences

For the spectral sequence associated with a filtered chain complex, there

exists a formal expression for the groups Erp,q, but they can only be deter-

mined in very simple situations.

However, the effective homology method allows us to determine, as a by-

product, this kind of spectral sequences, obtaining a real algorithm.

Computing spectral sequences

For the spectral sequence associated with a filtered chain complex, there

exists a formal expression for the groups Erp,q, but they can only be deter-

mined in very simple situations.

However, the effective homology method allows us to determine, as a by-

product, this kind of spectral sequences, obtaining a real algorithm.

This algorithm has been implemented as a set of programs (about 2500

lines) enhancing the Kenzo system, that determine the groups Erp,q and

also the differential maps drp,q for every level r.

Computing spectral sequences

Computing spectral sequences

• If the filtered complex is effective, then the spectral sequence can be

computed through elementary methods with integer matrices.

Computing spectral sequences

• If the filtered complex is effective, then the spectral sequence can be

computed through elementary methods with integer matrices.

• Otherwise, the effective homology is needed to compute it by means

of an analogous spectral sequence deduced of an appropriate filtration

on the associated effective complex.

Eilenberg-Moore spectral sequence

Eilenberg-Moore spectral sequence

Given a simplicial set X and its loop space ΩX, there exists a spectral

sequence Erp,q that converges to the homology groups H∗(ΩX).

Eilenberg-Moore spectral sequence

Given a simplicial set X and its loop space ΩX, there exists a spectral

sequence Erp,q that converges to the homology groups H∗(ΩX).

Example:

X = ΩS3 ∪2 e3

ΩX = Ω(ΩS3 ∪2 e3)

Example

X = ΩS3 ∪2 e3

ΩX = Ω(ΩS3 ∪2 e3)

Example

X = ΩS3 ∪2 e3

ΩX = Ω(ΩS3 ∪2 e3)

Using the effective homology method

Example

X = ΩS3 ∪2 e3

ΩX = Ω(ΩS3 ∪2 e3)

Using the effective homology method

• Kenzo computes the homology groups Hi(ΩX)

Example

X = ΩS3 ∪2 e3

ΩX = Ω(ΩS3 ∪2 e3)

Using the effective homology method

• Kenzo computes the homology groups Hi(ΩX)

• the new programs compute the Eilenberg-Moore spectral sequence be-

tween X = ΩS3 ∪2 e3 and ΩX = Ω(ΩS3 ∪2 e3)

Example

X = ΩS3 ∪2 e3

ΩX = Ω(ΩS3 ∪2 e3)

Example

X = ΩS3 ∪2 e3

ΩX = Ω(ΩS3 ∪2 e3)

i 0 1 2 3 4 5

Hi(ΩX) Z Z2 Z2 Z⊕ Z22 Z4

2 Z⊕ Z62

Example

X = ΩS3 ∪2 e3

ΩX = Ω(ΩS3 ∪2 e3)

i 0 1 2 3 4 5

Hi(ΩX) Z Z2 Z2 Z⊕ Z22 Z4

2 Z⊕ Z62

i 6 7 8 9

Hi(ΩX) Z122 ⊕ Z6 Z20

2 Z⊕ Z352 ⊕ Z10 · · ·

q Z2

Z62

14 Z112 Z2

13 Z112 Z5

2

12 Z52 Z7

2 Z2

11 0 Z62 Z4

2

10 Z2 ⊕ Z10 ⊕ Z Z2 Z42 Z2

9 0 0 Z32 Z3

2

8 0 0 Z6 Z22 Z2

7 0 0 0 Z22

6 0 Z Z2 Z2

5 0 0 Z2

4 0 Z Z2

3 0 02 0 Z2

1 00 Z

0 1 2 3 4 5 6 7 8 p