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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
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