Basic Perturbation Lemma and effective homology:
application to the computation of homology of 2-types
Ana Romero
Universidad de La Rioja (Spain)
(Joint work with J. Rubio and F. Sergeraert)
Homological Perturbation Theory Workshop
Galway, December 2014
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
1 / 29
Introduction
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
2 / 29
Introduction
The Basic Perturbation Lemma was discovered by Shih Weishu in
1960, and the abstract modern form was given by Ronnie Brown in
1964 (after some unpublished results by M. Barrat).
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
2 / 29
Introduction
The Basic Perturbation Lemma was discovered by Shih Weishu in
1960, and the abstract modern form was given by Ronnie Brown in
1964 (after some unpublished results by M. Barrat).
We have used it combined with the effective homology method, in
order to determine:
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
2 / 29
Introduction
The Basic Perturbation Lemma was discovered by Shih Weishu in
1960, and the abstract modern form was given by Ronnie Brown in
1964 (after some unpublished results by M. Barrat).
We have used it combined with the effective homology method, in
order to determine:
Homology of cones, bicomplexes, twisted Cartesian products, loop
spaces, classifying spaces...
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
2 / 29
Introduction
The Basic Perturbation Lemma was discovered by Shih Weishu in
1960, and the abstract modern form was given by Ronnie Brown in
1964 (after some unpublished results by M. Barrat).
We have used it combined with the effective homology method, in
order to determine:
Homology of cones, bicomplexes, twisted Cartesian products, loop
spaces, classifying spaces...
Homotopy groups of spaces by means of Whitehead and Postnikov
towers.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
2 / 29
Introduction
The Basic Perturbation Lemma was discovered by Shih Weishu in
1960, and the abstract modern form was given by Ronnie Brown in
1964 (after some unpublished results by M. Barrat).
We have used it combined with the effective homology method, in
order to determine:
Homology of cones, bicomplexes, twisted Cartesian products, loop
spaces, classifying spaces...
Homotopy groups of spaces by means of Whitehead and Postnikov
towers.
Homology of digital images by means of Discrete Vector Fields.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
2 / 29
Introduction
The Basic Perturbation Lemma was discovered by Shih Weishu in
1960, and the abstract modern form was given by Ronnie Brown in
1964 (after some unpublished results by M. Barrat).
We have used it combined with the effective homology method, in
order to determine:
Homology of cones, bicomplexes, twisted Cartesian products, loop
spaces, classifying spaces...
Homotopy groups of spaces by means of Whitehead and Postnikov
towers.
Homology of digital images by means of Discrete Vector Fields.
Spectral sequences associated with filtered complexes (including Serre
and Eilenberg-Moore spectral sequences).
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
2 / 29
Introduction
The Basic Perturbation Lemma was discovered by Shih Weishu in
1960, and the abstract modern form was given by Ronnie Brown in
1964 (after some unpublished results by M. Barrat).
We have used it combined with the effective homology method, in
order to determine:
Homology of cones, bicomplexes, twisted Cartesian products, loop
spaces, classifying spaces...
Homotopy groups of spaces by means of Whitehead and Postnikov
towers.
Homology of digital images by means of Discrete Vector Fields.
Spectral sequences associated with filtered complexes (including Serre
and Eilenberg-Moore spectral sequences).
Persistent homology.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
2 / 29
Introduction
The Basic Perturbation Lemma was discovered by Shih Weishu in
1960, and the abstract modern form was given by Ronnie Brown in
1964 (after some unpublished results by M. Barrat).
We have used it combined with the effective homology method, in
order to determine:
Homology of cones, bicomplexes, twisted Cartesian products, loop
spaces, classifying spaces...
Homotopy groups of spaces by means of Whitehead and Postnikov
towers.
Homology of digital images by means of Discrete Vector Fields.
Spectral sequences associated with filtered complexes (including Serre
and Eilenberg-Moore spectral sequences).
Persistent homology.
Koszul homology.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
2 / 29
Introduction
The Basic Perturbation Lemma was discovered by Shih Weishu in
1960, and the abstract modern form was given by Ronnie Brown in
1964 (after some unpublished results by M. Barrat).
We have used it combined with the effective homology method, in
order to determine:
Homology of cones, bicomplexes, twisted Cartesian products, loop
spaces, classifying spaces...
Homotopy groups of spaces by means of Whitehead and Postnikov
towers.
Homology of digital images by means of Discrete Vector Fields.
Spectral sequences associated with filtered complexes (including Serre
and Eilenberg-Moore spectral sequences).
Persistent homology.
Koszul homology.
Bousfield-Kan spectral sequence for computing homotopy groups of
spaces.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
2 / 29
Introduction
The Basic Perturbation Lemma was discovered by Shih Weishu in
1960, and the abstract modern form was given by Ronnie Brown in
1964 (after some unpublished results by M. Barrat).
We have used it combined with the effective homology method, in
order to determine:
Homology of cones, bicomplexes, twisted Cartesian products, loop
spaces, classifying spaces...
Homotopy groups of spaces by means of Whitehead and Postnikov
towers.
Homology of digital images by means of Discrete Vector Fields.
Spectral sequences associated with filtered complexes (including Serre
and Eilenberg-Moore spectral sequences).
Persistent homology.
Koszul homology.
Bousfield-Kan spectral sequence for computing homotopy groups of
spaces.
Homology of groups.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
2 / 29
Introduction
The Basic Perturbation Lemma was discovered by Shih Weishu in
1960, and the abstract modern form was given by Ronnie Brown in
1964 (after some unpublished results by M. Barrat).
We have used it combined with the effective homology method, in
order to determine:
Homology of cones, bicomplexes, twisted Cartesian products, loop
spaces, classifying spaces...
Homotopy groups of spaces by means of Whitehead and Postnikov
towers.
Homology of digital images by means of Discrete Vector Fields.
Spectral sequences associated with filtered complexes (including Serre
and Eilenberg-Moore spectral sequences).
Persistent homology.
Koszul homology.
Bousfield-Kan spectral sequence for computing homotopy groups of
spaces.
Homology of groups.
Homology of 2-types.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
2 / 29
Effective homology
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
3 / 29
Effective homology
Definition
A reduction ρ between two chain complexes C∗ and D∗ (denoted by
ρ : C∗ ⇒
⇒ D∗ ) is a triple ρ = (f , g , h)
h
f
C∗ k
+
D∗
g
satisfying the following relations:
1) fg = IdD∗ ;
2) dC h + hdC = IdC∗ −gf ;
3) fh = 0;
Ana Romero
hg = 0;
hh = 0.
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
3 / 29
Effective homology
Definition
A reduction ρ between two chain complexes C∗ and D∗ (denoted by
ρ : C∗ ⇒
⇒ D∗ ) is a triple ρ = (f , g , h)
h
f
C∗ k
+
D∗
g
satisfying the following relations:
1) fg = IdD∗ ;
2) dC h + hdC = IdC∗ −gf ;
3) fh = 0;
hg = 0;
hh = 0.
If C∗ ⇒
⇒ D∗ , then C∗ ∼
= D∗ ⊕ A∗ , with A∗ acyclic, which implies that
Hn (C∗ ) ∼
= Hn (D∗ ) for all n.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
3 / 29
Effective homology
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
4 / 29
Effective homology
Definition
A (strong chain) equivalence ε between C∗ and D∗ , ε : C∗ ⇐
⇐⇒
⇒ D∗ , is a
triple ε = (B∗ , ρ, ρ0 ) where B∗ is a chain complex, ρ : B∗ ⇒
⇒ C∗ and
ρ0 : B∗ ⇒
⇒ D∗ .
42
B∗
30
C∗
Ana Romero
s{ s{
#+ #+
D∗
14
10
t| t|
Basic Perturbation Lemma, effective homology and homology of 2-types
"* "* 21
15
HPT Workshop
4 / 29
Effective homology
Definition
A (strong chain) equivalence ε between C∗ and D∗ , ε : C∗ ⇐
⇐⇒
⇒ D∗ , is a
triple ε = (B∗ , ρ, ρ0 ) where B∗ is a chain complex, ρ : B∗ ⇒
⇒ C∗ and
ρ0 : B∗ ⇒
⇒ D∗ .
42
B∗
30
C∗
s{ s{
#+ #+
D∗
14
10
t| t|
"* "* 21
15
Definition
An object with effective homology is a quadruple (X , C∗ (X ), EC∗ , ε) where
EC∗ is an effective chain complex and ε : C∗ (X ) ⇐
⇐⇒
⇒ EC∗ .
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
4 / 29
Effective homology
Definition
A (strong chain) equivalence ε between C∗ and D∗ , ε : C∗ ⇐
⇐⇒
⇒ D∗ , is a
triple ε = (B∗ , ρ, ρ0 ) where B∗ is a chain complex, ρ : B∗ ⇒
⇒ C∗ and
ρ0 : B∗ ⇒
⇒ D∗ .
42
B∗
30
C∗
s{ s{
#+ #+
D∗
14
10
t| t|
"* "* 21
15
Definition
An object with effective homology is a quadruple (X , C∗ (X ), EC∗ , ε) where
EC∗ is an effective chain complex and ε : C∗ (X ) ⇐
⇐⇒
⇒ EC∗ .
This implies that Hn (X ) ∼
= Hn (EC∗ ) for all n.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
4 / 29
Effective homology
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
5 / 29
Effective homology
Meta-theorem
Let X1 , . . . , Xk be a collection of objects with effective homology and Φ be
a reasonable construction process:
Φ : (X1 , . . . , Xk ) → X .
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
5 / 29
Effective homology
Meta-theorem
Let X1 , . . . , Xk be a collection of objects with effective homology and Φ be
a reasonable construction process:
Φ : (X1 , . . . , Xk ) → X .
Then there exists a version with effective homology ΦEH
ΦEH : ((X1 , C (X1 ), EC1 , ε1 ), . . . , (Xk , C (Xk ), ECk , εk )) → (X , C (X ), EC , ε)
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
5 / 29
Effective homology
Meta-theorem
Let X1 , . . . , Xk be a collection of objects with effective homology and Φ be
a reasonable construction process:
Φ : (X1 , . . . , Xk ) → X .
Then there exists a version with effective homology ΦEH
ΦEH : ((X1 , C (X1 ), EC1 , ε1 ), . . . , (Xk , C (Xk ), ECk , εk )) → (X , C (X ), EC , ε)
The process is perfectly stable and can be again used with X for further
calculations.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
5 / 29
Effective homology
Meta-theorem
Let X1 , . . . , Xk be a collection of objects with effective homology and Φ be
a reasonable construction process:
Φ : (X1 , . . . , Xk ) → X .
Then there exists a version with effective homology ΦEH
ΦEH : ((X1 , C (X1 ), EC1 , ε1 ), . . . , (Xk , C (Xk ), ECk , εk )) → (X , C (X ), EC , ε)
The process is perfectly stable and can be again used with X for further
calculations.
Examples: twisted Cartesian products, loop spaces, suspensions, simplicial
Abelian groups generated by simplicial sets, . . . .
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
5 / 29
The Kenzo system
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
6 / 29
The Kenzo system
The Kenzo system uses the notion of object with effective homology to
compute homology groups of some complicated spaces.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
6 / 29
The Kenzo system
The Kenzo system uses the notion of object with effective homology to
compute homology groups of some complicated spaces.
If the complex is effective, then its homology groups can be
determined by means of diagonalization algorithms on matrices.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
6 / 29
The Kenzo system
The Kenzo system uses the notion of object with effective homology to
compute homology groups of some complicated spaces.
If the complex is effective, then its homology groups can be
determined by means of diagonalization algorithms on matrices.
Otherwise, the program uses the effective homology.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
6 / 29
The Kenzo system
The Kenzo system uses the notion of object with effective homology to
compute homology groups of some complicated spaces.
If the complex is effective, then its homology groups can be
determined by means of diagonalization algorithms on matrices.
Otherwise, the program uses the effective homology.
Example:
X = Ω(Ω(Ω(P ∞ R/P 3 R) ∪4 D 4 ) ∪2 D 2 )
H5 (X ) = Z23
2 ⊕ Z8 ⊕ Z16
3
3
H6 (X ) = Z52
2 ⊕ Z4 ⊕ Z
3
H7 (X ) = Z113
2 ⊕ Z4 ⊕ Z8 ⊕ Z16 ⊕ Z32 ⊕ Z
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
6 / 29
Perturbation theorems
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
7 / 29
Perturbation theorems
Definition
Let (C∗ , d) be a chain complex. A perturbation δ : C∗ → C∗−1 is an
operator of degree −1 satisfying (d + δ) ◦ (d + δ) = 0.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
7 / 29
Perturbation theorems
Definition
Let (C∗ , d) be a chain complex. A perturbation δ : C∗ → C∗−1 is an
operator of degree −1 satisfying (d + δ) ◦ (d + δ) = 0.
This produces a new perturbed chain complex (C∗ , d + δ)
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
7 / 29
Perturbation theorems
Definition
Let (C∗ , d) be a chain complex. A perturbation δ : C∗ → C∗−1 is an
operator of degree −1 satisfying (d + δ) ◦ (d + δ) = 0.
This produces a new perturbed chain complex (C∗ , d + δ)
Let ρ = (f , g , h) be a reduction
h
(C∗ , dC ) m
Ana Romero
f
-
(D∗ , dD )
g
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
7 / 29
Perturbation theorems
Definition
Let (C∗ , d) be a chain complex. A perturbation δ : C∗ → C∗−1 is an
operator of degree −1 satisfying (d + δ) ◦ (d + δ) = 0.
This produces a new perturbed chain complex (C∗ , d + δ)
Let ρ = (f , g , h) be a reduction
h
(C∗ , dC ) m
f
-
(D∗ , dD )
g
What happens if we perturb dC or dD ?
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
7 / 29
Perturbation theorems
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
8 / 29
Perturbation theorems
Theorem (Trivial Perturbation Lemma, TPL)
Let ρ = (f , g , h) : C∗ ⇒
⇒ D∗ be a reduction, and δD a perturbation of dD .
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
8 / 29
Perturbation theorems
Theorem (Trivial Perturbation Lemma, TPL)
Let ρ = (f , g , h) : C∗ ⇒
⇒ D∗ be a reduction, and δD a perturbation of dD .
h
Then we have a new reduction: (C∗ , dC + δC ) n
f
.
(D∗ , dD + δD )
g
where δC = g ◦ δD ◦ f .
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
8 / 29
Perturbation theorems
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
9 / 29
Perturbation theorems
Theorem (Basic Perturbation Lemma, BPL)
Let ρ = (f , g , h) : C∗ ⇒
⇒ D∗ be a reduction, and δC a perturbation of dC
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
9 / 29
Perturbation theorems
Theorem (Basic Perturbation Lemma, BPL)
Let ρ = (f , g , h) : C∗ ⇒
⇒ D∗ be a reduction, and δC a perturbation of dC
such that the composition h ◦ δC is pointwise nilpotent.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
9 / 29
Perturbation theorems
Theorem (Basic Perturbation Lemma, BPL)
Let ρ = (f , g , h) : C∗ ⇒
⇒ D∗ be a reduction, and δC a perturbation of dC
such that the composition h ◦ δC is pointwise nilpotent. Then we have a
h0
new reduction: (C∗ , dC + δC ) n
Ana Romero
f0
.
(D∗ , dD + δD )
g0
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
9 / 29
Perturbation theorems
Theorem (Basic Perturbation Lemma, BPL)
Let ρ = (f , g , h) : C∗ ⇒
⇒ D∗ be a reduction, and δC a perturbation of dC
such that the composition h ◦ δC is pointwise nilpotent. Then we have a
h0
f0
new reduction: (C∗ , dC + δC ) n
.
(D∗ , dD + δD ) where
g0
δD = f ◦ δC ◦ φ ◦ g = f ◦ ψ ◦ δC ◦ g ;
g0 = φ ◦ g;
f 0 = f ◦ ψ = f ◦ (IdC∗ −δC ◦ φ ◦ h);
h0 = φ ◦ h = h ◦ ψ;
with the operators φ and ψ defined by
φ=
∞
X
i=0
Ana Romero
(−1)i (h ◦ δC )i ,
ψ=
∞
X
(−1)i (δC ◦ h)i = IdC∗ −δC ◦ φ ◦ h
i=0
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
9 / 29
Algebraic cone construction
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
10 / 29
Algebraic cone construction
Definition
Let Φ : (C∗ , dC ) → (D∗ , dD ) be a chain complex morphism. The Cone
of Φ, Cone(Φ)∗ ≡ (A∗ , dA ), is a chain complex given by An = Cn ⊕ Dn+1 ,
with differential map dA (c, d) = (dC (c), Φ(c) − dD (d)).
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
10 / 29
Algebraic cone construction
Definition
Let Φ : (C∗ , dC ) → (D∗ , dD ) be a chain complex morphism. The Cone
of Φ, Cone(Φ)∗ ≡ (A∗ , dA ), is a chain complex given by An = Cn ⊕ Dn+1 ,
with differential map dA (c, d) = (dC (c), Φ(c) − dD (d)).
... o
. . . o
Ana Romero
dCn+2
Cn−1 o
dCn
Cn o
dCn+1
An−1
Φn
An
Φn+1 An+1 Φn+2 An+2
Dn o
−dDn+1
Dn+1 o
−dDn+2
Cn+1 o
−dDn+3
Dn+2 o
Cn+2 o
Dn+3 o
Basic Perturbation Lemma, effective homology and homology of 2-types
...
...
HPT Workshop
10 / 29
Algebraic cone construction
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
11 / 29
Algebraic cone construction
Theorem
A general algorithm can be produced:
Input: Φ : C∗ → D∗ and effective homologies for C∗ and D∗ .
Output: An effective homology for A∗ = Cone(Φ).
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
11 / 29
Algebraic cone construction
Theorem
A general algorithm can be produced:
Input: Φ : C∗ → D∗ and effective homologies for C∗ and D∗ .
Output: An effective homology for A∗ = Cone(Φ).
Proof:
Ana Romero
.
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
11 / 29
Algebraic cone construction
Theorem
A general algorithm can be produced:
Input: Φ : C∗ → D∗ and effective homologies for C∗ and D∗ .
Output: An effective homology for A∗ = Cone(Φ).
.
Proof:
h
? C∗O
f
D∗O _
g
f0
EC∗
h0
g0
ED∗
.
d
0
Φ
−d
DA
Ana Romero
d
0
Φ
−d
DA0
d
0
Φ
−d
F
d
0
Φ
−d
d
0
G
Basic Perturbation Lemma, effective homology and homology of 2-types
Φ
−d
H
HPT Workshop
11 / 29
Algebraic cone construction
Theorem
A general algorithm can be produced:
Input: Φ : C∗ → D∗ and effective homologies for C∗ and D∗ .
Output: An effective homology for A∗ = Cone(Φ).
.
Proof:
1
Particular case Φ = 0
(direct sum).
h
f
/
0
? C∗O
g
D∗O _
f0
g0
/ ED∗
0
EC∗
h0
.
d
0
Φ
−d
DA
Ana Romero
d
0
Φ
−d
DA0
d
0
Φ
−d
F
d
0
Φ
−d
d
0
G
Basic Perturbation Lemma, effective homology and homology of 2-types
Φ
−d
H
HPT Workshop
11 / 29
Algebraic cone construction
Theorem
A general algorithm can be produced:
Input: Φ : C∗ → D∗ and effective homologies for C∗ and D∗ .
Output: An effective homology for A∗ = Cone(Φ).
.
Proof:
1
Particular case Φ = 0
(direct sum).
f
/
0
? C∗O
h
g
D∗O _
f0
g0
/ ED∗
0
EC∗
h0
.
dC
0
0
−dD
DA
Ana Romero
dEC
0
0
−dED
DA0
f
0
0
f0
F
g
0
0
g0
G
Basic Perturbation Lemma, effective homology and homology of 2-types
h
0
0
−h0
H
HPT Workshop
11 / 29
Algebraic cone construction
Theorem
A general algorithm can be produced:
Input: Φ : C∗ → D∗ and effective homologies for C∗ and D∗ .
Output: An effective homology for A∗ = Cone(Φ).
.
Proof:
1
2
Particular case Φ = 0
(direct sum).
f
We install Φ.
g
dC
Φ
0
−dD
DA
Ana Romero
0
−dED
DA0
h0
g0
/ ED∗
0
EC∗
dEC
0
D∗O _
f0
The reduction is not valid.
/
Φ
? C∗O
h
.
f
0
0
f0
F
g
0
0
g0
G
Basic Perturbation Lemma, effective homology and homology of 2-types
h
0
0
−h0
H
HPT Workshop
11 / 29
Algebraic cone construction
Theorem
A general algorithm can be produced:
Input: Φ : C∗ → D∗ and effective homologies for C∗ and D∗ .
Output: An effective homology for A∗ = Cone(Φ).
.
h0 Φh
Proof:
1
2
3
dC
Φ
Particular case Φ = 0
(direct sum).
f
We install Φ.
g
0
−dD
DA0
0
−dED
7
h0 Φg
EC∗
dEC
f 0 Φg
/
Φ
D∗O _
f 0 Φh
f 0 Φg
We apply the BPL.
DA
Ana Romero
? C∗O
h
'/
f0
h0
g0
ED∗
.
f
f 0 Φh
F
0
f0
g
−h0 Φg
0
g0
G
Basic Perturbation Lemma, effective homology and homology of 2-types
h
0
−h0
h0 Φh
H
HPT Workshop
11 / 29
Bicomplex
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Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
12 / 29
Bicomplex
Definition
A bicomplex C∗,∗ is a bigraded free Z-module C∗,∗ = {Cp,q }p,q∈Z provided
0 :C
00
with morphisms dp,q
p,q → Cp−1,q and dp,q : Cp,q → Cp,q−1 satisfying
0
0
00
00
0
00 + d 00
0
dp−1,q ◦ dp,q = 0, dp,q−1 ◦ dp,q = 0, and dp,q−1 ◦ dp,q
p−1,q ◦ dp,q = 0.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
12 / 29
Bicomplex
Definition
A bicomplex C∗,∗ is a bigraded free Z-module C∗,∗ = {Cp,q }p,q∈Z provided
0 :C
00
with morphisms dp,q
p,q → Cp−1,q and dp,q : Cp,q → Cp,q−1 satisfying
0
0
00
00
0
00 + d 00
0
dp−1,q ◦ dp,q = 0, dp,q−1 ◦ dp,q = 0, and dp,q−1 ◦ dp,q
p−1,q ◦ dp,q = 0.
The total (chain) complex
L T∗ = T∗ (C∗,∗ ) = (Tn , dn )n∈Z is the chain
complex given by Tn = p+q=n Cp,q and differential map
0 (x) + d 00 (x) for x ∈ C
dn (x) = dp,q
p,q .
p,q
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
12 / 29
Bicomplex
Definition
A bicomplex C∗,∗ is a bigraded free Z-module C∗,∗ = {Cp,q }p,q∈Z provided
0 :C
00
with morphisms dp,q
p,q → Cp−1,q and dp,q : Cp,q → Cp,q−1 satisfying
0
0
00
00
0
00 + d 00
0
dp−1,q ◦ dp,q = 0, dp,q−1 ◦ dp,q = 0, and dp,q−1 ◦ dp,q
p−1,q ◦ dp,q = 0.
The total (chain) complex
L T∗ = T∗ (C∗,∗ ) = (Tn , dn )n∈Z is the chain
complex given by Tn = p+q=n Cp,q and differential map
0 (x) + d 00 (x) for x ∈ C
dn (x) = dp,q
p,q .
p,q
O
q
o
C1,3
o
C1,2
o
C1,1
o
C1,0
C0,3
C0,2
C0,1
C0,0
Ana Romero
o
o
o
o
o
C3,3
o
C3,2
o
C3,1
o
C3,0
C2,3
d0
C2,2
d 00
C2,1
C2,0
p
/
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
12 / 29
Bicomplex
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
13 / 29
Bicomplex
Theorem
A general algorithm can be produced:
Input: A bounded bicomplex C∗ and effective homologies of each
column.
Output: An effective homology for C∗ .
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
13 / 29
Bicomplex
Theorem
A general algorithm can be produced:
Input: A bounded bicomplex C∗ and effective homologies of each
column.
Output: An effective homology for C∗ .
Proof:
1
We consider only
the vertical
arrows.
O
C0,3
C1,3
C2,3
C3,3
C0,2
C1,2
00
d1,2
C0,1
C0,0
Ana Romero
O
q
C2,2
C3,2
00
d2,2
C1,1
C1,0
C2,1
C2,0
EC30
0
EC2
EC10
C3,1
C3,0
q
p
/
Basic Perturbation Lemma, effective homology and homology of 2-types
EC00
EC31
1
EC2
d21
EC11
EC01
EC32
EC33
EC2
EC23
EC12
EC13
EC02
EC03
2
d22
HPT Workshop
p
/
13 / 29
Bicomplex
Theorem
A general algorithm can be produced:
Input: A bounded bicomplex C∗ and effective homologies of each
column.
Output: An effective homology for C∗ .
Proof:
1
We consider only
the vertical
arrows.
O
h
C0,3
C1,3
C0,2
C0,0
C1,1
C1,0
C2,3
C3,3
h00
C1,2
00
d1,2
C0,1
Ana Romero
O
q
k
C2,2
C3,2
00
d2,2
C2,1
C2,0
f 00
EC30
0
EC2
EC10
C3,1
C3,0
q
/
p g 00
Basic Perturbation Lemma, effective homology and homology of 2-types
EC00
EC31
EC32
EC33
EC2
EC23
EC12
EC13
EC02
EC03
1 + 2
EC2
d21
EC11
EC01
d22
HPT Workshop
p
/
13 / 29
Bicomplex
Theorem
A general algorithm can be produced:
Input: A bounded bicomplex C∗ and effective homologies of each
column.
Output: An effective homology for C∗ .
Proof:
1
2
Ana Romero
We consider only
the vertical
arrows.
We perturb by
adding the
horizontal maps.
O
O
q
o
C0,3
o
C1,3
o
C2,3
d0 C0,2 o C1,2 o C2,2 o
00
00
od1,2 od2,2 o
EC2
C1,1
C2,1
C3,1
o
C1,0
o
C2,0
o
C3,0
C0,0
0
C3,2
C0,1
q
EC30
C3,3
EC10
p
/
Basic Perturbation Lemma, effective homology and homology of 2-types
EC00
EC31
1
EC2
d21
EC11
EC01
EC32
EC33
EC2
EC23
EC12
EC13
EC02
EC03
2
d22
HPT Workshop
p
/
13 / 29
Bicomplex
Theorem
A general algorithm can be produced:
Input: A bounded bicomplex C∗ and effective homologies of each
column.
Output: An effective homology for C∗ .
Proof:
1
2
3
Ana Romero
We consider only
the vertical
arrows.
We perturb by
adding the
horizontal maps.
O
O
q
o
C0,3
o
C1,3
o
C2,3
d0 C0,2 o C1,2 o C2,2 o
00
00
od1,2 od2,2 o
C3,2
C1,1
C2,1
C3,1
o
C1,0
o
C2,0
o
C3,0
C0,0
f og
0o
EC2 f
0o
C0,1
q
o EC32 o
1o d0 2o
EC2 g EC2
d21
d22 EC1
EC11 go EC12 o
EC00 o EC01 o EC02 o
EC30
C3,3
p
/
EC31
EC33
EC23
EC13
EC03
p
/
We apply the
BPL.
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
13 / 29
Bicomplex
Theorem
A general algorithm can be produced:
Input: A bounded bicomplex C∗ and effective homologies of each
column.
Output: An effective homology for C∗ .
Proof:
1
2
3
Ana Romero
We consider only
the vertical
arrows.
We perturb by
adding the
horizontal maps.
O
O
q
f
C0,3
C0,2
C0,1
C0,0
n
C1,3
h
C2,3
C3,3
h
f
C1,2
C2,2
p
C2,1
C1,1
C1,0
h
C2,0
f
C3,2
h
g
s
C3,1
h
C3,0
g
p
/
g
q
0
EC30
EC31
EC32
EC20
f
EC21
EC22
EC23
EC10
EC11
EC12
+EC 3
EC00
EC01
EC02
EC03
-
EC33
1
p
/
We apply the
BPL.
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
13 / 29
Twisted Eilenberg-Zilber Theorem
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
14 / 29
Twisted Eilenberg-Zilber Theorem
Theorem (Eilenberg-Zilber)
Given two simplicial sets G and B, there exists a reduction
ρ = (f , g , h) : C∗ (G × B) ⇒
⇒ C∗ (G ) ⊗ C∗ (B)
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
14 / 29
Twisted Eilenberg-Zilber Theorem
Theorem (Eilenberg-Zilber)
Given two simplicial sets G and B, there exists a reduction
ρ = (f , g , h) : C∗ (G × B) ⇒
⇒ C∗ (G ) ⊗ C∗ (B)
Theorem (Twisted Eilenberg-Zilber)
Given two simplicial sets G and B and a twisting operator τ : B → G , it is
possible to construct a reduction
ρ = (f , g , h) : C∗ (G ×τ B) ⇒
⇒ C∗ (G ) ⊗t C∗ (B)
where C∗ (G ) ⊗t C∗ (B) is a chain complex with the same underlying
graded module as the tensor product C∗ (G ) ⊗ C∗ (B), but the differential
is modified to take account of the twisting operator τ .
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
14 / 29
Twisted Eilenberg-Zilber Theorem
Theorem (Eilenberg-Zilber)
Given two simplicial sets G and B, there exists a reduction
ρ = (f , g , h) : C∗ (G × B) ⇒
⇒ C∗ (G ) ⊗ C∗ (B)
Theorem (Twisted Eilenberg-Zilber)
Given two simplicial sets G and B and a twisting operator τ : B → G , it is
possible to construct a reduction
ρ = (f , g , h) : C∗ (G ×τ B) ⇒
⇒ C∗ (G ) ⊗t C∗ (B)
where C∗ (G ) ⊗t C∗ (B) is a chain complex with the same underlying
graded module as the tensor product C∗ (G ) ⊗ C∗ (B), but the differential
is modified to take account of the twisting operator τ .
Proof: BPL.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
14 / 29
Effective homology of twisted Cartesian products
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
15 / 29
Effective homology of twisted Cartesian products
Theorem
A general algorithm can be produced:
Input: two simplicial sets G and B (where B is 1-reduced), a twisting
operator τ : B → G , and effective homologies for G and B.
Output: An effective homology for E = G ×τ B.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
15 / 29
Effective homology of twisted Cartesian products
Theorem
A general algorithm can be produced:
Input: two simplicial sets G and B (where B is 1-reduced), a twisting
operator τ : B → G , and effective homologies for G and B.
Output: An effective homology for E = G ×τ B.
Proof: It is constructed as the composition of two equivalences:
C∗ (G ×τ B)
s{ s{
Id
C∗ (G ×τ B)
DG∗ ⊗t DB∗
ρ1
ρ2
$, $,
rz rz
C∗ (G ) ⊗t C∗ (B)
ρ3
#+ #+
EG∗ ⊗t EB∗
where ρ2 and ρ3 are obtained by applying the TPL and the BPL
respectively.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
15 / 29
Effective homology of the fiber of a fibration
Theorem
A general algorithm can be produced:
Input: two simplicial sets G and B (where B is 1-reduced) and a
twisting operator τ : B → G , and effective homologies for B and
E = G ×τ B.
Output: An effective homology for G .
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
16 / 29
Effective homology of the fiber of a fibration
Theorem
A general algorithm can be produced:
Input: two simplicial sets G and B (where B is 1-reduced) and a
twisting operator τ : B → G , and effective homologies for B and
E = G ×τ B.
Output: An effective homology for G .
Proof: It is constructed as the composition of two equivalences:
CobarC∗ (B) (C∗ (G ) ⊗t C∗ (B), Z)
u} u}
C∗ (G )
Ana Romero
Id
&. &.
^
Cobar
DB∗
rz rz
(DE∗ , Z)
((
^
CobarC∗ (B) (C∗ (G ) ⊗t C∗ (B), Z) Cobar
Basic Perturbation Lemma, effective homology and homology of 2-types
EB∗
(EE∗ , Z)
HPT Workshop
16 / 29
Effective homology of the fiber of a fibration
Theorem
A general algorithm can be produced:
Input: two simplicial sets G and B (where B is 1-reduced) and a
twisting operator τ : B → G , and effective homologies for B and
E = G ×τ B.
Output: An effective homology for G .
Proof: It is constructed as the composition of two equivalences:
CobarC∗ (B) (C∗ (G ) ⊗t C∗ (B), Z)
u} u}
C∗ (G )
Id
&. &.
^
Cobar
DB∗
rz rz
(DE∗ , Z)
((
^
CobarC∗ (B) (C∗ (G ) ⊗t C∗ (B), Z) Cobar
EB∗
(EE∗ , Z)
In particular, it can be applied for computing the effective homology of a
loop space Ω(X ), which is the fiber of a fibration
Ω(X ) ,→ Ω(X ) ×τ X → X where the total space E = Ω(X ) ×τ X is
contractible, such that a reduction C∗ (Ω(X ) ×τ X ) ⇒
⇒ Z can be built.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
16 / 29
Discrete Morse theory
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
17 / 29
Discrete Morse theory
Definition
Let C∗ = (Cp , dp )p∈Z a free chain complex with distinguished Z-basis
βp ⊂ Cp . A discrete vector field V on C∗ is a collection of pairs
V = {(σi ; τi )}i∈I satisfying the conditions:
Every σi is some element of βp , in which case τi ∈ βp+1 . The degree p
depends on i and in general is not constant.
Every component σi is a regular face of the corresponding τi .
Each generator (cell) of C∗ appears at most one time in V .
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
17 / 29
Discrete Morse theory
Definition
Let C∗ = (Cp , dp )p∈Z a free chain complex with distinguished Z-basis
βp ⊂ Cp . A discrete vector field V on C∗ is a collection of pairs
V = {(σi ; τi )}i∈I satisfying the conditions:
Every σi is some element of βp , in which case τi ∈ βp+1 . The degree p
depends on i and in general is not constant.
Every component σi is a regular face of the corresponding τi .
Each generator (cell) of C∗ appears at most one time in V .
Definition
A V -path of degree p and length m is a sequence π = ((σik , τik ))0≤k<m
satisfying:
Every pair ((σik , τik )) is a component of V and τik is a p-cell.
For every 0 < k < m, the component σik is a face of τik−1 , non necessarily
regular, but different from σik−1 .
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
17 / 29
Discrete Morse theory
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
18 / 29
Discrete Morse theory
Definition
A discrete vector field V is admissible if for every p ∈ Z, a function
λp : βp → N is provided satisfying the following property: every V -path
starting from σ ∈ βp has a length bounded by λp (σ).
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
18 / 29
Discrete Morse theory
Definition
A discrete vector field V is admissible if for every p ∈ Z, a function
λp : βp → N is provided satisfying the following property: every V -path
starting from σ ∈ βp has a length bounded by λp (σ).
Definition
A cell σ which does not appear in a discrete vector field V is called a
critical cell.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
18 / 29
Discrete Morse theory
Definition
A discrete vector field V is admissible if for every p ∈ Z, a function
λp : βp → N is provided satisfying the following property: every V -path
starting from σ ∈ βp has a length bounded by λp (σ).
Definition
A cell σ which does not appear in a discrete vector field V is called a
critical cell.
Theorem
Let C∗ = (Cp , dp )p∈Z be a free chain complex and V = {(σi ; τi )}i∈I be an
admissible discrete vector field on C∗ . Then the vector field V defines a
canonical reduction ρ = (f , g , h) : (Cp , dp ) ⇒
⇒ (Cpc , dp0 ) where Cpc = Z[βpc ]
is the free Z-module generated by the critical p-cells.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
18 / 29
Discrete Morse theory
Definition
A discrete vector field V is admissible if for every p ∈ Z, a function
λp : βp → N is provided satisfying the following property: every V -path
starting from σ ∈ βp has a length bounded by λp (σ).
Definition
A cell σ which does not appear in a discrete vector field V is called a
critical cell.
Theorem
Let C∗ = (Cp , dp )p∈Z be a free chain complex and V = {(σi ; τi )}i∈I be an
admissible discrete vector field on C∗ . Then the vector field V defines a
canonical reduction ρ = (f , g , h) : (Cp , dp ) ⇒
⇒ (Cpc , dp0 ) where Cpc = Z[βpc ]
is the free Z-module generated by the critical p-cells.
Proof: Uses BPL.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
18 / 29
Discrete Morse theory and digital images
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16 vertices
24 edges
8 squares
Ana Romero
•
1 vertex
1 edge
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
19 / 29
Discrete Morse theory and digital images
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24 edges
8 squares
Ana Romero
•
1 vertex
1 edge
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
19 / 29
Discrete Morse theory and digital images
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8 squares
Ana Romero
•
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1 edge
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
19 / 29
Other applications of BPL and effective homology
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
20 / 29
Other applications of BPL and effective homology
Homotopy groups of spaces by means of Whitehead and Postnikov
towers.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
20 / 29
Other applications of BPL and effective homology
Homotopy groups of spaces by means of Whitehead and Postnikov
towers.
Spectral sequences of filtered complexes.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
20 / 29
Other applications of BPL and effective homology
Homotopy groups of spaces by means of Whitehead and Postnikov
towers.
Spectral sequences of filtered complexes.
Persistent homology.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
20 / 29
Other applications of BPL and effective homology
Homotopy groups of spaces by means of Whitehead and Postnikov
towers.
Spectral sequences of filtered complexes.
Persistent homology.
Koszul homology.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
20 / 29
Other applications of BPL and effective homology
Homotopy groups of spaces by means of Whitehead and Postnikov
towers.
Spectral sequences of filtered complexes.
Persistent homology.
Koszul homology.
Bousfield-Kan spectral sequence.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
20 / 29
Other applications of BPL and effective homology
Homotopy groups of spaces by means of Whitehead and Postnikov
towers.
Spectral sequences of filtered complexes.
Persistent homology.
Koszul homology.
Bousfield-Kan spectral sequence.
Homology of groups.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
20 / 29
Other applications of BPL and effective homology
Homotopy groups of spaces by means of Whitehead and Postnikov
towers.
Spectral sequences of filtered complexes.
Persistent homology.
Koszul homology.
Bousfield-Kan spectral sequence.
Homology of groups.
Homology of 2-types.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
20 / 29
Homology of groups
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
21 / 29
Homology of groups
Definition
A 2-type X is a topological space with πi (X ) = 0 for i 6= 1, 2.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
21 / 29
Homology of groups
Definition
A 2-type X is a topological space with πi (X ) = 0 for i 6= 1, 2.
Definition
A resolution F∗ for a group G is an acyclic chain complex of ZG -modules
d
d
ε
2
1
· · · −→ F2 −→
F1 −→
F0 −→ F−1 = Z −→ 0
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
21 / 29
Homology of groups
Definition
A 2-type X is a topological space with πi (X ) = 0 for i 6= 1, 2.
Definition
A resolution F∗ for a group G is an acyclic chain complex of ZG -modules
d
d
ε
2
1
· · · −→ F2 −→
F1 −→
F0 −→ F−1 = Z −→ 0
A chain complex of Abelian groups is obtained: Z ⊗ZG F∗
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
21 / 29
Homology of groups
Definition
A 2-type X is a topological space with πi (X ) = 0 for i 6= 1, 2.
Definition
A resolution F∗ for a group G is an acyclic chain complex of ZG -modules
d
d
ε
2
1
· · · −→ F2 −→
F1 −→
F0 −→ F−1 = Z −→ 0
A chain complex of Abelian groups is obtained: Z ⊗ZG F∗
Theorem
Let G be a group and F∗ , F∗0 two free resolutions of G . Then
Hn (Z ⊗ZG F∗ ) ∼
= Hn (Z ⊗ZG F∗0 ) for all n ∈ N
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
21 / 29
Homology of groups
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
22 / 29
Homology of groups
Definition
Given a group G , the homology groups Hn (G ) are defined as
Hn (G ) = Hn (Z ⊗ZG F∗ ), n ∈ N, where F∗ is any free resolution for G .
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
22 / 29
Homology of groups
Definition
Given a group G , the homology groups Hn (G ) are defined as
Hn (G ) = Hn (Z ⊗ZG F∗ ), n ∈ N, where F∗ is any free resolution for G .
One can always consider the bar resolution B∗ = Bar∗ (G )
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
22 / 29
Homology of groups
Definition
Given a group G , the homology groups Hn (G ) are defined as
Hn (G ) = Hn (Z ⊗ZG F∗ ), n ∈ N, where F∗ is any free resolution for G .
One can always consider the bar resolution B∗ = Bar∗ (G ), which satisfies
Z ⊗ZG B∗ ≡ C∗ (K (G , 1)).
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
22 / 29
Homology of groups
Definition
Given a group G , the homology groups Hn (G ) are defined as
Hn (G ) = Hn (Z ⊗ZG F∗ ), n ∈ N, where F∗ is any free resolution for G .
One can always consider the bar resolution B∗ = Bar∗ (G ), which satisfies
Z ⊗ZG B∗ ≡ C∗ (K (G , 1)). Drawback: for n > 1, K (G , 1)n = G n .
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
22 / 29
Homology of groups
Definition
Given a group G , the homology groups Hn (G ) are defined as
Hn (G ) = Hn (Z ⊗ZG F∗ ), n ∈ N, where F∗ is any free resolution for G .
One can always consider the bar resolution B∗ = Bar∗ (G ), which satisfies
Z ⊗ZG B∗ ≡ C∗ (K (G , 1)). Drawback: for n > 1, K (G , 1)n = G n .
For some particular cases, small (or minimal) resolutions can be directly
constructed.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
22 / 29
Homology of groups
Definition
Given a group G , the homology groups Hn (G ) are defined as
Hn (G ) = Hn (Z ⊗ZG F∗ ), n ∈ N, where F∗ is any free resolution for G .
One can always consider the bar resolution B∗ = Bar∗ (G ), which satisfies
Z ⊗ZG B∗ ≡ C∗ (K (G , 1)). Drawback: for n > 1, K (G , 1)n = G n .
For some particular cases, small (or minimal) resolutions can be directly
constructed.
For instance, let G = Cm with generator t. The resolution F∗
t−1
N
t−1
· · · −→ ZG −→ ZG −→ ZG −→Z −→ 0
produces

if i = 0
 Z
Z/mZ if i is odd
Hi (G ) =

0
if i is even, i > 0
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
22 / 29
Algorithm computing the effective homology of a group
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
23 / 29
Algorithm computing the effective homology of a group
Given G a group, F∗ a (small) free ZG -resolution with a contracting
homotopy hn : Fn → Fn+1 .
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
23 / 29
Algorithm computing the effective homology of a group
Given G a group, F∗ a (small) free ZG -resolution with a contracting
homotopy hn : Fn → Fn+1 .
Goal: an equivalence C∗ (K (G , 1)) ⇐
⇐⇒
⇒ E∗ where E∗ is an effective chain
complex.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
23 / 29
Algorithm computing the effective homology of a group
Given G a group, F∗ a (small) free ZG -resolution with a contracting
homotopy hn : Fn → Fn+1 .
Goal: an equivalence C∗ (K (G , 1)) ⇐
⇐⇒
⇒ E∗ where E∗ is an effective chain
complex.
We consider the bar resolution B∗ = Bar∗ (G ) for G with contracting
homotopy h0 .
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
23 / 29
Algorithm computing the effective homology of a group
Given G a group, F∗ a (small) free ZG -resolution with a contracting
homotopy hn : Fn → Fn+1 .
Goal: an equivalence C∗ (K (G , 1)) ⇐
⇐⇒
⇒ E∗ where E∗ is an effective chain
complex.
We consider the bar resolution B∗ = Bar∗ (G ) for G with contracting
homotopy h0 .
It is well known that there exists a morphism of chain complexes of
ZG -modules f : B∗ → F∗ which is a homotopy equivalence.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
23 / 29
Algorithm computing the effective homology of a group
Given G a group, F∗ a (small) free ZG -resolution with a contracting
homotopy hn : Fn → Fn+1 .
Goal: an equivalence C∗ (K (G , 1)) ⇐
⇐⇒
⇒ E∗ where E∗ is an effective chain
complex.
We consider the bar resolution B∗ = Bar∗ (G ) for G with contracting
homotopy h0 .
It is well known that there exists a morphism of chain complexes of
ZG -modules f : B∗ → F∗ which is a homotopy equivalence. An algorithm
has been designed constructing the explicit expressions of f and the
corresponding maps g , h and k
h
f
B∗ k
+
k
F∗
g
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
23 / 29
Algorithm computing the effective homology of a group
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
24 / 29
Algorithm computing the effective homology of a group
Applying the functor Z ⊗ZG − we obtain an equivalence of chain
complexes (of Z-modules):
h
Z ⊗ZG B∗ m
Ana Romero
f
g
-
k
Z ⊗ZG F∗
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
24 / 29
Algorithm computing the effective homology of a group
Applying the functor Z ⊗ZG − we obtain an equivalence of chain
complexes (of Z-modules):
h
f
Z ⊗ZG B∗ m
g
-
k
Z ⊗ZG F∗
In order to obtain a strong chain equivalence we make use of the mapping
cylinder construction.
ρ0
ρ
Z ⊗ZG B∗ ⇐
⇐ Cylinder(f )∗ ⇒
⇒ Z ⊗ZG F∗
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
24 / 29
Algorithm computing the effective homology of a group
Applying the functor Z ⊗ZG − we obtain an equivalence of chain
complexes (of Z-modules):
h
f
Z ⊗ZG B∗ m
g
-
k
Z ⊗ZG F∗
In order to obtain a strong chain equivalence we make use of the mapping
cylinder construction.
ρ0
ρ
Z ⊗ZG B∗ ⇐
⇐ Cylinder(f )∗ ⇒
⇒ Z ⊗ZG F∗
Finally we observe that the left chain complex Z ⊗ZG B∗ is equal to
C∗ (K (G , 1)).
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
24 / 29
Algorithm computing the effective homology of a group
Applying the functor Z ⊗ZG − we obtain an equivalence of chain
complexes (of Z-modules):
h
f
Z ⊗ZG B∗ m
g
-
k
Z ⊗ZG F∗
In order to obtain a strong chain equivalence we make use of the mapping
cylinder construction.
ρ0
ρ
Z ⊗ZG B∗ ⇐
⇐ Cylinder(f )∗ ⇒
⇒ Z ⊗ZG F∗
Finally we observe that the left chain complex Z ⊗ZG B∗ is equal to
C∗ (K (G , 1)). Moreover, if the initial resolution F∗ is of finite type (and
small), then the right chain complex Z ⊗ZG F∗ ≡ E∗ is effective.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
24 / 29
Algorithm computing the effective homology of a group
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
25 / 29
Algorithm computing the effective homology of a group
Theorem
A general algorithm can be produced:
Input: a group G and a free resolution F∗ of finite type with
contracting homotopy.
Output: the effective homology of K (G , 1), that is, a (strong chain)
equivalence C∗ (K (G , 1)) ⇐
⇐⇒
⇒ E∗ where E∗ is an effective chain
complex.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
25 / 29
Algorithm computing the effective homology of a group
Theorem
A general algorithm can be produced:
Input: a group G and a free resolution F∗ of finite type with
contracting homotopy.
Output: the effective homology of K (G , 1), that is, a (strong chain)
equivalence C∗ (K (G , 1)) ⇐
⇐⇒
⇒ E∗ where E∗ is an effective chain
complex.
Implemented in Common Lisp, enhancing the Kenzo system.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
25 / 29
Algorithm computing the effective homology of a group
Theorem
A general algorithm can be produced:
Input: a group G and a free resolution F∗ of finite type with
contracting homotopy.
Output: the effective homology of K (G , 1), that is, a (strong chain)
equivalence C∗ (K (G , 1)) ⇐
⇐⇒
⇒ E∗ where E∗ is an effective chain
complex.
Implemented in Common Lisp, enhancing the Kenzo system.
It allows to compute homology of groups
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
25 / 29
Algorithm computing the effective homology of a group
Theorem
A general algorithm can be produced:
Input: a group G and a free resolution F∗ of finite type with
contracting homotopy.
Output: the effective homology of K (G , 1), that is, a (strong chain)
equivalence C∗ (K (G , 1)) ⇐
⇐⇒
⇒ E∗ where E∗ is an effective chain
complex.
Implemented in Common Lisp, enhancing the Kenzo system.
It allows to compute homology of groups and, what is more
important,
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
25 / 29
Algorithm computing the effective homology of a group
Theorem
A general algorithm can be produced:
Input: a group G and a free resolution F∗ of finite type with
contracting homotopy.
Output: the effective homology of K (G , 1), that is, a (strong chain)
equivalence C∗ (K (G , 1)) ⇐
⇐⇒
⇒ E∗ where E∗ is an effective chain
complex.
Implemented in Common Lisp, enhancing the Kenzo system.
It allows to compute homology of groups and, what is more
important, to use the space K (G , 1) in other constructions allowing
new computations.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
25 / 29
Homology of 2-types
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
26 / 29
Homology of 2-types
Let G = C3 , A = Z/3Z with trivial G -action, and [f ] ∈ H 3 (G , A) = Z/3Z
a non-trivial cohomology class.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
26 / 29
Homology of 2-types
Let G = C3 , A = Z/3Z with trivial G -action, and [f ] ∈ H 3 (G , A) = Z/3Z
a non-trivial cohomology class. This corresponds to a 2-type with π1 = G
and π2 = A, which can be seen as X = K (A, 2) ×f K (G , 1).
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
26 / 29
Homology of 2-types
Let G = C3 , A = Z/3Z with trivial G -action, and [f ] ∈ H 3 (G , A) = Z/3Z
a non-trivial cohomology class. This corresponds to a 2-type with π1 = G
and π2 = A, which can be seen as X = K (A, 2) ×f K (G , 1).
Finite (small) resolutions are known for G and A.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
26 / 29
Homology of 2-types
Let G = C3 , A = Z/3Z with trivial G -action, and [f ] ∈ H 3 (G , A) = Z/3Z
a non-trivial cohomology class. This corresponds to a 2-type with π1 = G
and π2 = A, which can be seen as X = K (A, 2) ×f K (G , 1).
Finite (small) resolutions are known for G and A.
The spaces K (G , 1) and K (A, 1) have effective homology thanks to
the previous algorithm.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
26 / 29
Homology of 2-types
Let G = C3 , A = Z/3Z with trivial G -action, and [f ] ∈ H 3 (G , A) = Z/3Z
a non-trivial cohomology class. This corresponds to a 2-type with π1 = G
and π2 = A, which can be seen as X = K (A, 2) ×f K (G , 1).
Finite (small) resolutions are known for G and A.
The spaces K (G , 1) and K (A, 1) have effective homology thanks to
the previous algorithm.
From the effective homology of K (A, 1), the effective homology of the
classifying space WK (A, 1) = K (A, 2) is built.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
26 / 29
Homology of 2-types
Let G = C3 , A = Z/3Z with trivial G -action, and [f ] ∈ H 3 (G , A) = Z/3Z
a non-trivial cohomology class. This corresponds to a 2-type with π1 = G
and π2 = A, which can be seen as X = K (A, 2) ×f K (G , 1).
Finite (small) resolutions are known for G and A.
The spaces K (G , 1) and K (A, 1) have effective homology thanks to
the previous algorithm.
From the effective homology of K (A, 1), the effective homology of the
classifying space WK (A, 1) = K (A, 2) is built.
From the effective homologies of K (A, 2) and K (G , 1), we construct
the effective homology of X = K (A, 2) ×f K (G , 1).
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
26 / 29
Homology of 2-types
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
27 / 29
Homology of 2-types
Kenzo computes the homology groups of the 2-type:
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
27 / 29
Homology of 2-types
Kenzo computes the homology groups of the 2-type:
.
> (setf K-C3-1 (K-Cm-n 3 1))
[K261 Abelian-Simplicial-Group]
> (setf chml-clss (chml-clss K-C3-1 3))
[K308 Cohomology-Class on K288 of degree 3]
> (setf tau (zp-whitehead 3 K-C3-1 chml-clss))
[K323 Fibration K261 -> K309]
> (setf x (fibration-total tau))
[K329 Kan-Simplicial-Set]
> (efhm x)
[K541 Homotopy-Equivalence K329 <= K531 => K527]
> (homology x 5)
Homology in dimension 5 :
Component Z/3Z
---done---
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
27 / 29
Conclusions
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
28 / 29
Conclusions
The Basic Perturbation Lemma is not basic.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
28 / 29
Conclusions
The Basic Perturbation Lemma is not basic.
Combined with the effective homology method, it can be used for
computing homology and homotopy groups of different spaces
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
28 / 29
Conclusions
The Basic Perturbation Lemma is not basic.
Combined with the effective homology method, it can be used for
computing homology and homotopy groups of different spaces and
other constructions of Algebraic Topology such as spectral sequences,
persistent homology, homology of groups. . .
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
28 / 29
Conclusions
The Basic Perturbation Lemma is not basic.
Combined with the effective homology method, it can be used for
computing homology and homotopy groups of different spaces and
other constructions of Algebraic Topology such as spectral sequences,
persistent homology, homology of groups. . .
In particular, it allows us to determine homology of 2-types.
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
28 / 29
Basic Perturbation Lemma and effective homology:
application to the computation of homology of 2-types
Ana Romero
Universidad de La Rioja (Spain)
(Joint work with J. Rubio and F. Sergeraert)
Homological Perturbation Theory Workshop
Galway, December 2014
Ana Romero
Basic Perturbation Lemma, effective homology and homology of 2-types
HPT Workshop
29 / 29