The Homological
Hexagonal Lemma
Francis Sergeraert, Institut Fourier, Grenoble
Homological Perturbation Theory, Galway, December 2014
1/35
1/8. Introduction.
Gauss Reduction
Hexagonal Lemma
Homological Hexagonal Lemma
Hom. Perturbation. Th.
Forman Vector-F. Th.
Constructive Homology
Eil.-MacLane Conjecture
2/35
2/8. Homological Hexagonal Lemma.
· · · CC3
· · · A3
⊕
· · · B3
⊕
· · · C3
CC4
ISO
A4
⊕
B4
⊕
ISO
C4
C4
A5
⊕
B5
⊕
ISO
C5
⇒
⇒
· · · C3
CC6 · · ·
CC5
A6 · · ·
⊕
B6 · · ·
⊕
C6 · · ·
H∗-Reduction
C5
C6 · · ·
3/35
Details of the final result:

−1 5
d23

0



−d521

g5 = 

1
A 5 ⊕ B5 ⊕ C5
f5 = (0
−
−1
d631 d621
C5
1)
d521
d5
d5 = d533 − d531 d521

−1 4
d23

0



−d421

g4 = 

1
A 4 ⊕ B4 ⊕ C4
f4 = (0


d5 = 

d511
d521
d531
d512
d522
d532
d513
d523
d533




−
−1
d531 d521
C4
1)
−1 5
d23
4/35
3/8. Elementary Hexagonal Lemma.
dn+2
dn+2
=
Cn+1
η
γ
⊕ Bn
An
hn−1 =
ε−1
0
0
0
ε
ϕ
β
ψ
1
gn−1 =
fn−1 =
α
Cn−2
dn−2
γ
Bn
fn = (0 1)
An−1 ⊕ Bn−1
δ
!
−ε−1 ϕ
gn =
!
Cn+1
0
!
β − ψε−1 ϕ
1
−ψε
−1
1
Bn−1
α
=
Cn−2
dn−2
5/35
Iterating the (Elementary) Hexagonal Lemma:
E.H.L.
A 6 ⊕ B6 ⊕ C6
=
A6 ⊕ B6 ⊕ C6
B6 ⊕ C6
C6
B5 ⊕ C5
C5
C5
A4 ⊕ C4
C4
C4
E.H.L.
A 5 ⊕ B5 ⊕ C5
E.H.L.
A 4 ⊕ B4 ⊕ C4
E.H.L.
A 3 ⊕ B3 ⊕ C3
=
A3 ⊕ B3 ⊕ C3
A3 ⊕ C3
C3
E.H.L.
6/35
Iterating the (Elementary) Hexagonal Lemma:
A 6 ⊕ B6 ⊕ C6
C6
A 5 ⊕ B5 ⊕ C5
Composition of Reductions
C5
A 4 ⊕ B4 ⊕ C4
Homological Hexagonal Lemma
C4
A 3 ⊕ B3 ⊕ C3
C3
7/35
4/8. Gauss reduction 7−→ Hexagonal Lemma
Giving the elementary linear system
a homological hexagonal shape:
Qa
ψx + βy = b
7−→
0
×ϕ
Qb
7−→
0
0
0
Gauss Reduction
Qx
×ψ
0
εx + ϕy = a
×ε
×β
0
Qy
Elementary Hexagonal Lemma
8/35
R = Unitary ring
ε, ϕ, ψ, β ∈ R with ε invertible.
Gauss discussion of (1) + (2):
(1)
εx + ϕy = a
(2)
ψx + βy = b
(2) − ψε−1(1) ⇒
(20)
(β − ψε−1ϕ)y = (b − ψε−1a)
⇒ (1) + (2) has a solution ⇔
(β − ψε−1ϕ) | (b − ψε−1a) ⇒
⇒
y = ···
x = ε−1a − ε−1ϕy
9/35
Matrix translation:
!
!
ε ϕ
x
ψ β
y
a
=
!
b
⇔
1
0
!
−ψε−1 1
|

ε


!
!
!
!
x
ε ϕ
1 −ε−1 ϕ
1 ε−1 ϕ
ψ β
{z
0
0
0 β − ψε−1 ϕ




1
0
}|
1
{z

x


+ε
−1
y
y

=
}
ϕy 


1
0
!
!
a
−ψε−1 1
|
{z



a
−ψε−1 a + b
⇔
ε(x + ε−1 ϕ y) = a
(β − ψε−1 ϕ) y = (b − ψε−1 a)
⇔
(β − ψε−1 ϕ) | (b − ψε−1 a) ⇒ . . .
b




}
10/35
Diagram translation:
1 −ε−1 ϕ
0
!
1
R2
R2
1 ε−1 ϕ
0
!
1
!
ε ϕ
ε
ψ β
0 β − ψε−1 ϕ
1
!
0
ψε−1 1
R2
R2
1
0
!
−ψε−1 1
0
!
11/35
Combined with an obvious reduction:
1 −ε−1 ϕ
0
R
R
0
ψ
ϕ
R
!
ε−1
h=
R
2
0
1
1
!
β
!
1
1
1 ε
ϕ
0
2
−1
ε
!
0
1
0
ψε−1
1
0
!
ε
0 β − ψε
0
−ψε−1
1
β − ψε−1 ϕ
ϕ
0
!
1
R
0
−1
!
2
1
!
0
!
2
R
0
1
⇒
12/35
⇒ Canonical reduction induced by ε invertible
g=
1
R2
R
f =
h=
ε−1 0
0
!
!
−ε−1 ϕ
ε
ϕ
0 1
!
β − ψε−1 ϕ
ψ β
0
g=
R
0
!
1
2
R
f =
−ψε−1
1
13/35
The same is valid with
R2 = R ⊕ R replaced by An ⊕ Bn = Cn
or by An−1 ⊕ Bn−1 = Cn−1
and:


ε ϕ
ψ β

 : An ⊕ Bn → An−1 ⊕ Bn−1
with ε : An → An−1 isomorphism.
⇒ Hexagonal lemma.
14/35
Hexagonal lemma
dn+2
=
Cn+1
dn+2
dn+2
=
Cn+1
Cn+1
η + ε−1 ϕγ = 0
η
γ
ε−1
ε
1
0
1
!
0
ϕ
⊕ Bn
An
1
ε−1 ϕ
0
1
(0 1)
ε−1
1
0
ψε−1
1
!
0
0
0
−ψε−1
1
Bn−1
!
0
1
=
∼
=
−1
α
α
δ + αψε
dn−2
0
!
An−1 ⊕ Bn−1
1
α
Cn−2
β − ψε−1 ϕ
β − ψε−1 ϕ
ε
1
An−1 ⊕ Bn−1
δ
Bn
!
β
ψ
!
1
⊕ Bn
An
γ
γ
−ε−1 ϕ
=0
Cn−2
=
dn−2
Cn−2
dn−2
reduction
⇒
⇒
15/35
dn+2
dn+2
=
Cn+1
η
γ
Cn+1
gn =
γ
!
−ε−1 ϕ
1
⊕ Bn
An
Bn
fn = (0 1)
h = ε−1
ε
ϕ
β
ψ
gn−1 =
0
β − ψε−1 ϕ
!
1
An−1 ⊕ Bn−1
Bn−1
fn−1 =
−ψε
−1
1
α
δ
Cn−2
α
=
dn−2
Cn−2
dn−2
Hexagonal lemma
16/35
1/8. Introduction.
Gauss Reduction
Hexagonal Lemma
Homological Hexagonal Lemma
Hom. Perturbation. Th.
Forman Vector-F. Th.
Constructive Homology
Eil.-MacLane Conjecture
17/35
5/8. Homological Reductions and HP theorem.
Definition: A (homological) reduction is a diagram:
ρ = (f, g, h) = h
b∗, db∗)
(C
g
f
(C∗, d∗)
with:
1.
b∗ and C∗ = chain complexes.
C
2.
f and g = chain complex morphisms.
3.
h = homotopy operator (degree +1).
4.
f g = idC∗ and dCb h + hdCb + gf = idCb∗ .
5.
f h = 0, hg = 0 and hh = 0.
18/35
Meaning = Reduction Diagram:
{· · ·
{· · ·
d
⊕
d
h
d
Am
d∼
=
h
⊕
bm+1
C
d∼
=
h
⊕
Bm
Bm+1
⊕
⊕
⊕
0
Cm−1
d
Cm−1
0
Cm
d
f ∼
=g
d
Cm
h
d
0
Cm+1
d∼
=
h
Cm+1
⊕
· · · } = B∗
⊕
d
· · · } = C∗0
f ∼
=g
f ∼
=g
d
b∗
···} = C
· · · } = A∗
Am+1
Bm−1
f ∼
=g
{· · ·
bm
C
=
d
Am−1
d∼
=
h
{· · ·
h
=
d
bm−1
C
=
h
=
{· · ·
d
· · · } = C∗
19/35
Homological Perturbation Theorem (HPT)
Definition: (C∗ , d) = given chain complex.
A perturbation δ : C∗ → C∗−1 is an operator of degree -1
satisfying (d + δ)2 = 0 (⇔ (dδ + δd + δ 2 ) = 0):
(C∗ , d) + (δ) 7→ (C∗ , d+δ).
Let ρ : h
b∗ , db∗ ) g (C∗ , d∗ ) be a given reduction
(C
f
and δb a perturbation of db
satisfying hδb pointwise nilpotent .
Theorem: The HPT determines a new reduction:
ρ0 : h + δh
b∗ , db∗ + δb∗ )
(C
g + δg
f + δf
(C∗ , d∗ + δd∗ )
20/35
Proof:
Reduction Diagram:
{· · ·
{· · ·
d
⊕
d
h
d
Am
d∼
=
h
⊕
bm+1
C
d∼
=
h
⊕
Bm
Bm+1
⊕
⊕
⊕
0
Cm−1
d
Cm−1
0
Cm
d
f ∼
=g
d
Cm
h
d
0
Cm+1
d∼
=
h
Cm+1
⊕
· · · } = B∗
⊕
d
· · · } = C∗0
f ∼
=g
f ∼
=g
d
b∗
···} = C
· · · } = A∗
Am+1
Bm−1
f ∼
=g
{· · ·
bm
C
=
d
Am−1
d∼
=
h
{· · ·
h
=
d
bm−1
C
=
h
=
{· · ·
d
· · · } = C∗
21/35
Main part:

0




g=
 0 
1
A5 ⊕ B5 ⊕ C5
h = d−1
21
d21
C5
d33
d33
A4 ⊕ B4 ⊕ C4
f = (0 0 1)
C4
with d21 = isomorphism.
22/35

Perturbation =
δ11 δ12 δ13



 δ21 δ22 δ23 :


δ31 δ32 δ33

0




g=
 0 
1
A5 ⊕ B5 ⊕ C5
C5
d21 +δ21
h = d−1
21
d33 +δ33
A4 ⊕ B4 ⊕ C4
f = (0 0 1)
d33
C4
23/35
Question: (d21 + δ21) again isomorphism?
(applying the Global Hexagonal Theorem possible ?)

0




g=
 0 
1
A5 ⊕ B5 ⊕ C5
C5
d21 +δ21
h = d−1
21
d33 +δ33
A4 ⊕ B4 ⊕ C4
f = (0 0 1)
d33
C4
24/35
But d21 invertible with d21 h = 1
⇒
d21 + δ21 = d21 + d21 hδ21 = d21 (1 + hδ21 )
⇒
d21 + δ21 invertible ⇔ (1 + hδ21 ) invertible.
A sufficient condition is hδ21 nilpotent, in which case:
(1 + hδ21 )
−1
=
∞
X
(−1)i (hδ21 )i
i=0
Then:
(d21 + δ21 )−1 =: h0 :=
∞
X
!
(−1)i (hδ21 )i
h
i=0
Remark:
∞
X
i=0
!
(−1)i (hδ21 )i
h=
∞
X
i=0
!
(−1)i (hδ)i
h
25/35
Global Hexagonal Theorem:

−1 5
d23

0



−d521

g5 = 

1
A 5 ⊕ B5 ⊕ C5
f5 = (0
−
−1
d631 d621
C5
1)
d521
d5
d5 = d533 − d531 d521

−1 4
d23

0



−d421

g4 = 

1
A 4 ⊕ B4 ⊕ C4
f4 = (0

d511 d512 d513



5
5 
5
d5 = 
d
d
d
 21 22 23 
d531 d532 d533
−
−1
d531 d521
C4
1)
−1 5
d23
26/35
Applying to our situation:

G.H.L.
−1 5
d23

0



−d521

g5 = 

Before perturbation:
0




g=
 0 
1
1
A5 ⊕ B5 ⊕ C5

A5 ⊕ B5 ⊕ C 5
C5
C5
d521
d5 = d533 − d531 d521
A4 ⊕ B4 ⊕ C4
f4 = (0
−
−1
d531 d521
h = d521
−1
d521
d533
d533
A4 ⊕ B4 ⊕ C 4
C4
f = (0 0 1)
C4
1)
5
=: d521
d521 7→ d521 + δ21
!
∞
X
−1
h = d521 7→
(−1)i (hδ)i h =: h0
i=0
−1 5
d23
g 7→ (1 − h0 δ)g =: g 0
f 7→ f (1 − δh0 ) =: f 0
d33 7→ (d33 + δ33 ) − f δh0 δg
= d33 + f δg − f δh0 δg =: d033
= Homological Perturbation Theorem
QED
27/35
6/8. The topological case.
Corollary: The HPT can easily be extended
to topological situations.
Example 1: Banach situations:
||hδ21|| < 1 ⇒(1 + hδ21) invertible ⇒ OK.
Example 2: Frechetic situations:
The Nash-Moser-Schwartz technology
often allows to prove (1 − hδ21) is invertible ⇒ OK.
28/35
7/8. Forman Theorems.
C∗ =
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Source cells
Target cells
Critical cells
Forman Reduction Theorem ⇒
ρ : C∗ ⇒
⇒
C∗c
=
dc1 =0
dc1
•
dc1
= Z ←− Z = Circle
29/35
Homological Hexagonal Lemma
⇒ Forman Reduction Theorem:
· · · A3
⊕
· · · B3
⊕
· · · C3
ISO
A4
⊕
B4
⊕
C4
ISO
A5
⊕
B5
⊕
C5
ISO
A6 · · ·
⊕
B6 · · ·
⊕
C6 · · ·
Forman Theorem = Particular case where:
ISO = Triangular Unimodular Invertible Matrix.
30/35
Toy example:
•
•
•
•
•
⇐
•
•
•
•
•
H∗-reduction
=
•
•
31/35
Toy example:
•
•
•
•
•
•
•
•
•
•
Z8
Z10
Z2
Z0
⊕
Z7
⊕
Z2
⊕
Z7
⊕
Z2
⊕
Z0
⊕
Z1
Z1
Z0
32/35
Toy example:
5
•
6
•
•
4’
5’
•
•
2
1’
10
d21 =
7
•
20
•
8
•
6’
7’
2’
•
3
3’
•
4
30 40
50
60
70
0 −1
0
0
0
2
1 −1
3
0
1 −1
0
0 −1
4
0
0
1
0
0
0 −1
5
0
0
0
1
0
0
0
6
0
0
0
0
1
0
0
7
0
0
0
0
0
1
0
8
0
0
0
0
0
0
1
0
33/35
Other example:
•
•
Z0 ?? Z3
Z3
Z0
Z0
Z0
•

d21 =
−1

 1

0
0
−1
1
1


0 

−1
Not invertible!!
34/35
8/8. Eilenberg-MacLane conjecture (1953) :
From Eilenberg-MacLane = Annals of Maths, 1953, vol.58, pp.55-106:
First proof = Pedro Real’s thesis, 1993.
Discrete vector fields + New understanding of Eilenberg-Zilber
⇒ Totally different simple new proof
⇒ Very efficient new algorithms
in computational Algebraic Topology.
35/35
Given G = reduced simplicial group,
there exists a canonical reduction:
C∗ (BG) ⇒
⇒ Bar(C∗ (G))
Proved by discrete vector fields and immediately implemented in 2012.
Application: Given X := ΩS 3 ∪2 D 3 :
π2 X = Z/2
π3 X = Z/2
π4 X = Z/4 + Z
π5 X = (Z/2)4
(1998)
π6 X = (Z/2)5 + Z
(2014)
The END
Francis Sergeraert, Institut Fourier, Grenoble
MAP 2014, IHP Paris, May 2014