Download A conjecture in Rational Homotopy

Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
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The conjecture H
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Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
A conjecture in Rational Homotopy Theory
My Ismail Mamouni
Classes Prépas, Rabat, Morocco
Join work with Mohamed Rachid Hilali
Univ. Casablanca, Morocco
The Spanish-Morrocan mathematical meeting
12-15 November 2008
Casablanca, Morocco
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
the goal of this talk is to give a brief outline of my 3
years’research (2006-2008) with my advisor professor
Mohamed Rachid Hilali, univ. Casablanca.
we will proceed as follows:
1 Introduction
2
Our tools
3
Resolved cases
4
Papers
5
Acknowledgements
6
References
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Field of resarch.
Rational Homotopy Theory
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Subject of research.
We are interested to the following result:
The sum of the Betti numbers of a 1-connected elliptic
space is greater than the total rank of its homotopy groups.
It was formulated in 1990, by M. R. Hilali, who like to call it the
Conjecture H.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Subject of research.
We are interested to the following result:
The sum of the Betti numbers of a 1-connected elliptic
space is greater than the total rank of its homotopy groups.
It was formulated in 1990, by M. R. Hilali, who like to call it the
Conjecture H.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
The conjecture H
The conjecture H: Topological version
If X is a 1-connected elliptic space then
dim H ∗ (X ,Q) ≥ dim (π∗ (X ) ⊗ Q)
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Glossary
Let us recall that:
A topological space X is said to be n-connected if and only
if πi (X ) ≡ 0 for all i ∈ {0, . . . ,n}.
In particular:
X is 0-connected if and only if it is path-connected.
A space is 1-connected if and only if it is simply connected.
A space X is called elliptic when both of H ∗ (X ,Q) and
π∗ (X ) ⊗ Q are finite-dimensional.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Glossary
Let us recall that:
A topological space X is said to be n-connected if and only
if πi (X ) ≡ 0 for all i ∈ {0, . . . ,n}.
In particular:
X is 0-connected if and only if it is path-connected.
A space is 1-connected if and only if it is simply connected.
A space X is called elliptic when both of H ∗ (X ,Q) and
π∗ (X ) ⊗ Q are finite-dimensional.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Glossary
Let us recall that:
A topological space X is said to be n-connected if and only
if πi (X ) ≡ 0 for all i ∈ {0, . . . ,n}.
In particular:
X is 0-connected if and only if it is path-connected.
A space is 1-connected if and only if it is simply connected.
A space X is called elliptic when both of H ∗ (X ,Q) and
π∗ (X ) ⊗ Q are finite-dimensional.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Graded vector spaces
A graded vector space is defined as a sequence
V = (Vn )n∈Z of vector spaces, where elements of Vn are
called of degree n.
If n ∈ Z− , we write V n instead of Vn and |v | will denote the
degree of v .
OtherM
standard denotations.
M
V∗ =
Vn and V ∗ =
V n.
n∈N
V ≥n
:=
n∈N
M
p
V and
V+
:= V ≥1 .
p≥n
V even
:=
M
V 2p and V odd :=
p≥n
M
V 2p+1 .
p≥0
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Graded vector spaces
A graded vector space is defined as a sequence
V = (Vn )n∈Z of vector spaces, where elements of Vn are
called of degree n.
If n ∈ Z− , we write V n instead of Vn and |v | will denote the
degree of v .
OtherM
standard denotations.
M
V∗ =
Vn and V ∗ =
V n.
n∈N
V ≥n
:=
n∈N
M
p
V and
V+
:= V ≥1 .
p≥n
V even
:=
M
V 2p and V odd :=
p≥n
M
V 2p+1 .
p≥0
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Graded vector spaces
A graded vector space is defined as a sequence
V = (Vn )n∈Z of vector spaces, where elements of Vn are
called of degree n.
If n ∈ Z− , we write V n instead of Vn and |v | will denote the
degree of v .
OtherM
standard denotations.
M
V∗ =
Vn and V ∗ =
V n.
n∈N
V ≥n
:=
n∈N
M
p
V and
V+
:= V ≥1 .
p≥n
V even
:=
M
V 2p and V odd :=
p≥n
M
V 2p+1 .
p≥0
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
cgda.
The definition.
We say that (A,d) is a commutative differential graded algebra
(cdga) if:
A is an algebra graded as a vector space.
a.b = (−1)|a||b| b.a for all a,b ∈ A.
d is a differential of degree 1 on A, i.e., d : Ap −→ Ap+1 .
d(a.b) = (da).b + (−1)|a| a.db
My Ismail Mamouni
(Leibniz rule).
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
cgda.
The definition.
We say that (A,d) is a commutative differential graded algebra
(cdga) if:
A is an algebra graded as a vector space.
a.b = (−1)|a||b| b.a for all a,b ∈ A.
d is a differential of degree 1 on A, i.e., d : Ap −→ Ap+1 .
d(a.b) = (da).b + (−1)|a| a.db
My Ismail Mamouni
(Leibniz rule).
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
cgda.
The definition.
We say that (A,d) is a commutative differential graded algebra
(cdga) if:
A is an algebra graded as a vector space.
a.b = (−1)|a||b| b.a for all a,b ∈ A.
d is a differential of degree 1 on A, i.e., d : Ap −→ Ap+1 .
d(a.b) = (da).b + (−1)|a| a.db
My Ismail Mamouni
(Leibniz rule).
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
cgda.
The definition.
We say that (A,d) is a commutative differential graded algebra
(cdga) if:
A is an algebra graded as a vector space.
a.b = (−1)|a||b| b.a for all a,b ∈ A.
d is a differential of degree 1 on A, i.e., d : Ap −→ Ap+1 .
d(a.b) = (da).b + (−1)|a| a.db
My Ismail Mamouni
(Leibniz rule).
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
cgda.
The definition.
We say that (A,d) is a commutative differential graded algebra
(cdga) if:
A is an algebra graded as a vector space.
a.b = (−1)|a||b| b.a for all a,b ∈ A.
d is a differential of degree 1 on A, i.e., d : Ap −→ Ap+1 .
d(a.b) = (da).b + (−1)|a| a.db
My Ismail Mamouni
(Leibniz rule).
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
cgda.
An example of construction.
From any differential and graded vector space (V ,d), we
construct the free commutative graded algebra ΛV as follows:
ΛV := TV hv ⊗ w − (−1)|v ||w| w ⊗ v i
It is easy to verify that
ΛV = Exterior algebra (V odd ) ⊗ Symmetric algebra (V even )
Denotations:
Λn V denotes the set of elements of ΛV of wordlength n.
L
Λ≥n V := k ≥n Λk V denotes the set of elements of ΛV of
wordlength at least n.
Λ+ V := Λ≥1 V denotes the ideal generated by V .
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
cgda.
An example of construction.
From any differential and graded vector space (V ,d), we
construct the free commutative graded algebra ΛV as follows:
ΛV := TV hv ⊗ w − (−1)|v ||w| w ⊗ v i
It is easy to verify that
ΛV = Exterior algebra (V odd ) ⊗ Symmetric algebra (V even )
Denotations:
Λn V denotes the set of elements of ΛV of wordlength n.
L
Λ≥n V := k ≥n Λk V denotes the set of elements of ΛV of
wordlength at least n.
Λ+ V := Λ≥1 V denotes the ideal generated by V .
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
cgda.
An example of construction.
From any differential and graded vector space (V ,d), we
construct the free commutative graded algebra ΛV as follows:
ΛV := TV hv ⊗ w − (−1)|v ||w| w ⊗ v i
It is easy to verify that
ΛV = Exterior algebra (V odd ) ⊗ Symmetric algebra (V even )
Denotations:
Λn V denotes the set of elements of ΛV of wordlength n.
L
Λ≥n V := k ≥n Λk V denotes the set of elements of ΛV of
wordlength at least n.
Λ+ V := Λ≥1 V denotes the ideal generated by V .
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
cgda.
An example of construction.
From any differential and graded vector space (V ,d), we
construct the free commutative graded algebra ΛV as follows:
ΛV := TV hv ⊗ w − (−1)|v ||w| w ⊗ v i
It is easy to verify that
ΛV = Exterior algebra (V odd ) ⊗ Symmetric algebra (V even )
Denotations:
Λn V denotes the set of elements of ΛV of wordlength n.
L
Λ≥n V := k ≥n Λk V denotes the set of elements of ΛV of
wordlength at least n.
Λ+ V := Λ≥1 V denotes the ideal generated by V .
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
cgda.
An example of construction.
From any differential and graded vector space (V ,d), we
construct the free commutative graded algebra ΛV as follows:
ΛV := TV hv ⊗ w − (−1)|v ||w| w ⊗ v i
It is easy to verify that
ΛV = Exterior algebra (V odd ) ⊗ Symmetric algebra (V even )
Denotations:
Λn V denotes the set of elements of ΛV of wordlength n.
L
Λ≥n V := k ≥n Λk V denotes the set of elements of ΛV of
wordlength at least n.
Λ+ V := Λ≥1 V denotes the ideal generated by V .
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Torsion
Let G be a group. An element g of G is called a torsion
element if g has finite order.
If all elements of G are torsion, then G is called a torsion
group.
If the only torsion element is the identity element, then the
group G is called torsion-free.
An abelian group G is torsion-free, if the set of its torsion
elements denoted T (M) is zero.
Vector spaces are torsion-free abelian groups.
The torsion of a topological space is that of its homotopy
groups.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Torsion
Let G be a group. An element g of G is called a torsion
element if g has finite order.
If all elements of G are torsion, then G is called a torsion
group.
If the only torsion element is the identity element, then the
group G is called torsion-free.
An abelian group G is torsion-free, if the set of its torsion
elements denoted T (M) is zero.
Vector spaces are torsion-free abelian groups.
The torsion of a topological space is that of its homotopy
groups.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Torsion
Let G be a group. An element g of G is called a torsion
element if g has finite order.
If all elements of G are torsion, then G is called a torsion
group.
If the only torsion element is the identity element, then the
group G is called torsion-free.
An abelian group G is torsion-free, if the set of its torsion
elements denoted T (M) is zero.
Vector spaces are torsion-free abelian groups.
The torsion of a topological space is that of its homotopy
groups.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Torsion
Let G be a group. An element g of G is called a torsion
element if g has finite order.
If all elements of G are torsion, then G is called a torsion
group.
If the only torsion element is the identity element, then the
group G is called torsion-free.
An abelian group G is torsion-free, if the set of its torsion
elements denoted T (M) is zero.
Vector spaces are torsion-free abelian groups.
The torsion of a topological space is that of its homotopy
groups.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Torsion
Let G be a group. An element g of G is called a torsion
element if g has finite order.
If all elements of G are torsion, then G is called a torsion
group.
If the only torsion element is the identity element, then the
group G is called torsion-free.
An abelian group G is torsion-free, if the set of its torsion
elements denoted T (M) is zero.
Vector spaces are torsion-free abelian groups.
The torsion of a topological space is that of its homotopy
groups.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Torsion
Let G be a group. An element g of G is called a torsion
element if g has finite order.
If all elements of G are torsion, then G is called a torsion
group.
If the only torsion element is the identity element, then the
group G is called torsion-free.
An abelian group G is torsion-free, if the set of its torsion
elements denoted T (M) is zero.
Vector spaces are torsion-free abelian groups.
The torsion of a topological space is that of its homotopy
groups.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
How to kill the torsion?
Let M be a Z-module, the structure theorem for finitely
generated modules over a principal ideal domain gives a
detailed description of the module M up to isomorphism. In
particular, it claims that M ' F ⊕ T (M) where F is a free
Z-module of finite rank (so without torsion) and depending
only on M.
Let X be a 1-connected and finite CW complex, we know
that πi (X ) is a direct sum of finitely many copies of Z and a
finite abelian group, i.e., πi (X ) = Zni ⊕ Ti where Ti is a
finite abelian group and ni = dim πi (X ) ⊗ Q.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
How to kill the torsion?
Let M be a Z-module, the structure theorem for finitely
generated modules over a principal ideal domain gives a
detailed description of the module M up to isomorphism. In
particular, it claims that M ' F ⊕ T (M) where F is a free
Z-module of finite rank (so without torsion) and depending
only on M.
Let X be a 1-connected and finite CW complex, we know
that πi (X ) is a direct sum of finitely many copies of Z and a
finite abelian group, i.e., πi (X ) = Zni ⊕ Ti where Ti is a
finite abelian group and ni = dim πi (X ) ⊗ Q.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Weak homotopy type
A continuous map f : X −→ Y is a weak homotopy type if
each πn (f ) : πn (Y ) −→ πn (X ) is a bijection.
[γ] 7−→ [γ ◦ f ]
Two spaces X and Y have the same weak homotopy type
if they are connected by a chain of weak homotopy
equivalences X ← Z1 → · · · ← Zn → Y .
Any space admits a cellular model, that is a CW complex
with the same weak homotopy type.
Weak homotopy equivalence between CW complexes is a
homotopy equivalence.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Weak homotopy type
A continuous map f : X −→ Y is a weak homotopy type if
each πn (f ) : πn (Y ) −→ πn (X ) is a bijection.
[γ] 7−→ [γ ◦ f ]
Two spaces X and Y have the same weak homotopy type
if they are connected by a chain of weak homotopy
equivalences X ← Z1 → · · · ← Zn → Y .
Any space admits a cellular model, that is a CW complex
with the same weak homotopy type.
Weak homotopy equivalence between CW complexes is a
homotopy equivalence.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Weak homotopy type
A continuous map f : X −→ Y is a weak homotopy type if
each πn (f ) : πn (Y ) −→ πn (X ) is a bijection.
[γ] 7−→ [γ ◦ f ]
Two spaces X and Y have the same weak homotopy type
if they are connected by a chain of weak homotopy
equivalences X ← Z1 → · · · ← Zn → Y .
Any space admits a cellular model, that is a CW complex
with the same weak homotopy type.
Weak homotopy equivalence between CW complexes is a
homotopy equivalence.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Weak homotopy type
A continuous map f : X −→ Y is a weak homotopy type if
each πn (f ) : πn (Y ) −→ πn (X ) is a bijection.
[γ] 7−→ [γ ◦ f ]
Two spaces X and Y have the same weak homotopy type
if they are connected by a chain of weak homotopy
equivalences X ← Z1 → · · · ← Zn → Y .
Any space admits a cellular model, that is a CW complex
with the same weak homotopy type.
Weak homotopy equivalence between CW complexes is a
homotopy equivalence.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Rational homotopy type
A simply connected space X is said to be rational if π∗ (X )
is a Q-vector space.
For any simply connected space X , we can associate a
rational space XQ , unique up to homotopy, such that
π∗ (XQ ) = πi (X ) ⊗ Q.
XQ is called the rationalization of X .
The rational homotopy type of a simply connected space X
is the weak homotopy type of its rationalization.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Rational homotopy type
A simply connected space X is said to be rational if π∗ (X )
is a Q-vector space.
For any simply connected space X , we can associate a
rational space XQ , unique up to homotopy, such that
π∗ (XQ ) = πi (X ) ⊗ Q.
XQ is called the rationalization of X .
The rational homotopy type of a simply connected space X
is the weak homotopy type of its rationalization.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Rational homotopy type
A simply connected space X is said to be rational if π∗ (X )
is a Q-vector space.
For any simply connected space X , we can associate a
rational space XQ , unique up to homotopy, such that
π∗ (XQ ) = πi (X ) ⊗ Q.
XQ is called the rationalization of X .
The rational homotopy type of a simply connected space X
is the weak homotopy type of its rationalization.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Rational homotopy type
A simply connected space X is said to be rational if π∗ (X )
is a Q-vector space.
For any simply connected space X , we can associate a
rational space XQ , unique up to homotopy, such that
π∗ (XQ ) = πi (X ) ⊗ Q.
XQ is called the rationalization of X .
The rational homotopy type of a simply connected space X
is the weak homotopy type of its rationalization.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Rational homotopy theory
Main goal
Rational homotopy theory is the study of the rational homotopy type of a space and of the properties of spaces
and maps that are invariant under rational homotopy equivalence.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Rational homotopy theory
Main refrences
That means roughly that one ignores all torsion in the
homotopy groups.
It was started by Dennis Sullivan (1977) and Daniel Quillen
(1969).
The standard textbook on rational homotopy theory is
[FHT01].
The basic reference on the history of rational homotopy
theory is [Hs99].
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Rational homotopy theory
Main refrences
That means roughly that one ignores all torsion in the
homotopy groups.
It was started by Dennis Sullivan (1977) and Daniel Quillen
(1969).
The standard textbook on rational homotopy theory is
[FHT01].
The basic reference on the history of rational homotopy
theory is [Hs99].
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Rational homotopy theory
Main refrences
That means roughly that one ignores all torsion in the
homotopy groups.
It was started by Dennis Sullivan (1977) and Daniel Quillen
(1969).
The standard textbook on rational homotopy theory is
[FHT01].
The basic reference on the history of rational homotopy
theory is [Hs99].
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Rational homotopy theory
Main refrences
That means roughly that one ignores all torsion in the
homotopy groups.
It was started by Dennis Sullivan (1977) and Daniel Quillen
(1969).
The standard textbook on rational homotopy theory is
[FHT01].
The basic reference on the history of rational homotopy
theory is [Hs99].
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Sullivan’s models
The interest
Rational homotopy types of simply connected spaces can be
identified with certain algebraic objects called minimal Sullivan
algebras, which are commutative differential graded algebras
over the rational numbers satisfying certain conditions.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Sullivan’s models
The definition
A Sullivan model is a cgda (ΛV ,d) with the property that for
some well ordered homogeneous basis (vα )α∈I of V we
have dvα ∈ ΛV<α where ΛV<α is the subalgebra generated
by {vβ ,β < α}.
The model is called minimal if α < β =⇒ |vα | ≤ |vβ |
When ΛV is 1-connected this nilpotence condition is
equivalent to that dV ⊂ Λ≥2 V .
This means that the differential on ΛV is with no linear
term.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Sullivan’s models
The definition
A Sullivan model is a cgda (ΛV ,d) with the property that for
some well ordered homogeneous basis (vα )α∈I of V we
have dvα ∈ ΛV<α where ΛV<α is the subalgebra generated
by {vβ ,β < α}.
The model is called minimal if α < β =⇒ |vα | ≤ |vβ |
When ΛV is 1-connected this nilpotence condition is
equivalent to that dV ⊂ Λ≥2 V .
This means that the differential on ΛV is with no linear
term.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Sullivan’s models
The definition
A Sullivan model is a cgda (ΛV ,d) with the property that for
some well ordered homogeneous basis (vα )α∈I of V we
have dvα ∈ ΛV<α where ΛV<α is the subalgebra generated
by {vβ ,β < α}.
The model is called minimal if α < β =⇒ |vα | ≤ |vβ |
When ΛV is 1-connected this nilpotence condition is
equivalent to that dV ⊂ Λ≥2 V .
This means that the differential on ΛV is with no linear
term.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Sullivan’s models
The definition
A Sullivan model is a cgda (ΛV ,d) with the property that for
some well ordered homogeneous basis (vα )α∈I of V we
have dvα ∈ ΛV<α where ΛV<α is the subalgebra generated
by {vβ ,β < α}.
The model is called minimal if α < β =⇒ |vα | ≤ |vβ |
When ΛV is 1-connected this nilpotence condition is
equivalent to that dV ⊂ Λ≥2 V .
This means that the differential on ΛV is with no linear
term.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Sullivan’s models
The main result
In [Su78], D. Sullivan associates to any 1-connected space
X of finite type (i.e. dim H k (X ,Q) ∀k ≥ 0), a minimal model
also of finite type, (ΛV ,d) (unique up to isomorphism)
called the minimal model of X .
This contravariant correspondence yields an equivalence
between the homotopy category of 1-connected rational
spaces of finite type and that of 1-connected rational
cgda’s of finite type.
More precisely
H ∗ (ΛV ,d) ∼
as graded algebras
= H ∗ (X ,Q)
V ∼
= π∗ (X ) ⊗ Q as graded vector spaces
A space X and its model (ΛV ,d) are called elliptic, when V
and H ∗ (ΛV ,d) are both finite-dimensional.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Sullivan’s models
The main result
In [Su78], D. Sullivan associates to any 1-connected space
X of finite type (i.e. dim H k (X ,Q) ∀k ≥ 0), a minimal model
also of finite type, (ΛV ,d) (unique up to isomorphism)
called the minimal model of X .
This contravariant correspondence yields an equivalence
between the homotopy category of 1-connected rational
spaces of finite type and that of 1-connected rational
cgda’s of finite type.
More precisely
H ∗ (ΛV ,d) ∼
as graded algebras
= H ∗ (X ,Q)
V ∼
= π∗ (X ) ⊗ Q as graded vector spaces
A space X and its model (ΛV ,d) are called elliptic, when V
and H ∗ (ΛV ,d) are both finite-dimensional.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Sullivan’s models
The main result
In [Su78], D. Sullivan associates to any 1-connected space
X of finite type (i.e. dim H k (X ,Q) ∀k ≥ 0), a minimal model
also of finite type, (ΛV ,d) (unique up to isomorphism)
called the minimal model of X .
This contravariant correspondence yields an equivalence
between the homotopy category of 1-connected rational
spaces of finite type and that of 1-connected rational
cgda’s of finite type.
More precisely
H ∗ (ΛV ,d) ∼
as graded algebras
= H ∗ (X ,Q)
V ∼
= π∗ (X ) ⊗ Q as graded vector spaces
A space X and its model (ΛV ,d) are called elliptic, when V
and H ∗ (ΛV ,d) are both finite-dimensional.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Sullivan’s models
The main result
In [Su78], D. Sullivan associates to any 1-connected space
X of finite type (i.e. dim H k (X ,Q) ∀k ≥ 0), a minimal model
also of finite type, (ΛV ,d) (unique up to isomorphism)
called the minimal model of X .
This contravariant correspondence yields an equivalence
between the homotopy category of 1-connected rational
spaces of finite type and that of 1-connected rational
cgda’s of finite type.
More precisely
H ∗ (ΛV ,d) ∼
as graded algebras
= H ∗ (X ,Q)
V ∼
= π∗ (X ) ⊗ Q as graded vector spaces
A space X and its model (ΛV ,d) are called elliptic, when V
and H ∗ (ΛV ,d) are both finite-dimensional.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Conjecture H
The algebraic version
In term of Sullivan’s model, the conjecture H can be written as
follows
Conjecture H (Algebraic version)
If ΛV is a 1-connected and elliptic Sullivan minimal model, then
dim H ∗ (ΛV ,d) ≥ dim V
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Lusternick-Schnirelmann category
Definitions.
A subspace Y of a topological space is called contractible in
X if the inculsion i : Y ,→ X is homotopic to constant map.
The LS category of X , denoted cat(X ), is the least integer
m (or ∞) such that X is the union of m + 1 open subsets Ui ,
each contactible in X .
The rational LS category of X or that of its model (ΛV ,d) is
that of the associated rationalization, i.e.
cat0 (X ) = cat0 (ΛV ,d) := cat(XQ ).
Main result. Y. Felix and S. Halperin, [FH82] If (ΛV ,d) is a
1-connected model, then dim V odd ≤ cat0 (ΛV ,d).
Consequence. If (ΛV ,d) is a 1-connected model, then
dim V ≤ 2cat0 (ΛV ,d).
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Lusternick-Schnirelmann category
Definitions.
A subspace Y of a topological space is called contractible in
X if the inculsion i : Y ,→ X is homotopic to constant map.
The LS category of X , denoted cat(X ), is the least integer
m (or ∞) such that X is the union of m + 1 open subsets Ui ,
each contactible in X .
The rational LS category of X or that of its model (ΛV ,d) is
that of the associated rationalization, i.e.
cat0 (X ) = cat0 (ΛV ,d) := cat(XQ ).
Main result. Y. Felix and S. Halperin, [FH82] If (ΛV ,d) is a
1-connected model, then dim V odd ≤ cat0 (ΛV ,d).
Consequence. If (ΛV ,d) is a 1-connected model, then
dim V ≤ 2cat0 (ΛV ,d).
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Lusternick-Schnirelmann category
Definitions.
A subspace Y of a topological space is called contractible in
X if the inculsion i : Y ,→ X is homotopic to constant map.
The LS category of X , denoted cat(X ), is the least integer
m (or ∞) such that X is the union of m + 1 open subsets Ui ,
each contactible in X .
The rational LS category of X or that of its model (ΛV ,d) is
that of the associated rationalization, i.e.
cat0 (X ) = cat0 (ΛV ,d) := cat(XQ ).
Main result. Y. Felix and S. Halperin, [FH82] If (ΛV ,d) is a
1-connected model, then dim V odd ≤ cat0 (ΛV ,d).
Consequence. If (ΛV ,d) is a 1-connected model, then
dim V ≤ 2cat0 (ΛV ,d).
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Lusternick-Schnirelmann category
Definitions.
A subspace Y of a topological space is called contractible in
X if the inculsion i : Y ,→ X is homotopic to constant map.
The LS category of X , denoted cat(X ), is the least integer
m (or ∞) such that X is the union of m + 1 open subsets Ui ,
each contactible in X .
The rational LS category of X or that of its model (ΛV ,d) is
that of the associated rationalization, i.e.
cat0 (X ) = cat0 (ΛV ,d) := cat(XQ ).
Main result. Y. Felix and S. Halperin, [FH82] If (ΛV ,d) is a
1-connected model, then dim V odd ≤ cat0 (ΛV ,d).
Consequence. If (ΛV ,d) is a 1-connected model, then
dim V ≤ 2cat0 (ΛV ,d).
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Lusternick-Schnirelmann category
Definitions.
A subspace Y of a topological space is called contractible in
X if the inculsion i : Y ,→ X is homotopic to constant map.
The LS category of X , denoted cat(X ), is the least integer
m (or ∞) such that X is the union of m + 1 open subsets Ui ,
each contactible in X .
The rational LS category of X or that of its model (ΛV ,d) is
that of the associated rationalization, i.e.
cat0 (X ) = cat0 (ΛV ,d) := cat(XQ ).
Main result. Y. Felix and S. Halperin, [FH82] If (ΛV ,d) is a
1-connected model, then dim V odd ≤ cat0 (ΛV ,d).
Consequence. If (ΛV ,d) is a 1-connected model, then
dim V ≤ 2cat0 (ΛV ,d).
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
H-space
M.R Hilali and M.I. Mamouni (2006)
Definition. An H-space is a topological space X (generally
assumed to be connected) together with a continuous map
µ : X × X −→ X with an homotopy identity element e such
that the maps µ(e,∗) and µ(∗,e) are homotopic to the
identity.
Theorem
If X is an elliptic space, then dim H ∗ (X ,Q) ≥ dim (π∗ (X ) ⊗ Q)
Argument. By ([FHT01]-page 143-example 3), H-spaces
have minimal models of the form (ΛV ,0), then
ΛV ∼
= H ∗ (ΛV ,d).
Remark. The origin’name is J.P. Serre in honor of Hopf.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
H-space
M.R Hilali and M.I. Mamouni (2006)
Definition. An H-space is a topological space X (generally
assumed to be connected) together with a continuous map
µ : X × X −→ X with an homotopy identity element e such
that the maps µ(e,∗) and µ(∗,e) are homotopic to the
identity.
Theorem
If X is an elliptic space, then dim H ∗ (X ,Q) ≥ dim (π∗ (X ) ⊗ Q)
Argument. By ([FHT01]-page 143-example 3), H-spaces
have minimal models of the form (ΛV ,0), then
ΛV ∼
= H ∗ (ΛV ,d).
Remark. The origin’name is J.P. Serre in honor of Hopf.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
H-space
M.R Hilali and M.I. Mamouni (2006)
Topological groups, in particulier Lie groups and
homogeneous spaces.
The spheres S0 , S1 ( complexes), S3 (quaternions), S7
(octanions).
Adams has proved that these are the only H-spaces
among the spheres.
RP 1 = S1 ± 1,RP 3 = S3 ± 1,RP 7 = S7 ± 1.
In general RP n is an H-espace in and only if n + 1 = 2α .
CP ∞ .
The loop space ΩX where X is based space.
The Eilenberg-MacLane spaces, K (G,n), where n ≥ 1 et G
abelian, since K (G,n) = ΩK (G,n + 1).
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Pure case
M.R. Hilali(1990)
A space X and its minimal model are called pure if
dV even = 0
dV odd ⊂ Λ≥2 V even
This case was resolved by M.R. Hilali in 1990.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Pure case
M.R. Hilali(1990)
A space X and its minimal model are called pure if
dV even = 0
dV odd ⊂ Λ≥2 V even
This case was resolved by M.R. Hilali in 1990.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Pure spaces
Key idea
{xi ,i = 1, . . . ,n} a basis of V even .
W0 = H 0 (ΛV ,d) ∼
= Q.
W1 the vector space spanned by ([xi ])1≤i≤n .
W2 the vector space spanned by [xi xj ] 1≤i≤j≤n .
Minimality of the model assures that W0 ⊕ W1 ⊕ W2 is a
direct sum in H even (ΛV ,d), and that dim W1 = n.
W2 ⊕ (Λ2 V even ∩ dV odd ) = Λ2 V even , then
n(n + 1)
dim W2 ≥
− n − p where p = dim V odd .
2
n(n + 1)
dim H even (ΛV ,d) ≥
− p + 1.
2
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Main tool used
Euler-Poincaré characteristic
For any 1-connected elliptic space X , with (ΛV ,d) as a model,
we define two invariant:
X
Cohomological invariant: χc :=
(−1)k dim H k (X ,Q).
k ≥0
Homotopic invariant: χπ :=
X
(−1)k dim(πk (X ) ⊗ Q).
k ≥0
Thus
χc = dim H even (X ,Q) − dim H odd (X ,Q).
χπ = dim V even − dim V odd .
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Main tool used
Euler-Poincaré characteristic
For any 1-connected elliptic space X , with (ΛV ,d) as a model,
we define two invariant:
X
Cohomological invariant: χc :=
(−1)k dim H k (X ,Q).
k ≥0
Homotopic invariant: χπ :=
X
(−1)k dim(πk (X ) ⊗ Q).
k ≥0
Thus
χc = dim H even (X ,Q) − dim H odd (X ,Q).
χπ = dim V even − dim V odd .
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Main tool used
Euler-Poincaré characteristic
For any 1-connected elliptic space X , with (ΛV ,d) as a model,
we define two invariant:
X
Cohomological invariant: χc :=
(−1)k dim H k (X ,Q).
k ≥0
Homotopic invariant: χπ :=
X
(−1)k dim(πk (X ) ⊗ Q).
k ≥0
Thus
χc = dim H even (X ,Q) − dim H odd (X ,Q).
χπ = dim V even − dim V odd .
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Main theorem used
Stephen Halperin
Theorem [Ha83]
If X is a 1-connected and elliptic space, then χc ≥ 0 and χπ ≤ 0
Morever, χc > 0 ⇐⇒ χπ = 0 ⇐⇒ H odd (X ,Q) = 0
In terms of Sullivan’s models that means that:
dim V even = n ≤ n + p = dim V odd , dim V = 2n + p.
χπ = −p ≤ 0, dim H even (X ,Q) ≥ dim H odd (X ,Q).
and that
p = 0 ⇐⇒ H ∗ (X ,Q) = H even (X ,Q).
p 6= 0 ⇐⇒ dim H ∗ (X ,Q) = 2 dim H even (X ,Q).
Let us recall if p = 0, the space is pure, then the remainder
case is p 6= 0.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Main theorem used
Stephen Halperin
Theorem [Ha83]
If X is a 1-connected and elliptic space, then χc ≥ 0 and χπ ≤ 0
Morever, χc > 0 ⇐⇒ χπ = 0 ⇐⇒ H odd (X ,Q) = 0
In terms of Sullivan’s models that means that:
dim V even = n ≤ n + p = dim V odd , dim V = 2n + p.
χπ = −p ≤ 0, dim H even (X ,Q) ≥ dim H odd (X ,Q).
and that
p = 0 ⇐⇒ H ∗ (X ,Q) = H even (X ,Q).
p 6= 0 ⇐⇒ dim H ∗ (X ,Q) = 2 dim H even (X ,Q).
Let us recall if p = 0, the space is pure, then the remainder
case is p 6= 0.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Main theorem used
Stephen Halperin
Theorem [Ha83]
If X is a 1-connected and elliptic space, then χc ≥ 0 and χπ ≤ 0
Morever, χc > 0 ⇐⇒ χπ = 0 ⇐⇒ H odd (X ,Q) = 0
In terms of Sullivan’s models that means that:
dim V even = n ≤ n + p = dim V odd , dim V = 2n + p.
χπ = −p ≤ 0, dim H even (X ,Q) ≥ dim H odd (X ,Q).
and that
p = 0 ⇐⇒ H ∗ (X ,Q) = H even (X ,Q).
p 6= 0 ⇐⇒ dim H ∗ (X ,Q) = 2 dim H even (X ,Q).
Let us recall if p = 0, the space is pure, then the remainder
case is p 6= 0.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Hyperelliptic case, under conditions
M.R Hilali and M.I. Mamouni (2006)
The goal: To resolve the conjecture H in a largest case
Definition
A 1-connected elliptic space X and its model (ΛV ,d) are called
hyperelliptic if dV even = 0
dV odd ⊂ ΛV even ⊗ ΛV odd
Main idea:
if |y | is odd, than dy = P + ω where P ∈ ΛV even and
ω ∈ ΛV even ⊗ Λ+ V odd .
The terms ω are the obstruction for an hyperelliptic model
to be pure, the one at least among the ω 0 s is nonull.
1
dim W2 ≥ n(n + 1) − n − p + 1.
2 My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Hyperelliptic case, under conditions
M.R Hilali and M.I. Mamouni (2006)
The goal: To resolve the conjecture H in a largest case
Definition
A 1-connected elliptic space X and its model (ΛV ,d) are called
hyperelliptic if dV even = 0
dV odd ⊂ ΛV even ⊗ ΛV odd
Main idea:
if |y | is odd, than dy = P + ω where P ∈ ΛV even and
ω ∈ ΛV even ⊗ Λ+ V odd .
The terms ω are the obstruction for an hyperelliptic model
to be pure, the one at least among the ω 0 s is nonull.
1
dim W2 ≥ n(n + 1) − n − p + 1.
2 My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Hyperelliptic case, under conditions
M.R Hilali and M.I. Mamouni (2006)
The goal: To resolve the conjecture H in a largest case
Definition
A 1-connected elliptic space X and its model (ΛV ,d) are called
hyperelliptic if dV even = 0
dV odd ⊂ ΛV even ⊗ ΛV odd
Main idea:
if |y | is odd, than dy = P + ω where P ∈ ΛV even and
ω ∈ ΛV even ⊗ Λ+ V odd .
The terms ω are the obstruction for an hyperelliptic model
to be pure, the one at least among the ω 0 s is nonull.
1
dim W2 ≥ n(n + 1) − n − p + 1.
2 My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Hyperelliptic case, under conditions
Main result
Theorem
If (ΛV ,d) is a 1-connected and hyperelliptic model such that
n≥
p
1
1 + 12p − 15
2
where
n = dim V even and p = −χπ
then
dim H ∗ (ΛV ,d) ≥ dim V
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Formal case
M.R Hilali and M.I. Mamouni (2008)
Definition. A 1-connected elliptic space, X is called formal
if its model is (H ∗ (X ,Q),0).
Examples of formal spaces include spheres, H-spaces,
symmetric spaces, and compact Kähler manifolds.
Theorem. If (ΛV ,d) is a 1-connected, formal and elliptic
model then dim H ∗ (ΛV ,d) ≥ dim V
Main idea: Similar that of that in pure case, but the model
has the form [FxH82]:
V = V0 ⊕ V1 .
dV0 = 0.
V1 = V1odd
dV1 ⊂ ΛV0 .
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Toral rank, particular case
M.R Hilali and M.I. Mamouni (2007)
Definition.
The toral rank of a space is defined as
rk (X ) := max{n such that Tn acts almost freely on X }.
The rational toral rank, rk0 (X ) := rk (X ).
Theorem. The conjecture H holds for any 1-connected and
hyperlliptic space X , if rk0 (X ) = −χπ − i, where i ∈ {0,1,2}.
Main results used.
C. Allday and S. Halperin, [AH78]. If X is an 1-connected
and finite CW-complex then rk0 (X ) ≤ −χπ and the space is
pure is the case of equality.
M.R. Hilali, [Hi00]. If X is a 1-connected finite and
hyperelliptic CW complex, then dim H ∗ (X ,Q) ≥ 2rk0 (X ) (∗)
The inequality (*), called Toral Rank Conjecture, was
conjectured by S. Halperin have in 1986.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Toral rank, particular case
M.R Hilali and M.I. Mamouni (2007)
Definition.
The toral rank of a space is defined as
rk (X ) := max{n such that Tn acts almost freely on X }.
The rational toral rank, rk0 (X ) := rk (X ).
Theorem. The conjecture H holds for any 1-connected and
hyperlliptic space X , if rk0 (X ) = −χπ − i, where i ∈ {0,1,2}.
Main results used.
C. Allday and S. Halperin, [AH78]. If X is an 1-connected
and finite CW-complex then rk0 (X ) ≤ −χπ and the space is
pure is the case of equality.
M.R. Hilali, [Hi00]. If X is a 1-connected finite and
hyperelliptic CW complex, then dim H ∗ (X ,Q) ≥ 2rk0 (X ) (∗)
The inequality (*), called Toral Rank Conjecture, was
conjectured by S. Halperin have in 1986.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Toral rank, particular case
M.R Hilali and M.I. Mamouni (2007)
Definition.
The toral rank of a space is defined as
rk (X ) := max{n such that Tn acts almost freely on X }.
The rational toral rank, rk0 (X ) := rk (X ).
Theorem. The conjecture H holds for any 1-connected and
hyperlliptic space X , if rk0 (X ) = −χπ − i, where i ∈ {0,1,2}.
Main results used.
C. Allday and S. Halperin, [AH78]. If X is an 1-connected
and finite CW-complex then rk0 (X ) ≤ −χπ and the space is
pure is the case of equality.
M.R. Hilali, [Hi00]. If X is a 1-connected finite and
hyperelliptic CW complex, then dim H ∗ (X ,Q) ≥ 2rk0 (X ) (∗)
The inequality (*), called Toral Rank Conjecture, was
conjectured by S. Halperin have in 1986.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Toral rank, particular case
M.R Hilali and M.I. Mamouni (2007)
Definition.
The toral rank of a space is defined as
rk (X ) := max{n such that Tn acts almost freely on X }.
The rational toral rank, rk0 (X ) := rk (X ).
Theorem. The conjecture H holds for any 1-connected and
hyperlliptic space X , if rk0 (X ) = −χπ − i, where i ∈ {0,1,2}.
Main results used.
C. Allday and S. Halperin, [AH78]. If X is an 1-connected
and finite CW-complex then rk0 (X ) ≤ −χπ and the space is
pure is the case of equality.
M.R. Hilali, [Hi00]. If X is a 1-connected finite and
hyperelliptic CW complex, then dim H ∗ (X ,Q) ≥ 2rk0 (X ) (∗)
The inequality (*), called Toral Rank Conjecture, was
conjectured by S. Halperin have in 1986.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Formal dimension, under conditions
Definition and results used
Definition. fd(X ) = max{k ≥ 0, such that H k (X ,Q) 6= 0}.
Main results used. Let X a 1-connected and elliptic space,
and (ΛV ,d) its model, then:
dim H fd(X ) (X ,Q) = 1.
J. Friedlander and S. Halperin in [FH79].
1
2
fd(X ) ≥ dim V
There exists an homogeneous basis {x1 , . . . ,xn } of V even and
a another {y1 , . . . ,yn+p } of V odd such that:
|x1 | ≤ · · · ≤ |xn | |yi | ≥ 2|xi | − 1 for all i ∈ {1,...,n}
n+p
n+p
n
n
X
X
X
X
|xi | ≤ fd(X )
|yi | ≤ 2fd(X ) − 1,
|yi | −
(|xi | − 1) = fd
i=1
i=1
i=1
M.R. Hilali in [Hi00]. dim H ∗ (X ,Q) ≥ 2rk0 (X ) , when
fd(X ) − rk0 (X ) ≤ 6.
My Ismail Mamouni
The conjecture H
i=1
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Formal dimension, under conditions
Definition and results used
Definition. fd(X ) = max{k ≥ 0, such that H k (X ,Q) 6= 0}.
Main results used. Let X a 1-connected and elliptic space,
and (ΛV ,d) its model, then:
dim H fd(X ) (X ,Q) = 1.
J. Friedlander and S. Halperin in [FH79].
1
2
fd(X ) ≥ dim V
There exists an homogeneous basis {x1 , . . . ,xn } of V even and
a another {y1 , . . . ,yn+p } of V odd such that:
|x1 | ≤ · · · ≤ |xn | |yi | ≥ 2|xi | − 1 for all i ∈ {1,...,n}
n+p
n+p
n
n
X
X
X
X
|xi | ≤ fd(X )
|yi | ≤ 2fd(X ) − 1,
|yi | −
(|xi | − 1) = fd
i=1
i=1
i=1
M.R. Hilali in [Hi00]. dim H ∗ (X ,Q) ≥ 2rk0 (X ) , when
fd(X ) − rk0 (X ) ≤ 6.
My Ismail Mamouni
The conjecture H
i=1
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Formal dimension, under conditions
Definition and results used
Definition. fd(X ) = max{k ≥ 0, such that H k (X ,Q) 6= 0}.
Main results used. Let X a 1-connected and elliptic space,
and (ΛV ,d) its model, then:
dim H fd(X ) (X ,Q) = 1.
J. Friedlander and S. Halperin in [FH79].
1
2
fd(X ) ≥ dim V
There exists an homogeneous basis {x1 , . . . ,xn } of V even and
a another {y1 , . . . ,yn+p } of V odd such that:
|x1 | ≤ · · · ≤ |xn | |yi | ≥ 2|xi | − 1 for all i ∈ {1,...,n}
n+p
n+p
n
n
X
X
X
X
|xi | ≤ fd(X )
|yi | ≤ 2fd(X ) − 1,
|yi | −
(|xi | − 1) = fd
i=1
i=1
i=1
M.R. Hilali in [Hi00]. dim H ∗ (X ,Q) ≥ 2rk0 (X ) , when
fd(X ) − rk0 (X ) ≤ 6.
My Ismail Mamouni
The conjecture H
i=1
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Formal dimension, under conditions
Definition and results used
Definition. fd(X ) = max{k ≥ 0, such that H k (X ,Q) 6= 0}.
Main results used. Let X a 1-connected and elliptic space,
and (ΛV ,d) its model, then:
dim H fd(X ) (X ,Q) = 1.
J. Friedlander and S. Halperin in [FH79].
1
2
fd(X ) ≥ dim V
There exists an homogeneous basis {x1 , . . . ,xn } of V even and
a another {y1 , . . . ,yn+p } of V odd such that:
|x1 | ≤ · · · ≤ |xn | |yi | ≥ 2|xi | − 1 for all i ∈ {1,...,n}
n+p
n+p
n
n
X
X
X
X
|xi | ≤ fd(X )
|yi | ≤ 2fd(X ) − 1,
|yi | −
(|xi | − 1) = fd
i=1
i=1
i=1
M.R. Hilali in [Hi00]. dim H ∗ (X ,Q) ≥ 2rk0 (X ) , when
fd(X ) − rk0 (X ) ≤ 6.
My Ismail Mamouni
The conjecture H
i=1
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Formal dimension, under conditions
Definition and results used
Definition. fd(X ) = max{k ≥ 0, such that H k (X ,Q) 6= 0}.
Main results used. Let X a 1-connected and elliptic space,
and (ΛV ,d) its model, then:
dim H fd(X ) (X ,Q) = 1.
J. Friedlander and S. Halperin in [FH79].
1
2
fd(X ) ≥ dim V
There exists an homogeneous basis {x1 , . . . ,xn } of V even and
a another {y1 , . . . ,yn+p } of V odd such that:
|x1 | ≤ · · · ≤ |xn | |yi | ≥ 2|xi | − 1 for all i ∈ {1,...,n}
n+p
n+p
n
n
X
X
X
X
|xi | ≤ fd(X )
|yi | ≤ 2fd(X ) − 1,
|yi | −
(|xi | − 1) = fd
i=1
i=1
i=1
M.R. Hilali in [Hi00]. dim H ∗ (X ,Q) ≥ 2rk0 (X ) , when
fd(X ) − rk0 (X ) ≤ 6.
My Ismail Mamouni
The conjecture H
i=1
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Formal dimension, under conditions
M.R Hilali and M.I. Mamouni (2007)
Theorem 1
If X is a 1-connected and elliptic space such that fd(X ) ≤ 10,
then dim H ∗ (X ,Q) ≥ dim(π∗ (X ) ⊗ Q).
Theorem 2
If X is a 1-connected and elliptic space X such that
fd(X ) − rk0 (X ) ≤ 6, then dim H ∗ (X ,Q) ≥ dim(π∗ (X ) ⊗ Q).
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Symplectic manifolds
M.R. Hilali and M.I. Mamouni (2008)
Definition. It is a smooth manifold X = M 2m , endowed with
a closed and nondegenerate 2-form, called the symplectic
form.
Arguments.
There is a nonnull cohomolgical class ω in H 2 (X ,Q).
The cup-product, ω k : H m−k (X ,Q) −→ H m+k (X ,Q) is an
isomorphism for any k ∈ {0, . . . ,m}.
ω 2k 6= 0 for all k ∈ {1, . . . ,m}.
dim H even (ΛV ,d) ≥ m.
dim H ∗ (ΛV ,d) = 2 dim H even (ΛV ,d) ≥ 2m ≥ fd(X ) ≥ dim V .
Examples. Kähler compact manifolds
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Symplectic manifolds
M.R. Hilali and M.I. Mamouni (2008)
Definition. It is a smooth manifold X = M 2m , endowed with
a closed and nondegenerate 2-form, called the symplectic
form.
Arguments.
There is a nonnull cohomolgical class ω in H 2 (X ,Q).
The cup-product, ω k : H m−k (X ,Q) −→ H m+k (X ,Q) is an
isomorphism for any k ∈ {0, . . . ,m}.
ω 2k 6= 0 for all k ∈ {1, . . . ,m}.
dim H even (ΛV ,d) ≥ m.
dim H ∗ (ΛV ,d) = 2 dim H even (ΛV ,d) ≥ 2m ≥ fd(X ) ≥ dim V .
Examples. Kähler compact manifolds
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Symplectic manifolds
M.R. Hilali and M.I. Mamouni (2008)
Definition. It is a smooth manifold X = M 2m , endowed with
a closed and nondegenerate 2-form, called the symplectic
form.
Arguments.
There is a nonnull cohomolgical class ω in H 2 (X ,Q).
The cup-product, ω k : H m−k (X ,Q) −→ H m+k (X ,Q) is an
isomorphism for any k ∈ {0, . . . ,m}.
ω 2k 6= 0 for all k ∈ {1, . . . ,m}.
dim H even (ΛV ,d) ≥ m.
dim H ∗ (ΛV ,d) = 2 dim H even (ΛV ,d) ≥ 2m ≥ fd(X ) ≥ dim V .
Examples. Kähler compact manifolds
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
co-symplectic manifolds
M.R. Hilali and M.I. Mamouni (2008)
Definition. A cosymplectic structure on a 2m + 1-dimension
manifold X = M 2m+1 , is the data of a closed 1-form θ and a
closed 2-form ω.
Argument.
There exists at least one nonnull cohomological classe in
H 1 (X ,Q) and another one in H 2 (X ,Q).
Morever, we know from [BG67] that dim H k (X ,Q) 6= 0 for all
k ∈ {0, . . . ,m}.
dim H ∗ (ΛV ,d) ≥ 2m + 2 > 2m + 1 ≥ fd(X ) ≥ dim V .
Remark. All the known examples of cosymplectic
manifolds are non 1-connected. The π1 is that of a torus of
dimension ≥ 1 or a nilpotent group. However their minimal
models satisfy the property fd(X ) ≥ dimV . ([FH79]).
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
co-symplectic manifolds
M.R. Hilali and M.I. Mamouni (2008)
Definition. A cosymplectic structure on a 2m + 1-dimension
manifold X = M 2m+1 , is the data of a closed 1-form θ and a
closed 2-form ω.
Argument.
There exists at least one nonnull cohomological classe in
H 1 (X ,Q) and another one in H 2 (X ,Q).
Morever, we know from [BG67] that dim H k (X ,Q) 6= 0 for all
k ∈ {0, . . . ,m}.
dim H ∗ (ΛV ,d) ≥ 2m + 2 > 2m + 1 ≥ fd(X ) ≥ dim V .
Remark. All the known examples of cosymplectic
manifolds are non 1-connected. The π1 is that of a torus of
dimension ≥ 1 or a nilpotent group. However their minimal
models satisfy the property fd(X ) ≥ dimV . ([FH79]).
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
co-symplectic manifolds
M.R. Hilali and M.I. Mamouni (2008)
Definition. A cosymplectic structure on a 2m + 1-dimension
manifold X = M 2m+1 , is the data of a closed 1-form θ and a
closed 2-form ω.
Argument.
There exists at least one nonnull cohomological classe in
H 1 (X ,Q) and another one in H 2 (X ,Q).
Morever, we know from [BG67] that dim H k (X ,Q) 6= 0 for all
k ∈ {0, . . . ,m}.
dim H ∗ (ΛV ,d) ≥ 2m + 2 > 2m + 1 ≥ fd(X ) ≥ dim V .
Remark. All the known examples of cosymplectic
manifolds are non 1-connected. The π1 is that of a torus of
dimension ≥ 1 or a nilpotent group. However their minimal
models satisfy the property fd(X ) ≥ dimV . ([FH79]).
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
co-symplectic manifolds
M.R. Hilali and M.I. Mamouni (2008)
Definition. A cosymplectic structure on a 2m + 1-dimension
manifold X = M 2m+1 , is the data of a closed 1-form θ and a
closed 2-form ω.
Argument.
There exists at least one nonnull cohomological classe in
H 1 (X ,Q) and another one in H 2 (X ,Q).
Morever, we know from [BG67] that dim H k (X ,Q) 6= 0 for all
k ∈ {0, . . . ,m}.
dim H ∗ (ΛV ,d) ≥ 2m + 2 > 2m + 1 ≥ fd(X ) ≥ dim V .
Remark. All the known examples of cosymplectic
manifolds are non 1-connected. The π1 is that of a torus of
dimension ≥ 1 or a nilpotent group. However their minimal
models satisfy the property fd(X ) ≥ dimV . ([FH79]).
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Nilmanifolds
M.R. Hilali and M.I. Mamouni (2008)
Definition. Nilmanifold is a quotient of a nilpotent Lie group
by a discrete cocompact subgroup.
Arguments.
We know from [Dix55] that dim H k (X ,Q) ≥ 2 for k ≥ 1.
dim H ∗ (ΛV ,d) ≥ 2fd(X ) ≥ dim V .
All the known examples of nilmanifolds are non
1-connected. however the π1 is a nilpotent group, and the
model satisaies fd(X ) ≥ dimV ([FH79]).
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Nilmanifolds
M.R. Hilali and M.I. Mamouni (2008)
Definition. Nilmanifold is a quotient of a nilpotent Lie group
by a discrete cocompact subgroup.
Arguments.
We know from [Dix55] that dim H k (X ,Q) ≥ 2 for k ≥ 1.
dim H ∗ (ΛV ,d) ≥ 2fd(X ) ≥ dim V .
All the known examples of nilmanifolds are non
1-connected. however the π1 is a nilpotent group, and the
model satisaies fd(X ) ≥ dimV ([FH79]).
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Nilmanifolds
M.R. Hilali and M.I. Mamouni (2008)
Definition. Nilmanifold is a quotient of a nilpotent Lie group
by a discrete cocompact subgroup.
Arguments.
We know from [Dix55] that dim H k (X ,Q) ≥ 2 for k ≥ 1.
dim H ∗ (ΛV ,d) ≥ 2fd(X ) ≥ dim V .
All the known examples of nilmanifolds are non
1-connected. however the π1 is a nilpotent group, and the
model satisaies fd(X ) ≥ dimV ([FH79]).
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Length of the differential, under conditions
M.R. Hilali and M.I. Mamouni (2008)
We say that ΛV has differential d of homegeneous-length l
if dV ⊂ Λl V .
When l = 2, ΛV is called coformal.
We say that ΛV has differential d of homegeneous-length
at least l if dV ⊂ Λ≥l V .
By minimality, l ≥ 2.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Length of the differential, under conditions
M.R. Hilali and M.I. Mamouni (2008)
We say that ΛV has differential d of homegeneous-length l
if dV ⊂ Λl V .
When l = 2, ΛV is called coformal.
We say that ΛV has differential d of homegeneous-length
at least l if dV ⊂ Λ≥l V .
By minimality, l ≥ 2.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Length of the differential, under conditions
M.R. Hilali and M.I. Mamouni (2008)
We say that ΛV has differential d of homegeneous-length l
if dV ⊂ Λl V .
When l = 2, ΛV is called coformal.
We say that ΛV has differential d of homegeneous-length
at least l if dV ⊂ Λ≥l V .
By minimality, l ≥ 2.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Length of the differential, under conditions
M.R. Hilali and M.I. Mamouni (2008)
Theorem 1
If an elliptic minimal model (ΛV ,d) has an homogeneous-length
differential and whose rational Hurewicz homorphism is
non-zero in some odd degree. Then dim H ∗ (ΛV ,d) ≥ dim V .
Proof.
dim H ∗ (ΛV ,d) ≥ 2cat0 (ΛV ) by [Lu02]
dim V even ≤ dim V odd ≤ cat0 (ΛV ) by [FH82].
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Length of the differential, under conditions
M.R. Hilali and M.I. Mamouni (2008)
Theorem 1
If an elliptic minimal model (ΛV ,d) has an homogeneous-length
differential and whose rational Hurewicz homorphism is
non-zero in some odd degree. Then dim H ∗ (ΛV ,d) ≥ dim V .
Proof.
dim H ∗ (ΛV ,d) ≥ 2cat0 (ΛV ) by [Lu02]
dim V even ≤ dim V odd ≤ cat0 (ΛV ) by [FH82].
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Length of the differential, under conditions
M.R. Hilali and M.I. Mamouni (2008)
Theorem 2
If an elliptic minimal model (ΛV ,d) has a differential,
homogeneous of length at least 3, then dim H ∗ (ΛV ,d) ≥ dim V .
Theorem 3
If an elliptic minimal model (ΛV ,d) has a differential,
homogeneous of length 2 (i.e: coformal) with odd degree
generators only, (i.e., V even = 0), then dim H ∗ (ΛV ,d) ≥ dim V .
Proof. Hk∗ (ΛV ,d) 6= 0 for each k = 0, · · · ,e where
e = dim V odd + (l − 2) dim V even . (ref. [Lu02])
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Length of the differential, under conditions
M.R. Hilali and M.I. Mamouni (2008)
Theorem 2
If an elliptic minimal model (ΛV ,d) has a differential,
homogeneous of length at least 3, then dim H ∗ (ΛV ,d) ≥ dim V .
Theorem 3
If an elliptic minimal model (ΛV ,d) has a differential,
homogeneous of length 2 (i.e: coformal) with odd degree
generators only, (i.e., V even = 0), then dim H ∗ (ΛV ,d) ≥ dim V .
Proof. Hk∗ (ΛV ,d) 6= 0 for each k = 0, · · · ,e where
e = dim V odd + (l − 2) dim V even . (ref. [Lu02])
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Open question
Micheline Vigué, Paris 13 (2007)
If F −→ E −→ B is a fibration where F and B are elliptic and
both verify the conjecture H, what conditions on the fibration will
guarantee that E will too?
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Papers.
This results are accepted for publication in:
Journal of Homotopy and Related Structures:
http://www.emis.de/journals/JHRS/
Topology and its Applications:
http://ees.elsevier.com/topol/
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Papers.
This results are accepted for publication in:
Journal of Homotopy and Related Structures:
http://www.emis.de/journals/JHRS/
Topology and its Applications:
http://ees.elsevier.com/topol/
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Papers.
This results are accepted for publication in:
Journal of Homotopy and Related Structures:
http://www.emis.de/journals/JHRS/
Topology and its Applications:
http://ees.elsevier.com/topol/
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Acknowledgements
It is for us a pleasure to thank the following professors for their
interest and for their several readings and corrections.
Mohamed Rachid Hilali: Univ. Casablanca, Morocco.
Barry Jessup: Univ. Ottawa, Canada.
Jean Claude Thomas: Univ. Angers, France.
Micheline Vigué: Univ. Paris 13, France.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Acknowledgements
It is for us a pleasure to thank the following professors for their
interest and for their several readings and corrections.
Mohamed Rachid Hilali: Univ. Casablanca, Morocco.
Barry Jessup: Univ. Ottawa, Canada.
Jean Claude Thomas: Univ. Angers, France.
Micheline Vigué: Univ. Paris 13, France.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Acknowledgements
It is for us a pleasure to thank the following professors for their
interest and for their several readings and corrections.
Mohamed Rachid Hilali: Univ. Casablanca, Morocco.
Barry Jessup: Univ. Ottawa, Canada.
Jean Claude Thomas: Univ. Angers, France.
Micheline Vigué: Univ. Paris 13, France.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Acknowledgements
It is for us a pleasure to thank the following professors for their
interest and for their several readings and corrections.
Mohamed Rachid Hilali: Univ. Casablanca, Morocco.
Barry Jessup: Univ. Ottawa, Canada.
Jean Claude Thomas: Univ. Angers, France.
Micheline Vigué: Univ. Paris 13, France.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
Acknowledgements
It is for us a pleasure to thank the following professors for their
interest and for their several readings and corrections.
Mohamed Rachid Hilali: Univ. Casablanca, Morocco.
Barry Jessup: Univ. Ottawa, Canada.
Jean Claude Thomas: Univ. Angers, France.
Micheline Vigué: Univ. Paris 13, France.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
References I
[AH78]:
C. Allday and S. Halperin,
Lie group actions on spaces of finite rank, Quar. J. Math.
Oxford 28 (1978), 69-76.
[BG67]: D.E. Blair and S.I. Goldberg,
Topology of almost contact manifolds,
Journal of Differential Geometry Vol. 1 (1967), Intelpress,
347-354.
[Dix55]: J. Dixmier,
Cohomologie des algèbres de Lie nilpotentes,
Acta Sci. Math, Szeged 16 (1955), 246-250.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
References II
[FHT01]: Y. Félix, S. Halperin and J-C Thomas,
Rational homotopy theory
Graduate Texts in Math, Vol. 205, Springer-Verlag, New
York, 2001.
[FH79]: J. Friedlander and S. Halperin,
An arithmetic characterization of the rational homotopy
groups of certain spaces,
Invent. Math. 53 (1979), 117-133.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
References III
[FxH82]: Y. Félix and S. Halperin,
Formal spaces with finite dimensional rational homotopy,
Transactions of the American Mathematical Society 270
(1982), 575-588.
[FH82]: Y. Felix and S. Halperin,
Rational LS category and its applications,
Trans. Amer. Math. Soc. 273 (1982), no. 1, 1-38.
[Ha83]: S. Halperin,
Finitness in the minimal models of Sullivan, TAMS. 230
(1983), 173-199.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
References IV
[Hs99]: K. Hess,
A history of rational homotopy theory,
History of topology, chapt. 27 (1999), 757-796, Elsevier
Science.
[Hi00]: M.R. Hilali,
Sur la conjecture de Halperin relative au rang torique,
Bull. Belg. Math. Soc. Simon Stevin Vol. 7, Num. 2 (2000),
221-227.
[Lu02]: G. Lupton,
The Rational Toomer Invariant and Certain Elliptic Spaces,
Contemporary Mathematics Vol. 316 (2002), 135-146,
arXiv:math/0309392v1.
My Ismail Mamouni
The conjecture H
Introduction
Our tools
Resolved cases
Papers
Acknowledgements
References
References V
[Su78]: D. Sullivan,
Infinitesimal computations of topology, Publ. Math. IHES 47
(1978), 269-331.
My Ismail Mamouni
The conjecture H