Floer
fundamental
group
J.F. Barraud

Floer fundamental groupof Lagrangian submanifolds

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

J.F. Barraud
IMT Toulouse Universty

May 15, 2016

Floerfundamental
group

Settings

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms(M, ω) : closed monotone symplectic manifold
L ⊂ M : Lagrangian submanifold.
- ω|π2 (M,L) = 0 (or NL > dim L + 1).
- [will discuss the monotone case later].

? ∈ L : base point
Auxiliary data :- H : M × [0, 1] → R,
- J : (time dependent) ω-compatible almost complex
structure,- cutoff functions...

Perspectives
Morphisms

φtH : Hamiltonian flow of H,
Lt = φtH (L)Floerfundamental
group

Settings

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms(M, ω) : closed monotone symplectic manifold
L ⊂ M : Lagrangian submanifold.
- ω|π2 (M,L) = 0 (or NL > dim L + 1).
- [will discuss the monotone case later].

? ∈ L : base point
Auxiliary data :- H : M × [0, 1] → R,
- J : (time dependent) ω-compatible almost complex
structure,- cutoff functions...

Perspectives
Morphisms

φtH : Hamiltonian flow of H,
Lt = φtH (L)Floerfundamental
group

Settings

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms(M, ω) : closed monotone symplectic manifold
L ⊂ M : Lagrangian submanifold.
- ω|π2 (M,L) = 0 (or NL > dim L + 1).
- [will discuss the monotone case later].

? ∈ L : base point
Auxiliary data :- H : M × [0, 1] → R,
- J : (time dependent) ω-compatible almost complex
structure,- cutoff functions...

Perspectives
Morphisms

φtH : Hamiltonian flow of H,
Lt = φtH (L)Floerfundamental
group

Settings

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms(M, ω) : closed monotone symplectic manifold
L ⊂ M : Lagrangian submanifold.
- ω|π2 (M,L) = 0 (or NL > dim L + 1).
- [will discuss the monotone case later].

? ∈ L : base point
Auxiliary data :- H : M × [0, 1] → R,
- J : (time dependent) ω-compatible almost complex
structure,- cutoff functions...

Perspectives
Morphisms

φtH : Hamiltonian flow of H,
Lt = φtH (L)Floerfundamental
group

Settings

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms(M, ω) : closed monotone symplectic manifold
L ⊂ M : Lagrangian submanifold.
- ω|π2 (M,L) = 0 (or NL > dim L + 1).
- [will discuss the monotone case later].

? ∈ L : base point
Auxiliary data :- H : M × [0, 1] → R,
- J : (time dependent) ω-compatible almost complex
structure,- cutoff functions...

Perspectives
Morphisms

φtH : Hamiltonian flow of H,
Lt = φtH (L)Floerfundamental
group

Moduli spaces

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Recall the Floer equation for u : R × [0, 1] → M :(u(s, 0) ∈ L0∂u∂u
+ Jt (u)
=0
and
∂s
∂t
u(s, 1) ∈ L1

Floerfundamental
group

Moduli spaces

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Recall the Floer equation for u : R × [0, 1] → M :(u(s, 0) ∈ L0∂u∂u
+ Jt (u)
=0
and
∂s
∂t
u(s, 1) ∈ Lβ(s)Floerfundamental
group

Moduli spaces

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

Recall the Floer equation for u : R × [0, 1] → M :(u(s, 0) ∈ L0∂u∂u
+ Jt (u)
=0
and
∂s
∂t
u(s, 1) ∈ Lβ(s)π1Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

M(y , x)

Floerfundamental
group

Moduli spaces

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

Recall the Floer equation for u : R × [0, 1] → M :(u(s, 0) ∈ L0∂u∂u
+ Jt (u)
=0
and
∂s
∂t
u(s, 1) ∈ Lβ(s)π1Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

M(y , x)M(y , ∅)

Floerfundamental
group

Moduli spaces

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

Recall the Floer equation for u : R × [0, 1] → M :(u(s, 0) ∈ L0∂u∂u
+ Jt (u)
=0
and
∂s
∂t
u(s, 1) ∈ Lβ(s)π1Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

M(y , x)M(y , ∅)

M(?, ∅)

Floerfundamental
group

Moduli spaces

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1

Recall the Floer equation for u : R × [0, 1] → M :(u(s, 0) ∈ L0∂u∂u
+ Jt (u)
=0
and
∂s
∂t
u(s, 1) ∈ Lβ(s)Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

M(y , x)M(y , ∅)

M(?, ∅)

Floerfundamental
group

Moduli spaces

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1

Recall the Floer equation for u : R × [0, 1] → M :(u(s, 0) ∈ L0∂u∂u
+ Jt (u)
=0
and
∂s
∂t
u(s, 1) ∈ Lβ(s)Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

M(y , x)M(y , ∅)

M(?, ∅)

M(?, x)Floerfundamental
group

Moduli spaces

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

Recall the Floer equation for u : R × [0, 1] → M :(u(s, 0) ∈ L0∂u∂u
+ Jt (u)
=0
and
∂s
∂t
u(s, 1) ∈ Lβ(s)π1Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

M(y , x)M(y , ∅)

M(?, ∅)

M(?, x)dim =|y |−|x|−1

dim = |y |dim = 1

dim = −|x|

Floerfundamental
group

Floer steps

J.F. Barraud
Settings

Definition

Floer Loops

A Floer step is an (oriented) component with non empty
boundary of one of the moduli spaces M(y , ∅) for |y | = 1 orM(?, ∅).

Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

M(y , ∅) :

Floerfundamental
group

Floer steps

J.F. Barraud
Settings

Definition

Floer Loops

A Floer step is an (oriented) component with non empty
boundary of one of the moduli spaces M(y , ∅) for |y | = 1 orM(?, ∅).

Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

M(y , ∅) :

Floerfundamental
group

Floer steps

J.F. Barraud
Settings

Definition

Floer Loops

A Floer step is an (oriented) component with non empty
boundary of one of the moduli spaces M(y , ∅) for |y | = 1 orM(?, ∅).

Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

M(y , ∅) :

Floerfundamental
group

Base point

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

What about M(?, ∅) ?

Floerfundamental
group

Base point

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

What about M(?, ∅) ?

Floerfundamental
group

Base point

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

What about M(?, ∅) ?

Floerfundamental
group

Base point

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

What about M(?, ∅) ?

Floerfundamental
group

Base point

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

What about M(?, ∅) ?

Floerfundamental
group

Floer Loops

J.F. Barraud

Definition
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

A Floer based loop is a sequence of consecutive Floer steps,that start and end at ?.

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

LH (L, ?) = {Floer loops}∼(F )M(y , ∅)uev

/L
/ u(+∞)

Floerfundamental
group

Floer Loops

J.F. Barraud
Settings

Definition

Floer Loops

A Floer based loop is a sequence of consecutive Floer steps,that start and end at ?.

Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

LH (L, ?) = {Floer loops}∼(F )Perspectives
Morphisms

M(y , ∅)uev

/L
/ u(+∞)

Floerfundamental
group

Floer Loops

J.F. Barraud
Settings

Definition

Floer Loops

A Floer based loop is a sequence of consecutive Floer steps,that start and end at ?.

Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

LH (L, ?) = {Floer loops}∼(F )Perspectives
Morphisms

M(y , ∅)uev

/L
/ u(+∞)

Floerfundamental
group

Floer Loops

J.F. Barraud
Settings

Definition

Floer Loops

A Floer based loop is a sequence of consecutive Floer steps,that start and end at ?.

Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

LH (L, ?) = {Floer loops}∼(F )Perspectives
Morphisms(F )LH (L, ?)(uτ ) 

ev

/ π1 (L, ?)/ (uτ (+∞))

Floerfundamental
group

Main statement

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

Theorem

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms(F )The group morphism LH (L, ?)ev

Postponethe proof
the description of the relations
to first revisit the Morse case...

/ π1 (L, ?) is onto.

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group
J.F. Barraud

The Morse caseLetf : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,

Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group

The Morse case

J.F. Barraud
Settings
Floer Loops

Let

Moduli spaces
Floer steps
Base point
Floer Loops

f : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group

The Morse case

J.F. Barraud
Settings
Floer Loops

Let

Moduli spaces
Floer steps
Base point
Floer Loops

f : L → R be Morse function (with a single minimum),g be a metric on L such that (f , g ) is Morse-Smale,π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Definition(M)

Lf(L) (Morse loops) is the group of words in the letters{y , y −1 }y ∈Crit1 (f ) .

(M)

Lf

(L)

ev / /

π1 (L, ?) .

Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the Morse equation :dγ= −Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the “interrupted” Morse equation :dγ= −β(t) Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

Unstable manifolds revisited

J.F. Barraud
Settings
Floer Loops

We need to understand W u (y ) for y ∈ Crit1 (f ): :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Recall the “interrupted” Morse equation :dγ= −β(t) Xγ .
dt

Perspectives
Morphisms

W u (y ) ' M̊(y , ∅)Floerfundamental
group

From Morse to Floer loops

J.F. Barraud

Pick f such that it has a single minimum at ?.
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

(F )

φ

(M)

(M)

ψ

(F )

LH (L, ?) −−→ Lf
Lf(L, ?) : push down using f gradient flow.

(L, ?) −−→ LH (L, ?) : ?

Floerfundamental
group

From Morse to Floer loops

J.F. Barraud

Pick f such that it has a single minimum at ?.
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

(F )

φ

(M)

(M)

ψ

(F )

LH (L, ?) −−→ Lf
Lf(L, ?) : push down using f gradient flow.

(L, ?) −−→ LH (L, ?) : “crocodile walk”.

Floerfundamental
group

From Morse to Floer loops

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

(M)

Lf

(F )

Perspectives
Morphisms(M)

Lf/ / π1 (L, ?)ψ

Lf (L, ?)Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms(L, ?)

id/ π1 (L, ?)φ(L, ?)id/ / π1 (L, ?)Floerfundamental
group

From Morse to Floer loops

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

(M)

Lf

(F )

Perspectives
Morphisms(M)

Lf/ / π1 (L, ?)ψ

Lf (L, ?)Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms(L, ?)

id

/ / π1 (L, ?)φ(L, ?)id/ / π1 (L, ?)Floerfundamental
group

Relations

J.F. Barraud(F )Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

ev

ker LH (L, ?) −→ π1 (L, ?)= ?

It only depends on L, ?, H, J, but we will resort to a Morsefunction to pick explicit generators.
Let
Rf = {∂W u (z), z ∈ Crit2 (f )},
and
(F )

RH = {ψ(γ) = 1, γ ∈ Rf } ∪ {ψ ◦ φ(γ) = γ, γ ∈ LH }Perspectives
Morphisms

Theorem(F )∼LH (L, ?)< R > −−−−→ π1 (L, ?)H
ev

Floerfundamental
group

Relations

J.F. Barraud(F )Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

ev

ker LH (L, ?) −→ π1 (L, ?)= ?

It only depends on L, ?, H, J, but we will resort to a Morsefunction to pick explicit generators.
Let
Rf = {∂W u (z), z ∈ Crit2 (f )},
and
(F )

RH = {ψ(γ) = 1, γ ∈ Rf } ∪ {ψ ◦ φ(γ) = γ, γ ∈ LH }Perspectives
Morphisms

Theorem(F )∼LH (L, ?)< R > −−−−→ π1 (L, ?)H
ev

Floerfundamental
group

Relations

J.F. Barraud(F )Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

ev

ker LH (L, ?) −→ π1 (L, ?)= ?

It only depends on L, ?, H, J, but we will resort to a Morsefunction to pick explicit generators.
Let
Rf = {∂W u (z), z ∈ Crit2 (f )},
and
(F )

RH = {ψ(γ) = 1, γ ∈ Rf } ∪ {ψ ◦ φ(γ) = γ, γ ∈ LH }Perspectives
Morphisms

Theorem(F )∼LH (L, ?)< R > −−−−→ π1 (L, ?)H
ev

Floerfundamental
group

Relations

J.F. Barraud(F )Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

ev

ker LH (L, ?) −→ π1 (L, ?)= ?

It only depends on L, ?, H, J, but we will resort to a Morsefunction to pick explicit generators.
Let
Rf = {∂W u (z), z ∈ Crit2 (f )},
and
(F )

RH = {ψ(γ) = 1, γ ∈ Rf } ∪ {ψ ◦ φ(γ) = γ, γ ∈ LH }Perspectives
Morphisms

Theorem(F )∼LH (L, ?)< R > −−−−→ π1 (L, ?)H
ev

Floerfundamental
group

Monotone case.

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Suppose now L is monotone.

Floerfundamental
group

Monotone case.

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Suppose now L is monotone.

Floerfundamental
group

Monotone case.

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Suppose now L is monotone.

Floerfundamental
group

Monotone case.

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Suppose now L is monotone.

Floerfundamental
group

Monotone case.

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Suppose now L is monotone.

Floerfundamental
group

Monotone case.

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Suppose now L is monotone.

Floerfundamental
group

Monotone case.

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Suppose now L is monotone.

Floerfundamental
group

Monotone case.

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Suppose now L is monotone with NL ≥ 3.

Floerfundamental
group

Monotone case.

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Suppose now L is monotone with NL ≥ 3.

Floerfundamental
group

Monotone case.

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Suppose now L is monotone with NL ≥ 3.

Floerfundamental
group

Monotone case.

J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

Suppose now L is monotone with NL ≥ 3.
Add a Morse function f and a metric to the auxiliary data and“pearlify” all the previous definitions (following Biran, Cornea,
Lalonde) :π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms(F )With this new definition, LH,f (L, ?) → π1 (L, ?) is still
surjective.

Floerfundamental
group

Relation to Floer Homology

J.F. Barraud
Settings
Floer Loops

The homology makes use of 1-dimensional moduli spaces, but
only to prove ∂ 2 = 0 :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

1-dimensional moduli spaces “upgrade” ∂ 2 z = 0 by providing
a cyclic order on its support...

Floerfundamental
group

Relation to Floer Homology

J.F. Barraud
Settings
Floer Loops

The homology makes use of 1-dimensional moduli spaces, but
only to prove ∂ 2 = 0 :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

1-dimensional moduli spaces “upgrade” ∂ 2 z = 0 by providing
a cyclic order on its support...

Floerfundamental
group

Relation to Floer Homology

J.F. Barraud
Settings
Floer Loops

The homology makes use of 1-dimensional moduli spaces, but
only to prove ∂ 2 = 0 :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

1-dimensional moduli spaces “upgrade” ∂ 2 z = 0 by providing
a cyclic order on its support...

Floerfundamental
group

Relation to Floer Homology

J.F. Barraud
Settings
Floer Loops

The homology makes use of 1-dimensional moduli spaces, but
only to prove ∂ 2 = 0 :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

1-dimensional moduli spaces “upgrade” ∂ 2 z = 0 by providing
a cyclic order on its support...

Floerfundamental
group

Relation to Floer Homology

J.F. Barraud
Settings
Floer Loops

The homology makes use of 1-dimensional moduli spaces, but
only to prove ∂ 2 = 0 :Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

1-dimensional moduli spaces “upgrade” ∂ 2 z = 0 by providing
a cyclic order on its support...

Floerfundamental
group

Variation on the Arnold conjecture.

J.F. Barraud
Settings

Suppose L is (weakly) exact. Define (for |y | = 1) :Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

ν(y ) = ]{ steps through y }ν(?) = ]{non canonical steps through ?}π1Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives

TheoremLet δ(π1 ) be the minimal number of generators of π1 (L).
Then :
X
ν(?) +
ν(y ) ≥ δ(π1 (L))|y |=1Morphisms

Remark : in the stable Morse setting, exemples are known
where ]Crit(f ) < δ(π1 ).

Floerfundamental
group

Variation on the Arnold conjecture.

J.F. Barraud

Suppose L is (weakly) exact. Define (for |y | = 1) :Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

1 X|M(y , x)| × |M(x, ∅)|
2
|x|=0

 1
1 X
|M(?, x)| × |M(x, ∅)| −ν(?) =
22

ν(y ) =

|x|=0

TheoremLet δ(π1 ) be the minimal number of generators of π1 (L).
Then :
X
ν(?) +
ν(y ) ≥ δ(π1 (L))|y |=1

Remark : in the stable Morse setting, exemples are known
where ]Crit(f ) < δ(π1 ).

Floerfundamental
group

Variation on the Arnold conjecture.

J.F. Barraud

Suppose L is (weakly) exact. Define (for |y | = 1) :Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

1 X|M(y , x)| × |M(x, ∅)|
2
|x|=0

 1
1 X
|M(?, x)| × |M(x, ∅)| −ν(?) =
22

ν(y ) =

|x|=0

TheoremLet δ(π1 ) be the minimal number of generators of π1 (L).
Then :
X
ν(?) +
ν(y ) ≥ δ(π1 (L))|y |=1

Remark : in the stable Morse setting, exemples are known
where ]Crit(f ) < δ(π1 ).

Floerfundamental
group

Variation on the Arnold conjecture.

J.F. Barraud

Suppose L is (weakly) exact. Define (for |y | = 1) :Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

1 X|M(y , x)| × |M(x, ∅)|
2
|x|=0

 1
1 X
|M(?, x)| × |M(x, ∅)| −ν(?) =
22

ν(y ) =

|x|=0

TheoremLet δ(π1 ) be the minimal number of generators of π1 (L).
Then :
X
ν(?) +
ν(y ) ≥ δ(π1 (L))|y |=1

Remark : in the stable Morse setting, exemples are known
where ]Crit(f ) < δ(π1 ).

Floerfundamental
group
J.F. Barraud
Settings

Lagrangian cobordisms
Let L− and L+ be two Lagrangian submfds in M, andW ,→ C × M a Lagrangian cobordism from L− to L+ .

Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Biran,Cornea : W monotone ⇒ HF∗ (L− ) = HF∗ (L+ ).

Proposition (with L. Simone Suarez)i

If W is (weakly) exact, then the inclusions L± ,−→ W induce
surjective maps π1 (L± )i∗/ / π1 (W ) .

Floerfundamental
group

Lagrangian cobordisms

J.F. Barraud
Settings
Floer Loops

Sketch of proof: Deform W ,Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

pick ? above the intersection point (and suitable J) : all the
strips are confined to this fiber.
Remark : this can also be derived from HF? with local
coefficients [B. Chantraı̂ne - L. Simone Suarez].
Q : is i∗ also injective ?

Floerfundamental
group

Lagrangian cobordisms

J.F. Barraud
Settings
Floer Loops

Sketch of proof: Deform W ,Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

pick ? above the intersection point (and suitable J) : all the
strips are confined to this fiber.
Remark : this can also be derived from HF? with local
coefficients [B. Chantraı̂ne - L. Simone Suarez].
Q : is i∗ also injective ?

Floerfundamental
group

Lagrangian cobordisms

J.F. Barraud
Settings
Floer Loops

Sketch of proof: Deform W ,Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

pick ? above the intersection point (and suitable J) : all the
strips are confined to this fiber.
Remark : this can also be derived from HF? with local
coefficients [B. Chantraı̂ne - L. Simone Suarez].
Q : is i∗ also injective ?

Floerfundamental
group
J.F. Barraud
Settings

Morphisms ?
An essential feature of the Floer homology is the comparison
morphisms associated to homotopies of the auxiliary data.

Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

H o (Hλ ) / H 0Φ/ HF∗ (L0 , L1 ) −→ HF∗ (L0 , L01 )π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

However, this construction
makes sens for small perturbations of generic situations,
may allow to cross non genericity walls in at least one
direction ?

Floerfundamental
group
J.F. Barraud
Settings

Morphisms ?
An essential feature of the Floer homology is the comparison
morphisms associated to homotopies of the auxiliary data.

Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

H o (Hλ ) / H 0Φ/ HF∗ (L0 , L1 ) −→ HF∗ (L0 , L01 )π1 (L, H) −→?

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

However, this construction
makes sens for small perturbations of generic situations,
may allow to cross non genericity walls in at least one
direction ?

Floerfundamental
group
J.F. Barraud
Settings

Morphisms ?
An essential feature of the Floer homology is the comparison
morphisms associated to homotopies of the auxiliary data.

Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

H o (Hλ ) / H 0Φ/ HF∗ (L0 , L1 ) −→ HF∗ (L0 , L01 )π1 (L, H) −→?

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

However, this construction
makes sens for small perturbations of generic situations,
may allow to cross non genericity walls in at least one
direction ?

Floerfundamental
group
J.F. Barraud
Settings

Morphisms ?
An essential feature of the Floer homology is the comparison
morphisms associated to homotopies of the auxiliary data.

Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

H o (Hλ ) / H 0Φ/ HF∗ (L0 , L1 ) −→ HF∗ (L0 , L01 )π1 (L, H) −→?

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

However, this construction
makes sens for small perturbations of generic situations,
may allow to cross non genericity walls in at least one
direction ?

Floerfundamental
group
J.F. Barraud
Settings

Morphisms ?
An essential feature of the Floer homology is the comparison
morphisms associated to homotopies of the auxiliary data.

Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

H o (Hλ ) / H 0Φ/ HF∗ (L0 , L1 ) −→ HF∗ (L0 , L01 )π1 (L, H) −→?

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

However, this construction
makes sens for small perturbations of generic situations,
may allow to cross non genericity walls in at least one
direction ?

Floerfundamental
group
J.F. Barraud
Settings
Floer Loops
Moduli spaces
Floer steps
Base point
Floer Loops

π1
Main statement
The Morse case
From Morse to
Floer
Relations
BubblesComments

Applications
Variation on the
Arnold conj.
Lagr.
Cobordisms

Perspectives
Morphisms

Thank you !

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

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group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Floerfundamental
group
J.F. Barraud

Thank you !