jos e gabriel carrasquel vera - cmup.fc.up.pt
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On the sectional category of certain maps
Jose Gabriel Carrasquel Vera
Universite catholique de Louvain, Belgique
XXIst Oporto Meeting on Geometry, Topology and PhysicsLisboa, 6 February 2015
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Rational homotopy
All spaces considered are rational simply connected spaces offinite type.
These spaces form a category whose homotopy category isequivalent to that of cdga.
cdga= simply connected commutative differential gradedQ-algebras of finite type.
The cofibrant objects of cdga are the Sullivan algebras(ΛV , d).
The cofibrations are relative Sullivan inclusionsA � (A⊗ ΛV ,D).
Here, cat(∗) = 0.
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Rational homotopy
All spaces considered are rational simply connected spaces offinite type.
These spaces form a category whose homotopy category isequivalent to that of cdga.
cdga= simply connected commutative differential gradedQ-algebras of finite type.
The cofibrant objects of cdga are the Sullivan algebras(ΛV , d).
The cofibrations are relative Sullivan inclusionsA � (A⊗ ΛV ,D).
Here, cat(∗) = 0.
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Rational homotopy
All spaces considered are rational simply connected spaces offinite type.
These spaces form a category whose homotopy category isequivalent to that of cdga.
cdga= simply connected commutative differential gradedQ-algebras of finite type.
The cofibrant objects of cdga are the Sullivan algebras(ΛV , d).
The cofibrations are relative Sullivan inclusionsA � (A⊗ ΛV ,D).
Here, cat(∗) = 0.
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Rational homotopy
All spaces considered are rational simply connected spaces offinite type.
These spaces form a category whose homotopy category isequivalent to that of cdga.
cdga= simply connected commutative differential gradedQ-algebras of finite type.
The cofibrant objects of cdga are the Sullivan algebras(ΛV , d).
The cofibrations are relative Sullivan inclusionsA � (A⊗ ΛV ,D).
Here, cat(∗) = 0.
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Rational homotopy
All spaces considered are rational simply connected spaces offinite type.
These spaces form a category whose homotopy category isequivalent to that of cdga.
cdga= simply connected commutative differential gradedQ-algebras of finite type.
The cofibrant objects of cdga are the Sullivan algebras(ΛV , d).
The cofibrations are relative Sullivan inclusionsA � (A⊗ ΛV ,D).
Here, cat(∗) = 0.
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Rational homotopy
All spaces considered are rational simply connected spaces offinite type.
These spaces form a category whose homotopy category isequivalent to that of cdga.
cdga= simply connected commutative differential gradedQ-algebras of finite type.
The cofibrant objects of cdga are the Sullivan algebras(ΛV , d).
The cofibrations are relative Sullivan inclusionsA � (A⊗ ΛV ,D).
Here, cat(∗) = 0.
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
A brief history of rational LS category
Definition
Let (ΛV , d) be a Sullivan algebra and consider the cdga morphism
ρm : ΛV → ΛV
Λ>mV.
Define:
cat (ΛV , d) as the smallest m such that ρm admits ahomotopy retraction as cdga.
mcat (ΛV , d) the smallest m such that ρm admits a homotopyretraction as (ΛV , d)-module.
Hcat (ΛV , d) the smallest m such that H(ρm) is injective.
Hcat (ΛV , d) ≤ mcat (ΛV , d) ≤ cat (ΛV , d)
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
A brief history of rational LS category
Definition
Let (ΛV , d) be a Sullivan algebra and consider the cdga morphism
ρm : ΛV → ΛV
Λ>mV.
Define:
cat (ΛV , d) as the smallest m such that ρm admits ahomotopy retraction as cdga.
mcat (ΛV , d) the smallest m such that ρm admits a homotopyretraction as (ΛV , d)-module.
Hcat (ΛV , d) the smallest m such that H(ρm) is injective.
Hcat (ΛV , d) ≤ mcat (ΛV , d) ≤ cat (ΛV , d)
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
A brief history of rational LS category
Definition
Let (ΛV , d) be a Sullivan algebra and consider the cdga morphism
ρm : ΛV → ΛV
Λ>mV.
Define:
cat (ΛV , d) as the smallest m such that ρm admits ahomotopy retraction as cdga.
mcat (ΛV , d) the smallest m such that ρm admits a homotopyretraction as (ΛV , d)-module.
Hcat (ΛV , d) the smallest m such that H(ρm) is injective.
Hcat (ΛV , d) ≤ mcat (ΛV , d) ≤ cat (ΛV , d)
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
A brief history of rational LS category
Definition
Let (ΛV , d) be a Sullivan algebra and consider the cdga morphism
ρm : ΛV → ΛV
Λ>mV.
Define:
cat (ΛV , d) as the smallest m such that ρm admits ahomotopy retraction as cdga.
mcat (ΛV , d) the smallest m such that ρm admits a homotopyretraction as (ΛV , d)-module.
Hcat (ΛV , d) the smallest m such that H(ρm) is injective.
Hcat (ΛV , d) ≤ mcat (ΛV , d) ≤ cat (ΛV , d)
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
A brief history of rational LS category
Definition
Let (ΛV , d) be a Sullivan algebra and consider the cdga morphism
ρm : ΛV → ΛV
Λ>mV.
Define:
cat (ΛV , d) as the smallest m such that ρm admits ahomotopy retraction as cdga.
mcat (ΛV , d) the smallest m such that ρm admits a homotopyretraction as (ΛV , d)-module.
Hcat (ΛV , d) the smallest m such that H(ρm) is injective.
Hcat (ΛV , d) ≤ mcat (ΛV , d) ≤ cat (ΛV , d)
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
A brief history of rational LS category
Theorem (Felix, Halperin, TAMS 1982)
If (ΛV , d) is a Sullivan model for X , then
cat(X ) = cat (ΛV , d) .
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
A brief history of rational LS category
Theorem (Jessup, TAMS 1990)
Denote ΛSn the rational model for the sphere Sn then
mcat((ΛV , d)⊗ ΛSn) = mcat (ΛV , d) + 1
Theorem (Hess, Topology 1991)
cat (ΛV , d) = mcat (ΛV , d) and thus cat(X ) = mcat (ΛV , d)
Theorem (Felix, Halperin, Lemaire, Topology 1998)
If H (ΛV , d) verifies Poincare duality then
Hcat (ΛV , d) = mcat (ΛV , d)
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
A brief history of rational LS category
Theorem (Jessup, TAMS 1990)
Denote ΛSn the rational model for the sphere Sn then
mcat((ΛV , d)⊗ ΛSn) = mcat (ΛV , d) + 1
Theorem (Hess, Topology 1991)
cat (ΛV , d) = mcat (ΛV , d) and thus cat(X ) = mcat (ΛV , d)
Theorem (Felix, Halperin, Lemaire, Topology 1998)
If H (ΛV , d) verifies Poincare duality then
Hcat (ΛV , d) = mcat (ΛV , d)
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
A brief history of rational LS category
Theorem (Jessup, TAMS 1990)
Denote ΛSn the rational model for the sphere Sn then
mcat((ΛV , d)⊗ ΛSn) = mcat (ΛV , d) + 1
Theorem (Hess, Topology 1991)
cat (ΛV , d) = mcat (ΛV , d) and thus cat(X ) = mcat (ΛV , d)
Theorem (Felix, Halperin, Lemaire, Topology 1998)
If H (ΛV , d) verifies Poincare duality then
Hcat (ΛV , d) = mcat (ΛV , d)
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
A brief history of rational LS category
FH*(He+Je):
Corollary: the rational Ganea conjecture
cat(X × Sn) = cat(X ) + 1
FH*(He+FHL):
Corollary
If X is a Poincare duality complex, then
cat(X ) = Hcat (ΛV , d) .
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
A brief history of rational LS category
FH*(He+Je):
Corollary: the rational Ganea conjecture
cat(X × Sn) = cat(X ) + 1
FH*(He+FHL):
Corollary
If X is a Poincare duality complex, then
cat(X ) = Hcat (ΛV , d) .
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Sectional category
One can talk about sectional category (or Schwarz genus) of anymap f : X → Y .
Examples of sectional category
cat(X ) = secat(∗ ↪→ X ).
TC(X ) = secat(∆: X ↪→ X × X ).
TCn(X ) = secat(∆: X ↪→ X n).
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Sectional category
One can talk about sectional category (or Schwarz genus) of anymap f : X → Y .
Examples of sectional category
cat(X ) = secat(∗ ↪→ X ).
TC(X ) = secat(∆: X ↪→ X × X ).
TCn(X ) = secat(∆: X ↪→ X n).
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Sectional category
One can talk about sectional category (or Schwarz genus) of anymap f : X → Y .
Examples of sectional category
cat(X ) = secat(∗ ↪→ X ).
TC(X ) = secat(∆: X ↪→ X × X ).
TCn(X ) = secat(∆: X ↪→ X n).
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Rational sectional category
Definition
Let ϕ : A � B be a surjective cdga morphism and consider themorphism
ρm : A −→ A
(kerϕ)m+1.
Define:
Secat(ϕ) as the smallest m such that ρm admits a homotopyretraction as cdga.
mSecat(ϕ) as the smallest m such that ρm admits ahomotopy retraction as A-module.
HSecat(ϕ) the smallest m such that H(ρm) is injective.
HSecat(ϕ) ≤ mSecat(ϕ) ≤ Secat(ϕ)
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Rational sectional category
Definition
Let ϕ : A � B be a surjective cdga morphism and consider themorphism
ρm : A −→ A
(kerϕ)m+1.
Define:
Secat(ϕ) as the smallest m such that ρm admits a homotopyretraction as cdga.
mSecat(ϕ) as the smallest m such that ρm admits ahomotopy retraction as A-module.
HSecat(ϕ) the smallest m such that H(ρm) is injective.
HSecat(ϕ) ≤ mSecat(ϕ) ≤ Secat(ϕ)
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Rational sectional category
Definition
Let ϕ : A � B be a surjective cdga morphism and consider themorphism
ρm : A −→ A
(kerϕ)m+1.
Define:
Secat(ϕ) as the smallest m such that ρm admits a homotopyretraction as cdga.
mSecat(ϕ) as the smallest m such that ρm admits ahomotopy retraction as A-module.
HSecat(ϕ) the smallest m such that H(ρm) is injective.
HSecat(ϕ) ≤ mSecat(ϕ) ≤ Secat(ϕ)
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Rational sectional category
Definition
Let ϕ : A � B be a surjective cdga morphism and consider themorphism
ρm : A −→ A
(kerϕ)m+1.
Define:
Secat(ϕ) as the smallest m such that ρm admits a homotopyretraction as cdga.
mSecat(ϕ) as the smallest m such that ρm admits ahomotopy retraction as A-module.
HSecat(ϕ) the smallest m such that H(ρm) is injective.
HSecat(ϕ) ≤ mSecat(ϕ) ≤ Secat(ϕ)
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Rational Sectional Category
Example: LS category
Let (ΛV , d) be a Sullivan model for X , then ∗ ↪→ X is modelled bythe augmentation morphism ε : (ΛV , d)→ Q.
Then ker ε = Λ+V and (ker ε)m+1 = Λ>mV .
Since cat(X ) = secat(∗ ↪→ X ) we can rewrite
Theorem (Felix, Halperin)
If ΛV is a model for X , then cat(X ) = Secat(ε).
Or even
Theorem (Felix, Halperin)
If ΛV is a model for X , then secat(∗ ↪→ X ) = Secat(ε).
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Rational Sectional Category
Example: LS category
Let (ΛV , d) be a Sullivan model for X , then ∗ ↪→ X is modelled bythe augmentation morphism ε : (ΛV , d)→ Q.Then ker ε = Λ+V and (ker ε)m+1 = Λ>mV .
Since cat(X ) = secat(∗ ↪→ X ) we can rewrite
Theorem (Felix, Halperin)
If ΛV is a model for X , then cat(X ) = Secat(ε).
Or even
Theorem (Felix, Halperin)
If ΛV is a model for X , then secat(∗ ↪→ X ) = Secat(ε).
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Rational Sectional Category
Example: LS category
Let (ΛV , d) be a Sullivan model for X , then ∗ ↪→ X is modelled bythe augmentation morphism ε : (ΛV , d)→ Q.Then ker ε = Λ+V and (ker ε)m+1 = Λ>mV .
Since cat(X ) = secat(∗ ↪→ X ) we can rewrite
Theorem (Felix, Halperin)
If ΛV is a model for X , then cat(X ) = Secat(ε).
Or even
Theorem (Felix, Halperin)
If ΛV is a model for X , then secat(∗ ↪→ X ) = Secat(ε).
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Rational Sectional Category
Example: LS category
Let (ΛV , d) be a Sullivan model for X , then ∗ ↪→ X is modelled bythe augmentation morphism ε : (ΛV , d)→ Q.Then ker ε = Λ+V and (ker ε)m+1 = Λ>mV .
Since cat(X ) = secat(∗ ↪→ X ) we can rewrite
Theorem (Felix, Halperin)
If ΛV is a model for X , then cat(X ) = Secat(ε).
Or even
Theorem (Felix, Halperin)
If ΛV is a model for X , then secat(∗ ↪→ X ) = Secat(ε).
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
The main result
Theorem
Let f be a map modelled by a cdga morphism ϕ : A→ Badmitting a section which is a cofibration. Then
secat(f ) = Secat(ϕ).
Explicitly, secat(f ) is the smallest m such that
ρm : A→ A
(kerϕ)m+1
admits a homotopy retraction.
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
The main result
Theorem
Let f be a map modelled by a cdga morphism ϕ : A→ Badmitting a section which is a cofibration. Then
secat(f ) = Secat(ϕ).
Explicitly, secat(f ) is the smallest m such that
ρm : A→ A
(kerϕ)m+1
admits a homotopy retraction.
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Applications: topological complexity
The diagonal inclusion ∆2 : X → X × X is modelled bymultiplication morphism µ2 : ΛV ⊗ ΛV → ΛV .
Since inclusion inthe first factor ΛV � ΛV ⊗ ΛV is a cofibration, previous theoremapplied to ∆2 we get a proof of the Jessup-Murillo-Parentconjecture:
Theorem
Let X be a space, then TC(X ) is the smallest m for which themorphism
ρm : ΛV ⊗ ΛV −→ ΛV ⊗ ΛV
(ker µ)m+1
admits a homotopy retraction.
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Applications: topological complexity
The diagonal inclusion ∆2 : X → X × X is modelled bymultiplication morphism µ2 : ΛV ⊗ ΛV → ΛV . Since inclusion inthe first factor ΛV � ΛV ⊗ ΛV is a cofibration, previous theoremapplied to ∆2 we get a proof of the Jessup-Murillo-Parentconjecture:
Theorem
Let X be a space, then TC(X ) is the smallest m for which themorphism
ρm : ΛV ⊗ ΛV −→ ΛV ⊗ ΛV
(ker µ)m+1
admits a homotopy retraction.
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Applications: topological complexity
The diagonal inclusion ∆2 : X → X × X is modelled bymultiplication morphism µ2 : ΛV ⊗ ΛV → ΛV . Since inclusion inthe first factor ΛV � ΛV ⊗ ΛV is a cofibration, previous theoremapplied to ∆2 we get a proof of the Jessup-Murillo-Parentconjecture:
Theorem
Let X be a space, then TC(X ) is the smallest m for which themorphism
ρm : ΛV ⊗ ΛV −→ ΛV ⊗ ΛV
(ker µ)m+1
admits a homotopy retraction.
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Applications: higher topological complexity
Our main result applied to the n-diagonal inclusion ∆n : X → X n
gives
Theorem
Let X be a space, then TCn(X ) is the smallest m for which themorphism
ρm(ΛV )⊗n −→ (ΛV )⊗n
(kerµn)m+1
admits a homotopy retraction.
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Applications: Iwase-Sakai conjecture
Using our main theorem we give a characterisation of the relativecategory of a map f , relcat(f ), in the sense of Doeraene-ElHaouari. This should help solve
The Doeraene-El Haouari conjecture
If f admits a homotopy retraction then secat(f ) = relcat(f ).
Theorem (C, Garcıa-Calcines, Vandembroucq)
The Doeraene-El Haouari conjecture includes the Iwase-Sakaiconjecture.
In particular, we have an effective way of computing TCM ofrational spaces.
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Applications: Iwase-Sakai conjecture
Using our main theorem we give a characterisation of the relativecategory of a map f , relcat(f ), in the sense of Doeraene-ElHaouari. This should help solve
The Doeraene-El Haouari conjecture
If f admits a homotopy retraction then secat(f ) = relcat(f ).
Theorem (C, Garcıa-Calcines, Vandembroucq)
The Doeraene-El Haouari conjecture includes the Iwase-Sakaiconjecture.
In particular, we have an effective way of computing TCM ofrational spaces.
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Applications: Iwase-Sakai conjecture
Using our main theorem we give a characterisation of the relativecategory of a map f , relcat(f ), in the sense of Doeraene-ElHaouari. This should help solve
The Doeraene-El Haouari conjecture
If f admits a homotopy retraction then secat(f ) = relcat(f ).
Theorem (C, Garcıa-Calcines, Vandembroucq)
The Doeraene-El Haouari conjecture includes the Iwase-Sakaiconjecture.
In particular, we have an effective way of computing TCM ofrational spaces.
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Applications: Iwase-Sakai conjecture
Using our main theorem we give a characterisation of the relativecategory of a map f , relcat(f ), in the sense of Doeraene-ElHaouari. This should help solve
The Doeraene-El Haouari conjecture
If f admits a homotopy retraction then secat(f ) = relcat(f ).
Theorem (C, Garcıa-Calcines, Vandembroucq)
The Doeraene-El Haouari conjecture includes the Iwase-Sakaiconjecture.
In particular, we have an effective way of computing TCM ofrational spaces.
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
More applications
A Jessup’s theorem for TC:
Theorem (Jessup-Murillo-Parent+C)
mTC(X × Sn) = mTC(X ) + mTC(Sn)
A generalised Felix-Halperin-Lemaire
Theorem (C, Kahl, Vandembroucq)
If X is a Poincare duality complex and f : Y → X , thenmsecat(f ) = Hsecat(f ). In particular mTC(X ) = HTC(X ).
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
More applications
A Jessup’s theorem for TC:
Theorem (Jessup-Murillo-Parent+C)
mTC(X × Sn) = mTC(X ) + mTC(Sn)
A generalised Felix-Halperin-Lemaire
Theorem (C, Kahl, Vandembroucq)
If X is a Poincare duality complex and f : Y → X , thenmsecat(f ) = Hsecat(f ). In particular mTC(X ) = HTC(X ).
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
More applications
Conjecture (Hess’ theorem for TC)
TC(X)=mTC(X).
Or more generally,
Conjecture (Generalised Hess’ theorem)
If ϕ is as in our main theorem,
Secat(ϕ) = mSecat(ϕ).
Consequences:
The Ganea conjecture for TC and perhaps TCn.
If f has a base verifying Poincare dualty,secat(f ) = Hsecat(ϕ). In particular, TCn(X ) = HTCn(X ).
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
More applications
Conjecture (Hess’ theorem for TC)
TC(X)=mTC(X).
Or more generally,
Conjecture (Generalised Hess’ theorem)
If ϕ is as in our main theorem,
Secat(ϕ) = mSecat(ϕ).
Consequences:
The Ganea conjecture for TC and perhaps TCn.
If f has a base verifying Poincare dualty,secat(f ) = Hsecat(ϕ). In particular, TCn(X ) = HTCn(X ).
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
More applications
Conjecture (Hess’ theorem for TC)
TC(X)=mTC(X).
Or more generally,
Conjecture (Generalised Hess’ theorem)
If ϕ is as in our main theorem,
Secat(ϕ) = mSecat(ϕ).
Consequences:
The Ganea conjecture for TC and perhaps TCn.
If f has a base verifying Poincare dualty,secat(f ) = Hsecat(ϕ). In particular, TCn(X ) = HTCn(X ).
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
More applications
Conjecture (Hess’ theorem for TC)
TC(X)=mTC(X).
Or more generally,
Conjecture (Generalised Hess’ theorem)
If ϕ is as in our main theorem,
Secat(ϕ) = mSecat(ϕ).
Consequences:
The Ganea conjecture for TC and perhaps TCn.
If f has a base verifying Poincare dualty,secat(f ) = Hsecat(ϕ). In particular, TCn(X ) = HTCn(X ).
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Thanks a lot for your attention
Jose Gabriel Carrasquel Vera On the sectional category of certain maps
Good question!
Consider
A =
(Λ(a4, b3)
(a2), d
),
with db = a. We have that H(A) =< 1, [ab] > and theaugmentation ϕ : A→ Q models the inclusion ∗ ↪→ S7. We have(kerϕ)2 = ab then
H
(ρm : A→ A
(kerϕ)2
)is not injective. Then secat(ϕ) ≥ 2 butsecat(∗ ↪→ S7) = cat(S7) = 1.
Jose Gabriel Carrasquel Vera On the sectional category of certain maps