Jos e Gabriel Carrasquel Vera - cmup.fc.up.pt

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.


Rational homotopy






Here, cat(∗) = 0.


Rational homotopy






Here, cat(∗) = 0.


Rational homotopy






Here, cat(∗) = 0.


Rational homotopy






Here, cat(∗) = 0.


Rational homotopy






Here, cat(∗) = 0.


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)



Definition


ρm : ΛV → ΛV

Λ>mV.

Define:







Definition


ρm : ΛV → ΛV

Λ>mV.

Define:







Definition


ρm : ΛV → ΛV

Λ>mV.

Define:







Definition


ρm : ΛV → ΛV

Λ>mV.

Define:







Theorem (Felix, Halperin, TAMS 1982)

If (ΛV , d) is a Sullivan model for X , then

cat(X ) = cat (ΛV , d) .



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)























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) .



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) .


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).


Sectional category




TC(X ) = secat(∆: X ↪→ X × X ).



Sectional category




TC(X ) = secat(∆: X ↪→ X × X ).



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)



Definition


ρm : A −→ A

(kerϕ)m+1.

Define:







Definition


ρm : A −→ A

(kerϕ)m+1.

Define:







Definition


ρm : A −→ A

(kerϕ)m+1.

Define:






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


If ΛV is a model for X , then secat(∗ ↪→ X ) = Secat(ε).




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 .




Or even










Or even










Or even




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.


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



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




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



(ker µ)m+1




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



(ker µ)m+1



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



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.


























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 ).


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 ).


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 ).


More applications


TC(X)=mTC(X).

Or more generally,




Consequences:




More applications


TC(X)=mTC(X).

Or more generally,




Consequences:




More applications


TC(X)=mTC(X).

Or more generally,




Consequences:




Thanks a lot for your attention


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.


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Jos e Gabriel Carrasquel Vera - cmup.fc.up.pt