On the sectional category of certain maps

Transcription

1 Université catholique de Louvain, Belgique XXIst Oporto Meeting on Geometry, Topology and Physics Lisboa, 6 February 2015

2 Rational homotopy All spaces considered are rational simply connected spaces of finite type.

3 Rational homotopy All spaces considered are rational simply connected spaces of finite type. These spaces form a category whose homotopy category is equivalent to that of cdga.

4 Rational homotopy All spaces considered are rational simply connected spaces of finite type. These spaces form a category whose homotopy category is equivalent to that of cdga. cdga= simply connected commutative differential graded Q-algebras of finite type.

5 Rational homotopy All spaces considered are rational simply connected spaces of finite type. These spaces form a category whose homotopy category is equivalent to that of cdga. cdga= simply connected commutative differential graded Q-algebras of finite type. The cofibrant objects of cdga are the Sullivan algebras (ΛV, d).

6 Rational homotopy All spaces considered are rational simply connected spaces of finite type. These spaces form a category whose homotopy category is equivalent to that of cdga. cdga= simply connected commutative differential graded Q-algebras of finite type. The cofibrant objects of cdga are the Sullivan algebras (ΛV, d). The cofibrations are relative Sullivan inclusions A (A ΛV, D).

7 Rational homotopy All spaces considered are rational simply connected spaces of finite type. These spaces form a category whose homotopy category is equivalent to that of cdga. cdga= simply connected commutative differential graded Q-algebras of finite type. The cofibrant objects of cdga are the Sullivan algebras (ΛV, d). The cofibrations are relative Sullivan inclusions A (A ΛV, D). Here, cat( ) = 0.

8 A brief history of rational LS category Definition Let (ΛV, d) be a Sullivan algebra and consider the cdga morphism ρ m : ΛV ΛV Λ >m V.

9 A brief history of rational LS category Definition Let (ΛV, d) be a Sullivan algebra and consider the cdga morphism Define: ρ m : ΛV ΛV Λ >m V. cat (ΛV, d) as the smallest m such that ρ m admits a homotopy retraction as cdga.

10 A brief history of rational LS category Definition Let (ΛV, d) be a Sullivan algebra and consider the cdga morphism Define: ρ m : ΛV ΛV Λ >m V. cat (ΛV, d) as the smallest m such that ρ m admits a homotopy retraction as cdga. mcat (ΛV, d) the smallest m such that ρ m admits a homotopy retraction as (ΛV, d)-module.

11 A brief history of rational LS category Definition Let (ΛV, d) be a Sullivan algebra and consider the cdga morphism Define: ρ m : ΛV ΛV Λ >m V. cat (ΛV, d) as the smallest m such that ρ m admits a homotopy retraction as cdga. mcat (ΛV, d) the smallest m such that ρ m admits a homotopy retraction as (ΛV, d)-module. Hcat (ΛV, d) the smallest m such that H(ρ m ) is injective.

12 A brief history of rational LS category Definition Let (ΛV, d) be a Sullivan algebra and consider the cdga morphism Define: ρ m : ΛV ΛV Λ >m V. cat (ΛV, d) as the smallest m such that ρ m admits a homotopy retraction as cdga. mcat (ΛV, d) the smallest m such that ρ m admits a homotopy retraction as (ΛV, d)-module. Hcat (ΛV, d) the smallest m such that H(ρ m ) is injective. Hcat (ΛV, d) mcat (ΛV, d) cat (ΛV, d)

13 A brief history of rational LS category Theorem (Félix, Halperin, TAMS 1982) If (ΛV, d) is a Sullivan model for X, then cat(x ) = cat (ΛV, d).

14 A brief history of rational LS category Theorem (Jessup, TAMS 1990) Denote ΛS n the rational model for the sphere S n then mcat((λv, d) ΛS n ) = mcat (ΛV, d) + 1

15 A brief history of rational LS category Theorem (Jessup, TAMS 1990) Denote ΛS n the rational model for the sphere S n then mcat((λv, d) ΛS n ) = mcat (ΛV, d) + 1 Theorem (Hess, Topology 1991) cat (ΛV, d) = mcat (ΛV, d) and thus cat(x ) = mcat (ΛV, d)

16 A brief history of rational LS category Theorem (Jessup, TAMS 1990) Denote ΛS n the rational model for the sphere S n then mcat((λv, d) ΛS n ) = mcat (ΛV, d) + 1 Theorem (Hess, Topology 1991) cat (ΛV, d) = mcat (ΛV, d) and thus cat(x ) = mcat (ΛV, d) Theorem (Félix, Halperin, Lemaire, Topology 1998) If H (ΛV, d) verifies Poincaré duality then Hcat (ΛV, d) = mcat (ΛV, d)

17 A brief history of rational LS category FH*(He+Je): Corollary: the rational Ganea conjecture cat(x S n ) = cat(x ) + 1

18 A brief history of rational LS category FH*(He+Je): Corollary: the rational Ganea conjecture cat(x S n ) = cat(x ) + 1 FH*(He+FHL): Corollary If X is a Poincaré duality complex, then cat(x ) = Hcat (ΛV, d).

19 Sectional category One can talk about sectional category (or Schwarz genus) of any map f : X Y. Examples of sectional category cat(x ) = secat( X ).

20 Sectional category One can talk about sectional category (or Schwarz genus) of any map f : X Y. Examples of sectional category cat(x ) = secat( X ). TC(X ) = secat( : X X X ).

21 Sectional category One can talk about sectional category (or Schwarz genus) of any map f : X Y. Examples of sectional category cat(x ) = secat( X ). TC(X ) = secat( : X X X ). TC n (X ) = secat( : X X n ).

22 Rational sectional category Definition Let ϕ: A B be a surjective cdga morphism and consider the morphism A ρ m : A (ker ϕ) m+1. Define: Secat(ϕ) as the smallest m such that ρ m admits a homotopy retraction as cdga.

23 Rational sectional category Definition Let ϕ: A B be a surjective cdga morphism and consider the morphism A ρ m : A (ker ϕ) m+1. Define: Secat(ϕ) as the smallest m such that ρ m admits a homotopy retraction as cdga. msecat(ϕ) as the smallest m such that ρ m admits a homotopy retraction as A-module.

24 Rational sectional category Definition Let ϕ: A B be a surjective cdga morphism and consider the morphism A ρ m : A (ker ϕ) m+1. Define: Secat(ϕ) as the smallest m such that ρ m admits a homotopy retraction as cdga. msecat(ϕ) as the smallest m such that ρ m admits a homotopy retraction as A-module. HSecat(ϕ) the smallest m such that H(ρ m ) is injective.

25 Rational sectional category Definition Let ϕ: A B be a surjective cdga morphism and consider the morphism A ρ m : A (ker ϕ) m+1. Define: Secat(ϕ) as the smallest m such that ρ m admits a homotopy retraction as cdga. msecat(ϕ) as the smallest m such that ρ m admits a homotopy retraction as A-module. HSecat(ϕ) the smallest m such that H(ρ m ) is injective. HSecat(ϕ) msecat(ϕ) Secat(ϕ)

26 Rational Sectional Category Example: LS category Let (ΛV, d) be a Sullivan model for X, then X is modelled by the augmentation morphism ɛ: (ΛV, d) Q.

27 Rational Sectional Category Example: LS category Let (ΛV, d) be a Sullivan model for X, then X is modelled by the augmentation morphism ɛ: (ΛV, d) Q. Then ker ɛ = Λ + V and (ker ɛ) m+1 = Λ >m V.

28 Rational Sectional Category Example: LS category Let (ΛV, d) be a Sullivan model for X, then X is modelled by the augmentation morphism ɛ: (ΛV, d) Q. Then ker ɛ = Λ + V and (ker ɛ) m+1 = Λ >m V. Since cat(x ) = secat( X ) we can rewrite Theorem (Félix, Halperin) If ΛV is a model for X, then cat(x ) = Secat(ɛ).

29 Rational Sectional Category Example: LS category Let (ΛV, d) be a Sullivan model for X, then X is modelled by the augmentation morphism ɛ: (ΛV, d) Q. Then ker ɛ = Λ + V and (ker ɛ) m+1 = Λ >m V. Since cat(x ) = secat( X ) we can rewrite Theorem (Félix, Halperin) If ΛV is a model for X, then cat(x ) = Secat(ɛ). Or even Theorem (Félix, Halperin) If ΛV is a model for X, then secat( X ) = Secat(ɛ).

30 The main result Theorem Let f be a map modelled by a cdga morphism ϕ: A B admitting a section which is a cofibration. Then secat(f ) = Secat(ϕ).

31 The main result Theorem Let f be a map modelled by a cdga morphism ϕ: A B admitting a section which is a cofibration. Then secat(f ) = Secat(ϕ). Explicitly, secat(f ) is the smallest m such that ρ m : A admits a homotopy retraction. A (ker ϕ) m+1

32 Applications: topological complexity The diagonal inclusion 2 : X X X is modelled by multiplication morphism µ 2 : ΛV ΛV ΛV.

33 Applications: topological complexity The diagonal inclusion 2 : X X X is modelled by multiplication morphism µ 2 : ΛV ΛV ΛV. Since inclusion in the first factor ΛV ΛV ΛV is a cofibration, previous theorem applied to 2 we get a proof of the Jessup-Murillo-Parent conjecture:

34 Applications: topological complexity The diagonal inclusion 2 : X X X is modelled by multiplication morphism µ 2 : ΛV ΛV ΛV. Since inclusion in the first factor ΛV ΛV ΛV is a cofibration, previous theorem applied to 2 we get a proof of the Jessup-Murillo-Parent conjecture: Theorem Let X be a space, then TC(X ) is the smallest m for which the morphism ΛV ΛV ρ m : ΛV ΛV (ker µ) m+1 admits a homotopy retraction.

35 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 TC n (X ) is the smallest m for which the morphism ρ m (ΛV ) n (ΛV ) n (ker µ n ) m+1 admits a homotopy retraction.

36 Applications: Iwase-Sakai conjecture Using our main theorem we give a characterisation of the relative category of a map f, relcat(f ), in the sense of Doeraene-El Haouari. This should help solve

37 Applications: Iwase-Sakai conjecture Using our main theorem we give a characterisation of the relative category of a map f, relcat(f ), in the sense of Doeraene-El Haouari. This should help solve The Doeraene-El Haouari conjecture If f admits a homotopy retraction then secat(f ) = relcat(f ).

38 Applications: Iwase-Sakai conjecture Using our main theorem we give a characterisation of the relative category of a map f, relcat(f ), in the sense of Doeraene-El Haouari. This should help solve The Doeraene-El Haouari conjecture If f admits a homotopy retraction then secat(f ) = relcat(f ). Theorem (C, García-Calcines, Vandembroucq) The Doeraene-El Haouari conjecture includes the Iwase-Sakai conjecture.

39 Applications: Iwase-Sakai conjecture Using our main theorem we give a characterisation of the relative category of a map f, relcat(f ), in the sense of Doeraene-El Haouari. This should help solve The Doeraene-El Haouari conjecture If f admits a homotopy retraction then secat(f ) = relcat(f ). Theorem (C, García-Calcines, Vandembroucq) The Doeraene-El Haouari conjecture includes the Iwase-Sakai conjecture. In particular, we have an effective way of computing TC M of rational spaces.

40 More applications A Jessup s theorem for TC: Theorem (Jessup-Murillo-Parent+C) mtc(x S n ) = mtc(x ) + mtc(s n )

41 More applications A Jessup s theorem for TC: Theorem (Jessup-Murillo-Parent+C) mtc(x S n ) = mtc(x ) + mtc(s n ) A generalised Félix-Halperin-Lemaire Theorem (C, Kahl, Vandembroucq) If X is a Poincaré duality complex and f : Y X, then msecat(f ) = Hsecat(f ). In particular mtc(x ) = HTC(X ).

42 More applications Conjecture (Hess theorem for TC) TC(X)=mTC(X).

43 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).

44 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 TC n.

45 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, Consequences: Secat(ϕ) = msecat(ϕ). The Ganea conjecture for TC and perhaps TC n. If f has a base verifying Poincaré dualty, secat(f ) = Hsecat(ϕ). In particular, TC n (X ) = HTC n (X ).

46 Thanks a lot for your attention

47 Good question! Consider ( ) Λ(a4, b 3 ) A = (a 2, d, ) with db = a. We have that H(A) =< 1, [ab] > and the augmentation ϕ: A Q models the inclusion S 7. We have (ker ϕ) 2 = ab then H ( ρ m : A is not injective. Then secat(ϕ) 2 but secat( S 7 ) = cat(s 7 ) = 1. A ) (ker ϕ) 2