YET ANOTHER HOPF INVARIANT 5 1 JEAN-PAULDOERAENEANDMOHAMMEDELHAOUARI 0 2 g u A Abstract. TheclassicalHopfinvariantisdefinedforamapf:Sr →X. Here wedefine‘hcat’whichissomekindofHopfinvariantbuiltwithaconstruction 7 in Ganea’s style, valid for maps not only on spheres but more generally on 2 a ‘relative suspension’ f: ΣAW → X. We study the relation between this invariant and the sectional category and the relative category of a map. In ] particular,forιX:A→Xbeingthe‘restriction’offonA,wehaverelcatιX 6 T hcatf 6relcatιX+1andrelcatf 6hcatf. A Our aim here is to make clearer the link between the Lusternik-Schnirelmann . category (cat), more generally the ‘relative category’ (relcat), closely related to h t James’sectional category (secat), and the Hopf invariants. In order to do this, we a introduce a new integer, namely hcat, that combines the Iwaze’s version of Hopf m invariant [3], based on the difference up to homotopy between two maps defined [ for a given section of a Ganea fibration, and the framework of the sectional and 4 relativecategories,searchingforthe least integer suchthatthe Ganeafibrationhas v a section, possibly with additional conditions. To do this combination, we simply 2 define our invariant hcat, as the least integer such that the Ganea fibration has a 1 sectionσ with additionalconditionthat the correspondingtwo maps(f◦σ andω 7 n in this paper) are homotopic. 3 0 It appears that for f: Sr → X or even for f: ΣW → X, we obtain an integer . that can be either cat(X), or cat(X)+1. More generally, for any f: Σ W →X, 1 A 0 wehaverelcat(f◦θ)6hcat(f)6relcat(f◦θ)+1,whereθ: A→ΣAW isthe map 5 arising in the construction of ΣAW. 1 Insection2,westudythe influenceofhcatinahomotopypushout. Insection3, : v weintroducethe‘strong’versionofourinvariant,andweobtainanotherimportant Xi inequality: for any f: ΣAW → X, we have relcat(f) 6 hcat(f). In section 4, we give alternative equivalent conditions to get hcat. Applications and examples are r given. a 1. The Hopf category We work in the category of pointed topological spaces. All constructions are made up to homotopy. A ‘homotopy commutative diagram’ has to be understood in the sense of [4]. Recall the following construction: 2010 Mathematics Subject Classification. 55M30. Key words and phrases. Ganeafibration,sectionalcategory, Hopfinvariant. 1 2 JEAN-PAULDOERAENEANDMOHAMMEDELHAOUARI Definition 1. For any map ι : A → X, the Ganea construction of ι is the X X following sequence of homotopy commutative diagrams (i>0): >>A❊ ηi ⑥⑥⑥⑥ ❊❊❊❊ ιX ⑥ ❊ Fi❅⑥β❅⑥❅i⑥❅❅❅❅❅ α③i③+γ③i1③③③③❊③❊G<<"" i+1 gi gi+1 77//''X G i where the outside square is a homotopy pullback, the inside square is a homotopy pushoutandthemapgi+1 =(gi,ιX): Gi+1 →X isthewhiskermapinducedbythis homotopy pushout. The iteration starts with g0 =ιX: A→X. We set α0 =idA. For any i > 0, there is a whisker map θ = (id ,α ): A → F induced by i A i i the homotopy pullback. Thus we have the sequence of maps A θi //Fi ηi //A and θi is a homotopy section of ηi. Moreover we have γi ◦αi ≃ αi+1, thus also αi+1 ≃γi◦γi−1◦···◦γ0. We denote by γi,j: Gi →Gj the composite γj−1◦···◦γi+1◦γi (for i< j) and set γ =id . i,i Gi Of course, everything in the Ganea construction depends on ι . We sometimes X denote G by G (ι ) to avoid ambiguity. i i X Definition 2. Let ι : A→X be any map. X 1)Thesectionalcategory ofι istheleastintegernsuchthatthemapg : G (ι )→ X n n X X has a homotopy section, i.e. there exists a map σ: X → G (ι ) such that n X g ◦σ ≃id . n X 2)Therelativecategory ofι istheleastintegernsuchthatthemapg : G (ι )→ X n n X X has a homotopy section σ and σ◦ι ≃α . X n 3)The relative category of order k ofι is the leastintegern suchthatthe map X g : G (ι )→X has a homotopy section σ and σ◦g ≃γ . n n X k k,n Wedenotethesectionalcategorybysecat(ι ),therelativecategorybyrelcat(ι ), X X and the relative category of order k by relcat (ι ). If A = ∗, secat(ι ) = k X X relcat(ι ) and is denoted simply by cat(X); this is the ‘normalized’ version of X the Lusternik-Schnirelmann category. Clearly, secat(ιX) 6 relcat(ιX). We have also relcat(ιX) 6 relcat1(ιX), see Proposition 6 below. In the sequel, we will consider a given homotopy pushout: W η //A β θ (cid:15)(cid:15) (cid:15)(cid:15) A //Σ W A θ Inother words,the map θ is a mapsuchthat Pushcatθ ≤1 in the sense of[2]. We call this homotopy pushout a ‘relative suspension’ because in some sense, A plays the role of the point in the ordinary suspension. We also consider any map f: Σ W →X, and set ι =f ◦θ. A X YET ANOTHER HOPF INVARIANT 3 We don’t assume η ≃ β in general. This is true, however, if θ is a homotopy monomorphism,andin this casewe can ‘think’ ofι asthe ‘restriction’of f onA. X For n>1, consider the following homotopy commutative diagram: W β //A Ayyrrrηrrrr ✤✤ θ // Σ W vv♥θ♥♥♥♥♥♥♥ ❖❖❖❖❖θ❖❖''Σ W ✤ A A (†) (cid:15)(cid:15)✤ ωn (cid:15)(cid:15) αn−1 Fn−1(ιX) //Gn−1(ιX) f Azztttttt //G ((cid:15)(cid:15)ι )ww♦♦♦♦γ♦n♦−1 ◆◆◆g◆n◆−◆◆1//&&X(cid:15)(cid:15) αn n X gn where the mapW →Fn−1 is induced by the bottom outer homotopypullback and the mapω : Σ W →G is induced by the top inner homotopypushout. We have n A n f ≃g ◦ω by the ‘Whiskersmapsinside a cube’lemma (see [2], Lemma49). Also n n notice that αn ≃ωn◦θ ≃γn−1◦αn−1; so ωn ≃(αn,αn) is the whisker map of two copies of α induced by the homotopy pushout Σ W. Finally, for all k > 1, we n A can see that ω ≃γ ◦ω . n k,n k Definition3. TheHopfcategoryoff istheleastintegern>1suchthatg : G (ι )→ n n X X has a homotopy section σ: X →G (ι ) such that σ◦f ≃ω . n X n We denote this integer by hcat(f). Actually, speaking of ‘Hopf category of f’ is a misuse of language. We should speak of ‘Hopf category of the datas η, β and f’. Example 4. Let X =Σ W and f ≃id . Then, asmight be expected, hcat(f)= A X 1. Indeed, in this case, as g1 ◦ω1 ≃ f ≃ idX, ω1 is a homotopy section of g1. Moreover, ω1◦f ≃ω1◦idX ≃ω1, so hcat(f)=1. Example 5. Let X 6≃ ∗ and W = A∨A, β ≃ pr : A∨A → A and η ≃ pr : 1 2 A∨A → A the obvious maps. Then Σ W ≃ ∗ and we have no choice for f that A must be the null map f: ∗ → X. In this case the condition σ◦f ≃ ω is always n satisfied, so hcat(f)=secat(ι )=cat(X). X Notice that relcat is a particularcaseofhcat: WhenW =A, η ≃β ≃id , then A ιX ≃ f, ωn ≃ αn and hcat(f) = relcat(ιX). Also relcat1 is a particular case of hcat: When W = F0, then ΣAW ≃ G1, θ ≃ γ0 ≃ α1, and if, moreover, f ≃ g1, then ωn ≃γ1,n and hcat(f)=relcat1(ιX). Thefollowingpropositionshowsthattheseparticularcasesareinfactlowerand upper bounds for hcat(f). Proposition 6. Whatever can be f (and ι =f ◦θ), we have X secat(f)6relcat(ιX)6hcat(f)6relcat1(ιX)6relcat(ιX)+1. Proof. Consider the following homotopy commutative diagram (n>1): αn 00:: Gn A θ //ΣAW✈✈✈ω✈✈n gn ιX ■■■f■■..$$ X(cid:10)(cid:10) 4 JEAN-PAULDOERAENEANDMOHAMMEDELHAOUARI We see that if there is a map σ: X → G such that ω ≃ σ◦f then α ≃ σ◦ι n n n X and this proves the second inequality. Now consider the following homotopy commutative diagram (n>1): ωn 00==Gn ΣAW ω1 //G1③③③γ③1,n gn ❉❉g1 ❉❉"" (cid:10)(cid:10) f ..X We see that if there is a map σ: X →Gn such that γ1,n ≃σ◦g1 then ωn ≃ σ◦f and this proves the third inequality. The first inequality comes from secat(f) 6 secat(ι ) 6 relcat(ι ), the first of X X these two inequalities comes from [2], Proposition 29. Finally, the fourth inequality is proved in [1]. (cid:3) So hcat(f) establishes a ‘dichotomy’ between maps f: Σ W →X: A – Either hcat(f)=relcat(ι ) andwe haveaσ suchthat f◦σ ≃ω already X n for n=secat(ι ); X – either hcat(f)=relcat(ι )+1 andwe haveaσ suchthatf◦σ≃ω only X n for n>secat(ι ) X Our last example of the section shows that the inequalities of Proposition6 can be strict, and even that two may be strict at the same time: Example 7. Let X = ∗, A 6≃ ∗ and consider ι∗: A → ∗. We have Gi(ι∗) ≃ A ⊲⊳ ... ⊲⊳ A, the join of i+1 copies of A. For any k, γ ≃ id, so it cannot factorize k,k through ∗; but γk,k+1 is homotopic to the null map, so relcatk(ι∗) = k+1. Now consider f ≃ g1(ι∗): A ⊲⊳ A → ∗. As said before, in this case we have hcat(f) = relcat1(ιX). So we get secat(f)=01,startingfromthe outsidecube oftheabovediagramand φ0 =φ, we can build a homotopy diagram: W ❑❑❑❑❑❑%% vv♥♥♥♥♥//♥Fn−1✤✤(ιX) {{✈✈✈✈✈✈//A❂❂❂❂❂ιX(cid:30)(cid:30) Gn−1(ιX) ✤ //Gn(ιX) //X (cid:15)(cid:15) φn−1 (cid:15)(cid:15)✤ φn (cid:15)(cid:15)φ B ❑❑❑❑❑❑%% (cid:15)(cid:15) vv♥♥♥♥♥//♥Fn−1(κY) (cid:15)(cid:15) {{✈✈✈✈✈✈//BκY❂❂❂❂❂(cid:30)(cid:30) (cid:15)(cid:15)χ Gn−1(κY) //Gn(κY) gn // Y wherethedashedanddottedmapsareinducedbythehomotopypullbackFn−1(κY) and the homotopy pushout G (ι ) respectively. n X So we obtain a homotopy commutative diagram: W // A ~~⑦⑦⑦⑦⑦ zz✉✉✉✉✉ ❄❄❄❄ιX(cid:31)(cid:31) A // G (ι ) //X n X gn (cid:15)(cid:15) (cid:15)(cid:15) B B χ (cid:15)(cid:15) ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ (cid:15)(cid:15) zz✉✉✉✉✉ ❄❄❄❄(cid:31)(cid:31) (cid:15)(cid:15) B //G (κ ) //Y n Y gn Finally take the homotopy pushout inside the upper and lower lefter squares to get the homotopy commutative diagram: Σ W // G (ι ) //X A ωn n X ρ χ (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) B //G (κ ) //Y n Y and this gives the required splitting of the original square. (cid:3) Proposition 9. If the square ‡ is a homotopy pushout, then relcat(κ )6hcat(f). Y Asaparticularcase,whenB ≃∗,Y isthehomotopycofibreoff,andrelcat(κ )= Y cat(Y). So the Proposition asserts that hcat(f)>cat(Y). 6 JEAN-PAULDOERAENEANDMOHAMMEDELHAOUARI Proof. Let hcat(f) 6 n, so we have a homotopy section σ of g (ι ) such that n X σ◦f ≃ω . First apply the ‘Whisker maps inside a cube’ lemma to the outer part n of the following homotopy commutative diagram: Σ W //B ωxxrrnrrr A a(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ■■■■α■n$$ G (ι ) // S //G (κ ) n X n Y b c X(cid:15)(cid:15) xxqqqqqqqΣAW // Y(cid:15)(cid:15) (cid:127)(cid:127)⑦⑦⑦⑦// B ❏❏❏❏❏❏$$Y(cid:15)(cid:15)gn where the inner horizontal squares are homotopy pushouts, and c and b are the whisker maps induced by the homotopy pushout S. Next build the following homotopy commutative diagram: Σ W //B A X xxqqqfqqqq //YκY(cid:127)(cid:127)⑦⑦⑦⑦ d σ (cid:15)(cid:15) ωxxrrnrrrΣAW (cid:15)(cid:15) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)a(cid:0)// B ■■■■α■n$$ G (ι ) // S // G (κ ) n X n Y b where d is the whisker map induced by the homotopy pushout Y. Let σ′ = b◦d. We haveg ◦σ′ ≃g ◦b◦d≃c◦d≃id and σ′◦κ ≃b◦d◦κ ≃b◦a≃α . (cid:3) n n C Y Y n Corollary 10. In the diagram ‡, if relcat(κ ) = relcat(ι )+1, then hcat(f) = Y X relcat(ι )+1. X Proof. ByProposition9,thehypothesisimpliesthathcat(f)>relcat(ι )+1. But X by Proposition 6, we have hcat(f)6relcat(ι )+1. So we have the equality. (cid:3) X It is now easy to exhibit examples of maps f with hcat(f) = relcat(ι ) + X 1. Indeed there are plenty examples of homotopy pushouts where relcat(κ ) = Y relcat(ι )+1: X Example 11. Let A=B =∗ andf: Sr →Sn be any ofthe Hopf maps S3 →S2, S7 →S4 or S15 →S8. So here relcat(ι )=cat(Sn)=1. On the other hand it is X well known that those maps have a homotopy cofibre Sn/Sr of category 2, so here relcat(κ )=cat(Sn/Sr)=2. By Corollary 10, we have hcat(f)=2. Y Example 12. Let f be the map u in the homotopy cofibration Z ⊲⊳Z //ΣZ∨ΣZ // ΣZ×ΣZ u t1 where Z ⊲⊳Z ≃Σ(Z ∧Z) is the join of two copies of Z and is also the suspension of the smash product of two copies of Z. Let A = B = ∗, ΣZ 6≃ ∗. We have relcat(ι ) = cat(ΣZ ∨ ΣZ) = 1 and relcat(κ ) = cat(ΣZ × ΣZ) = 2, so by X Y Corollary 10 again, we have hcat(u)=2. YET ANOTHER HOPF INVARIANT 7 Example 13. For i>1, let f be the map β in the Ganea construction: i A θi // F ηi // A ❅ i ❅ ❅ ❅ αi❅❅❅❅(cid:31)(cid:31) (cid:15)(cid:15)βi (cid:15)(cid:15)αi+1 Gi γi // Gi+1 Actually Fi is a join over A of i+1 copies of F0, and also a relative suspension Σ W where W is a relative smash product. For any i 6 relcat(ι ), we have A X relcat(α )=i, see [2], Proposition 23. So by Corollary 10 again, if i relcat(f). Consequently, if relcat(f)>relcat(ι ), then hcat(f)=relcat(ι )+1. X X Definition 14. The strong Hopf category of a map f: Σ W → X is the least A integer n>1 such that: – there are maps ι0: A → X0 and a homotopy inverse λ: X0 → A, i.e. ι0◦λ≃idX0 and λ◦ι0 ≃idA; – for each i, 06ihcat(f)=1. Proposition 17. In the diagram ♮, we have Relcat(ζ )6i i As ω is a particular case of ζ , this implies Relcat(ω )6i. i i i Proof. Fori>0,letbuildthefollowinghomotopydiagramwherethethreesquares are homotopy pushouts: η W //Zi−1 ((//A β zi−1 θ (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) A //Ci−1 //ΣAW ιi ci−1 ζi ιi−1 '' (cid:15)(cid:15) (cid:3)(cid:3) (cid:15)(cid:15) (cid:3)(cid:3) Xi−1 //Xi andwherethemapci−1 =(ιi−1,zi−1)isthewhiskermapinducedbythehomotopy pushout. Wehavesecat(ιi−1)6Pushcat(ιi−1)6i−1by[2],Theorem18. Sosecat(ci−1)6 i−1 by [2], Proposition 29. So Relcat(ci−1)6(i−1)+1=i by [2], Theorem 18. And this implies Relcat(ζ )6i by [2], Lemma 11. (cid:3) i Theorem 18. For any f: Σ W →X, we have A Relcat(f)6Hcat(f) and relcat(f)6hcat(f) Proof. If Hcat(f) = n, then we have f ≃ ζ in ♮. So Relcat(f) = Relcat(ζ ) 6 n n n by Proposition 17. If hcat(f) = n, then f is relatively dominated by ω . As Relcat(ω ) 6 n, we n n have relcat(f)6n by [2], Proposition 10. (cid:3) As a corollary, we get an indirect proof of Proposition 9 because relcat(κ ) 6 Y relcat(f)by[2],Lemma 11,thatassertsthata homotopypushoutdoesn’tincrease the relative category. It is not difficult to find an example where these inequalities are strict: Example 19. Let f be the map t1 in the homotopy cofibration Z ⊲⊳Z // ΣZ∨ΣZ // ΣZ×ΣZ u t1 LetA=∗,ΣZ 6≃∗. Ast1 isahomotopycofibre,wehaverelcat(t1)6Relcat(t1)6 1, see [2], Proposition 9. On the other hand, we have Hcat(t1) > hcat(t1) > relcat(ι )=cat(ΣZ×ΣZ)=2 by Proposition 6. X 10 JEAN-PAULDOERAENEANDMOHAMMEDELHAOUARI 4. Equivalent conditions to get the Hopf category Let be given any map f: Σ W → X with secat(ι ) 6 n and any homotopy A X section σ: X →G of g : G →X. Consider the following homotopy pullbacks: n n n Q π //Σ W π′ (cid:15)(cid:15) θnWA(cid:15)(cid:15) ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ Σ W //H // Σ W A σ¯ n ηW A n f fn f (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) X // G //X σ n gn where θnW = (ωn,idΣAX) is the whisker map induced by the homotopy pullback H . By the ‘Prism lemma’ (see [2], Lemma 46, for instance ), we know that the n hnoomticoetotphyatpπul≃lbaπc′ksionfcσe πan≃dηfWn i◦sθiWnd◦eeπd≃ΣAηWW,◦aσ¯n◦dπt′h≃atπη′nW. ◦σ¯ ≃ idΣAW. Also n n n Proposition20. Letbegiven anymap f: Σ W →X with secat(ι )6n andany A X homotopy section σ: X →G (ι ) of g : G (ι )→X. With the same definitions n X n n X and notations as above, the following conditions are equivalent: (i) σ◦f ≃ω . n (ii) π has a homotopy section. (iii) π is a homotopy epimorphism. (iv) θW ≃σ¯. n Proof. We have the following sequence of implications: (i) =⇒ (ii): Since σ ◦f ≃ ωn ≃ fn ◦θnW ◦idΣAW, we have a whisker map (f,idΣAW): ΣAW →Q induced by the homotopy pullback Q which is a homotopy section of π. (ii) =⇒ (iii): Obvious. (iii) =⇒ (iv): We have θW ◦π ≃σ¯◦π since π ≃π′. Thus θW ≃σ¯ since π is a n n homotopy epimorphism. (iv) =⇒ (i): We have σ◦f ≃f ◦σ¯ ≃f ◦θW ≃ω . (cid:3) n n n n Theorem 21. Let be a (q−1)-connected map ι : A → X with secatι 6 n. If X X Σ W is a CW-complex with dimΣ W < (n+1)q −1 then σ ◦f ≃ ω for any A A n homotopy section σ of g . n Proof. Recall that g is the (i+1)-fold join of ι . Thus by [4], Theorem 47, we i X obtain that, for each i > 0, g : G → X is (i+1)q−1-connected. As g and ηW i i i i have the same homotopy fibre, the Five lemma implies that ηW: H → Σ W is i i A (i+1)q −1-connected, too. By [5], Theorem IV.7.16, this means that for every CW-complexK withdimK <(i+1)q−1,ηW inducesaone-to-onecorrespondence i [K,H ] → [K,Σ W]. Apply this to K = Σ W and i = n: Since θW and σ¯ are i A A n both homotopy sections of ηW, we obtain θW ≃σ¯, and Proposition 20 implies the n n desired result. (cid:3) Example 22. Let A = ∗ and W = Sr−1, so Σ W = Sr, and X = Sm. In this A case secatι = catSm = 1. Hence Theorem 21 means that if r < 2m−1, we X have σ◦f ≃ω1, whatever can be f and σ: X →G1(ιX), so hcatf =1 and we get

Yet another Hopf Invariant

Similar Yet another Hopf Invariant