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A NOTE ON THE HILALI CONJECTURE MANUEL AMANN Abstract. In this short note we observe that the Hilali conjecture holds for 2-stage spaces, i.e. we argue that the dimension of the rational 5 cohomologyisatleastaslargeasthedimensionoftherationalhomotopy 1 groups for these spaces. We also prove the Hilali conjecture for a class 0 of spaces which puts it into the context of fibrations. 2 n a Introduction J 3 There are a lot of prominent conjectures in Rational Homotopy Theory 1 centering around the question of how large the cohomology algebra of a space has to be under certain conditions. The toral rank conjecture poses ] T this question in the context of (almost) free torus actions, the Halperin A conjecture—interpreted in the right way—asks this for the cohomology of . the total space of certain fibrations. A conjecture by Hilali poses such a h t question based upon the size of the rational homotopy groups of a space (see a m [4]). It is formulated for rationally elliptic spaces, i.e. for simply connected spaces X satisfying dimπ (X)⊗Q < ∞ and dimH∗(X;Q) < ∞. [ ∗ 1 Conjecture (Hilali). Let X be a simply-connected rationally elliptic space. v Then it holds that 5 7 dimH∗(X;Q) ≥ dimπ (X)⊗Q ∗ 9 2 We remark that in terms of minimal Sullivan models (ΛW,d) this conjec- 0 ture can be stated equivalently as . 1 0 dimH(ΛW,d) ≥ dimW 5 1 In this article we shall essentially use this transition. : v The Hilali conjecture was confirmed in several cases. It is known if X is a i nilmanifold, if dimX ≤ 16, if X is a formal or a coformal space, or if it is a X symplectic or cosymplectic manifold (see [10], [7], [5], [6], [8]), respectively r a for hyperelliptic spaces (see [3]). In this note we consider the two-stage case as well as a class of spaces which involves fibration constructions. This is the class (I) of elliptic spaces, characterised by the subsequent properties of the minimal Sullivan model (ΛW,d). For this we draw upon Date: January 12th, 2015. 2010 Mathematics Subject Classification. 55Q52 (Primary), 55P62 (Secondary). Key words and phrases. Hilali conjecture, two-stage space, rational cohomology groups, rational homotopy groups. 2 MANUELAMANN (cid:76) the n-stage decomposition of (ΛW,d) = Λ( W ,d) with d(W )⊆ΛW 0≤i i i <i for all i. Besides, recall the definition of the Wang derivation θ by wi w ·θ = d−d¯ i wi where we define d¯ = d| as the usual differential followed by evaluation wi wi=0 of w = 0. i Let (ΛV,d(cid:48)) be an elliptic minimal Sullivan algebra which satisfies the Hilali conjecture and admits a n -stage decomposition for some n ∈ N . Let 2 2 0 W = V for 0 ≤ i ≤ n . Let W ⊕...⊕W ⊕W ⊕...⊕W be n1+1+i i 2 0 n1 n1+n2+1 n3 concentratedinodd(ordinary)degree. TheclassI isdefinedviatheexistence of a homogeneous basis (w ) —homogeneous with respect to ordinary degree i i deg and lower degree lowdeg—for W satisfying that d| = d(cid:48) ΛWn1,...,n2 and one of the following two conditions. (1) - (dw )2 = 0 for lowdegw > n +n and i i 1 2 - θ2 (w ) = 0forlowdegw ≤ n andlowdegw ≤ wi j i 1 j (I) n1+n2, or (2) - (dw )2| = 0 for lowdegw > n +n and i Λ(W>n1) i 1 2 - θ2 = 0 for lowdegw ≤ n . wi i 1 where | denotes the projection w (cid:55)→ 0 for i ≤ n . Λ(W>n1) i 1 The second class of spaces we consider are the so-called 2-stage algebras, which are characterised by (1) d(U) = 0 (S ) 2 (2) d(V)⊆ΛU (with a priori no further restrictions on the parity of the degree.) Note that neither all two-stage spaces are hyperelliptic nor that all hyperelliptic spaces are two-stage (see Section 2 of the Appendix). However, both classes contain pure spaces and the proofs in both cases are similar. Theorem A. The Hilali conjecture holds for the classes (I) and (S ). 2 Corollary. Let (ΛW,d) be a Sullivan algebra with W concentrated in odd degree. Suppose W admits a basis w ,...,w satisfying that dw is a monomial 1 n i in the w for all 1 ≤ i ≤ n. Then (ΛW,d) satisfies the Hilali conjecture. j Remark. We note that [3, Estimate (9), p. 32] also proves the toral rank conjecture for hyperelliptic spaces. In fact, on rationally elliptic spaces X it holds that rk (X) ≤ −χ (X) = dimWodd −dimWeven where (ΛW,d) 0 π is a minimal Sullivan model for X (see [2, Theorem 7.13, p. 279]). In the hyperelliptic case the estimate says that dimH(ΛW,d) ≥ 2−χπ(X). (cid:30) Remark. We observe that [3, Estimate (9), p. 32] can also be obtained by the following elementary arguments. In the terminology of [3, Section 4.2, 32] one may compare the dimension of the E -term of the Leray–Serre 2 spectral sequence associated to the rational fibration (ΛV,d) (cid:44)→ (ΛW,d) = (Λ(V ⊕(cid:104)y¯ (cid:105)),d) → (Λ(cid:104)y¯ (cid:105),0) 1 1 with dy¯ = x ∈ V, namely 2dimH(ΛV,d), to the dimension of its infinity 1 1 term, which is dimH(ΛW,d). Proceeding inductively directly yields the inequality. (cid:30) A NOTE ON THE HILALI CONJECTURE 3 1. The First Class of Spaces As usual we write W for W ⊕W ⊕...⊕W etc. ≤n1+n2 0 1 n1+n2 The main characteristic of the class I is indeed that it allows a decomposition into rational fibrations. In the first case we obtain the fibrations (ΛW ,d) (cid:44)→ (ΛW,d) → (ΛW ,d¯) ≤n1+n2 >n2 and (ΛW ,d) (cid:44)→ (ΛW ,d) → (ΛW ,d¯) ≤n1 ≤n1+n2 n1+1,...,n1+n2 and in the second one the fibrations (ΛW ,d) (cid:44)→ (ΛW,d) → (ΛW ,d¯) ≤n1 >n1 and (ΛW ,d) (cid:44)→ (ΛW ,d) → (ΛW ,d¯) n1+1,...,n1+n2 ≥n1 >n2 That is, in any case the space results from two fibrations involving the space (ΛV,d(cid:48)). In the first case this space will first play the role of the fibre in a fibration, before the resulting totla space will serve as a base space in a second fibration. Inthesecondcase(ΛV,d(cid:48))willplaytheroleofthebaseinthefirstfibration, before the new total space will act as a fibre in a second fibration. Proof of Theorem A–Second assertion. Let (ΛW,d) be an algebra of type I. We shall only consider the first scenario depicted above, the second one following from completely analogous arguments. We consider the fibration (ΛW ,d) (cid:44)→ (ΛW ,d) → (ΛW ,d¯) ≤n1 ≤n1+n2 n1+1,...,n1+n2 and observe that we may replace it by a sequence of fibrations as follows. Let w ,...,w be the basis elements from above spanning W in increasing 1 k ≤n1 lower degree. We obtain a fibration (Λ(cid:104)w (cid:105),d) (cid:44)→ (ΛW ,d) → (Λ(W ⊕(cid:104)w ,...,w (cid:105)),d¯) 1 ≤n1+n2 n1+1,...,n1+n2 2 k and consider its associated Wang sequence. From [9, Remark 2.2, p. 195] we recall that the connecting homomorphism of the sequence corresponds to the Wang derivation from above. More precisely, we have the following long exact sequence. ... →− Hi(ΛW ,d) → Hi(Λ(W ⊕(cid:104)w ,...,w (cid:105)),d¯) ≤n1+n2 n1+1,...,n1+n2 2 k θ∗ −−w→1 Hi−degw1+1(Λ(W ⊕(cid:104)w ,...,w (cid:105)),d¯) n1+1,...,n1+n2 2 k → Hi+1(ΛW ,d) → ... ≤n1+n2 where it is easy to see that θ induces θ∗ on cohomology (see [9, p. 194]). By w1 assumption also the morphism θ∗ satisfies (θ∗ )2 = 0. We have that w1 H(Λ(W ⊕(cid:104)w ,...,w (cid:105)),d¯) ∼= kerθ∗ ⊕imθ∗ n1+1,...,n1+n2 2 k w1 w1 The Wang sequence splits as (1) 0 → (cokerθ∗ )i−degw1 → Hi(ΛW ,d) → (kerθ∗ )i → 0 w1 ≤n1+n2 w1 4 MANUELAMANN Consequently, every element in kerθ∗ contributes to the cohomology of w1 H(ΛW ,d). Choose a complement to kerθ∗ isomorphic to imθ∗ . ≤n1+n2 w1 w1 Since (θ∗ )2 = 0 every such element also contributes to the cohomology via w1 the kernel, namely in its isomorphic form from imθ∗ , on which θ∗ vanishes. w1 w1 This shows that H(ΛW ,d) ≤n1+n2 =dimH(Λ(W ⊕(cid:104)w ,...,w (cid:105)),d) n1+1,...,n1+n2 1 k ≥dimH(Λ(W ⊕(cid:104)w ,...,w (cid:105)),d¯) n1,...,n1+n2 2 k We need to argue that strict inequality holds. For this we concentrate on the contribution by the cokernel in (4). Indeed, strict equality holds unless cokerθ∗ = 0 and θ∗ is surjective. Since degw (cid:54)= 0, the morphism w1 w1 1 θ∗ strictly reduces the degree. As we are dealing with rationally elliptic w1 spaces, which have finite-dimensional cohomology, in particular, it follows that surjectivity can never hold. This proves that dimH(Λ(W ⊕(cid:104)w ,...,w (cid:105)),d) n1,...,n1+n2 1 k ≥dimH(Λ(W ⊕(cid:104)w ,...,w (cid:105)),d¯)+1 n1,...,n1+n2 2 k Using the analogous arguments we proceed by considering the morphisms θ∗ , θ∗ , ..., θ∗ . Eventually, this yields w2 w3 wk (2) dimH(Λ(W ,d) ≥ dimH(ΛW ,d¯)+k ≤n1+n2 n1+1,...,n1+n2 Let us now deal with the fibration (ΛW ,d) (cid:44)→ (ΛW,d) → (ΛW ,d¯) ≤n1+n2 >n2 This time let w ,...,w denote a basis of W ordered by increasing lower 1 k >n2 degree. Again, we shall decompose this fibration into a sequence of fibrations, this time with spherical fibres, beginning with (Λ(W ⊕(cid:104)w ,...,w (cid:105)),d) (cid:44)→ (ΛW,d) → (Λ(cid:104)w (cid:105),d¯) ≤n1+n2 1 k−1 k We consider the associated Gysin sequence in order to prove that dimH(ΛW,d) =dimH(ΛW ⊕(cid:104)w ,...,w (cid:105),d) ≤n1+n2 1 k ≥dimH(ΛW ⊕(cid:104)w ,...,w (cid:105),d)+1 ≤n1+n2 1 k−1 . . (3) . ≥dimH(ΛW ,d)+k ≤n1+n2 For this we argue in an analogous way to the line of reasoning with the Wang sequence above. We only present the arguments for the first spherical fibrations, all the other ones being completely analogous. It is essential to observe that the Euler class of the sphere bundle is given as the class [dw ] of dw in the cohomology of the base space (Λ(W ⊕ 1 1 ≤n1+n2 (cid:104)w ,...,w (cid:105)),d)—see [1, Example 4, p. 202]. By assumption (dw )2 = 0. 1 k−1 1 The connecting homomorphism of the Gysin sequence is the cup product A NOTE ON THE HILALI CONJECTURE 5 with the Euler class. Thus we may mimic exactly the arguments from above by considering the short exact sequence (4) 0 → (coker([dw ]∪(·)))i → Hi(ΛW,d) → (ker([dw ]∪(·)))i−degwk → 0 1 1 This yields Inequality (3). Now we combine Inequalities (3) and (2) together with the assumption from the assertion that dimH(ΛW ,d¯) n1+1,...,n1+n2 =dimH(ΛV,d) ≥dimπ (ΛV,d) ∗ =dimV =dimW n1+1,...,n1+n2 in order to prove the assertion, namely dimH(ΛW,d) ≥dimH(ΛW ,d)+dimW ≤n1+n2 >n2 ≥dimH(ΛW ,d¯)+dimW +dimW n1+1,...,n1+n2 ≤n1 >n2 ≥dimW +dimW +dimW n1+1,...,n1+n2 ≤n1 >n2 =dimW (cid:3) Remark 1.1. The Gysin and Wang sequences are not restricted to odd- dimensional fibres or bases. In the case of a fibration with fibre S2, the Halperin conjecture, which is confirmed in this case, however, yields that the cohomology modules of the total space splits as a product and the Hilali conjecture for it holds true trivially once the base space satisfies it. We leave it to the reader to consider variations of the presented arguments whenever the base space of the fibration is an even sphere. (cid:30) Remark 1.2. We only used that the cohomology classes of the differentials and the Wang derivations vanish; thus the class (I) may be presented in slightly larger generality. (cid:30) The proof of the corollary to Theorem A is obvious now: Once W is concentrated in odd degrees, the square of every monomial in the w vanishes. i 2. The Second Class of Spaces Let (Λ(U ⊕V),d) be a 2-stage algebra. We essentially use the inequality (5) dimH(Λ(U ⊕V),d) ≥ 2dimV−dimUeven from [9, Theorem 2.3, p. 195]. Proof of Theorem A–Second assertion. Wechooseabasis(x ) i 1≤i≤n of U and (y ) of V such that i 1≤i≤n1+r 6 MANUELAMANN - n = n +n 1 2 - degx is even for 1 ≤ i ≤ n i 1 - degx is odd for n +1 ≤ i ≤ n +n i 1 1 2 - degy is odd for 1 ≤ i ≤ n +r i 1 Let us justify these statements: Passing to the associated pure model of (Λ(U ⊕V),d) we make the following classical observations: - Since this pure model has finite dimensional cohomology if and only so has the 2-stage model, V is concentrated in odd degrees. - Since the pure model has finite dimensional cohomology, there need to exist at least n many basis elements of V (mapping to a regular 1 sequence in ΛU under d , the associated pure differential). σ Thus we need to prove that (6) dimH(Λ(U ⊕V),d) ≥ n+n +r 1 By definition we have that dimH(Λ(U ⊕V),d) ≥ n+1 (since dU = 0 and taking into account zeroth cohomology). Every element in Λ2U represents a cohomology class. Every relation imposed on this vector space comes from an element of V for reasons of word-length. In other words: The cohomology vector space represented by elements from Λ2U has dimension dimΛ2U −dimimd| ∩Λ2U V (cid:18)(cid:18) (cid:19) (cid:19) n ≥ +n −(n +r) 1 1 2 n2−n = −r 2 Thus the cohomology represented by elements of word-length at most two is bounded from below by n2−n n2+n (7) 1+n+ −r = 1−r+ 2 2 Let us prove that except for a few cases of (n ,n ,r) one of the following 1 2 two assertions holds (cid:40) 1−r+ n2+n or n+n +r ≤ 2 1 2r (cid:40) n +2r ≤ 1+ n2−n or ⇐⇒ 1 2 n+n +r ≤ 2r 1 By Observation (6) this will prove the result in nearly all the cases due to Inequality (7) in the first case respectively due to Inequality (5) (and r = dimV −dimUeven) in the second one. So assume that r ≥ n2−3n and note that n2−3n ≤ 1 + n2−n − n1. Under 4 4 2 4 2 this assumption we verify that the second condition is satisfied for n ≥ 6. Thus we assume n ≤ n ≤ 5. In this case n+n +r ≤ 2r is satisfied if r ≥ 4. 1 1 Thus we have to assume that n ≤ n ≤ 5 and r ≤ 3. A computer based 1 A NOTE ON THE HILALI CONJECTURE 7 check then reveals that none of the conditions is satisfied only if (n ,n ,r) ∈ {(1,0,1),(1,1,1),(1,1,2),(1,2,2),(2,0,1),(2,0,2),(2,1,2), 1 2 (2,2,3),(3,0,1),(3,0,2),(3,0,3),(3,1,3),(4,0,2),(4,0,3)} One may do the respective simple checks in each case, or one may cite [3], which proves the Hilali conjecture in the hyperelliptic case. Indeed, we may neglect all those triples with n = 0, as this is the pure case. Moreover, we 2 mayalsoignorethecaseswithn = 1,sinceinthiscasethespaceisnecessarily 2 hyperelliptic. Thus it remains to deal with (n ,n ,r) ∈ {(1,2,2),(2,2,3)}. 1 2 Let us deal with (1,2,2) first. In this case dimπ (Λ(U⊕V),d) = dim(U⊕ ∗ V) = 6. Note that dimVodd = 2 and dim(Λ2V)odd = 2. Since V is concentratedinodddegrees,itfollowsthatdimHodd(Λ(U⊕V),d) ≥ 4. Since the space has vanishing Euler characteristic, it follows that dimHeven(Λ(U⊕ V),d) ≥ 4 and dimH(Λ(U ⊕V),d) ≥ 8. In the case of the triple (2,2,3) we argue similarly: The odd-degree subspace of the cohomology algebra represented by word-length one elements is 2-dimensional, the odd-degree subspace represented by word-length 2 is 4-dimensional. Vanishing Euler characteristic implies that dimH(Λ(U ⊕ V),d) ≥ 2·6 = 12. (cid:3) Remark 2.1. The arguments concerning the special cases in the last part of the proof generalise: Since these elements have odd degree, the elements x x i j with 1 ≤ i ≤ n , n +1 ≤ j ≤ n, represent linearly independent elements in 1 1 cohomology. So the odd-degree cohomology is at least n ·n -dimensional 1 2 and due to the vanishing Euler characteristic for n > n the cohomology will 1 be at least 2n n -dimensional. 1 2 (cid:30) Appendix A. Separating Examples We briefly sketch some algebras by which one may tell apart the different classesofspacesweconsider. Denoteby(H)theclassofhyperelliptic algebras (ΛW,d), i.e. elliptic algebras satisfying that (1) U = Weven, V = Wodd (2) d(U) = 0 (3) d(V)⊆Λ>0U ⊗ΛV WeprovidethreeminimalSullivanalgebrashavingthefollowingproperties: • (ΛX,d) is in (I), but neither in (H) nor in (S ). 2 • (ΛY,d) is in (S ), but neither in (H) nor in (I). 2 • (ΛZ,d) is in (H), but not in (S ). 2 WedefinethefirstalgebrabyX = (cid:104)x ,x ,x ,x (cid:105)withdegx = degx = 3, 1 2 3 4 1 2 degx = 5, degx = 7, dx = dx = 0, dx = x x , dx = x x . Then, by 3 4 1 2 3 1 2 4 1 3 construction, this algebra is 3-stage (and not 2-stage), and, since Xeven = 0, but not all differentials are trivial, it is not hyperelliptic. A direct check shows that it is in (I), however. The second algebra is given by Y = (cid:104)y ,y ,y ,y ,y (cid:105) with degy = 1 2 3 4 5 1 degy = degy = degy = 3, degy = 5, dy = dy = dy = dy = 0, 2 3 4 5 1 2 3 4 8 MANUELAMANN dy = y y +y y +y y +y y +y y +y y . Obviously, this algebra is not 5 1 2 1 3 1 4 2 3 2 4 3 4 hyperelliptic. A direct check then also shows that it does not satisfy any of the vanishing conditions for the differential or the Wang derivation required for (I). (For example, (dy )2 = y y y y .) 5 1 2 3 4 Define the third algebra by Z = (cid:104)z ,z ,z ,z ,z ,z ,z (cid:105) with degz = 1 2 3 4 5 6 7 1 degz = degz = 2, degz = degz = degz = 3, degz = 7, dz = dz = 2 3 4 5 6 7 1 2 dz = dz = 0, dz = z z , dz = z z , dz = (z z −z z )z . It is 3-stage 3 6 4 1 2 5 1 3 4 3 4 2 5 6 and hyperelliptic. References [1] Y. Félix, S. Halperin, and J.-C. Thomas. Rational homotopy theory, volume 205 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2001. [2] Y. Félix, J. Oprea, and D. Tanré. Algebraic models in geometry, volume 17 of Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, 2008. [3] J.FernańdezdeBobilla,J.Fresań,V.Muno˜z,andA.Murillo.TheHilaliconjecturefor hyperelliptic spaces. In Mathematics without boundaries, Surv. in Pure Mathematics, pages 21–36. Springer, 2014. [4] M. R. Hilali. Action du tore Tn sur les espaces simplement connexes. PhD thesis, Universite catholique de Louvain, 1980. [5] M. R. Hilali and M. I. Mamouni. A conjectured lower bound for the cohomological dimension of elliptic spaces. J. Homotopy Relat. Struct., 3(1):379–384, 2008. [6] M. R. Hilali and M. I. Mamouni. A lower bound of cohomologic dimension for an elliptic space. Topology Appl., 156(2):274–283, 2008. [7] M. R. Hilali and M. I. Mamouni. A conjectured lower bound for the cohomological dimension of elliptic spaces – recent results in some simple cases. arXiv:0803.3821v1, 2013. [8] M. R. Hilali and M. I. Mamouni. La conjecture H: Une minoration de la dimension cohomologique pour un espace elliptique. arXiv:0712.3784v2, 2013. [9] B. Jessup and G. Lupton. Free torus actions and two-stage spaces. Math. Proc. Cambridge Philos. Soc., 137(1):191–207, 2004. [10] O. Nakamura and T. Yamaguchi. Lower bounds of Betti numbers of elliptic spaces with certain formal dimensions. Kochi J. Math., 6:9–28, 2011. Manuel Amann Fakulta¨t fu¨r Mathematik Institut fu¨r Algebra und Geometrie Karlsruher Institut fu¨r Technologie Kaiserstraße 89–93 76133 Karlsruhe Germany [email protected] http://topology.math.kit.edu/21 54.php