A conjecture in Rational Homotopy Theory CIMPA, FEBRUARY 2012

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1 UNIV. LIBAN, BEYROUTH A conjecture in Rational Homotopy Theory My Ismail Mamouni, CPGE-CPR, Rabat Professeur Agrégé-Docteur en Math Master 1 en Sc de l éducation, Univ. Rouen mamouni.new.fr mamouni.myismail@gmail.com CIMPA, FEBRUARY 2012 My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 1 / 25

2 Aknowledgements Participants My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 2 / 25

3 Aknowledgements Sponsors My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 2 / 25

4 Aknowledgements Organizers My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 2 / 25

5 Aknowledgements All kind people of Lebanon for their hospitality, kindness and amiability My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 2 / 25

6 Aim of the talk is to discover : Rational Homotopy Theory Conjecture H The symplectic and co-symplectic manifold cases Join work in 2008 with M.R. Hilali Interaction between RHT and Geometry And finally to invite you to Morocco My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 3 / 25

7 Aim of the talk is to discover : Rational Homotopy Theory Conjecture H The symplectic and co-symplectic manifold cases Join work in 2008 with M.R. Hilali Interaction between RHT and Geometry And finally to invite you to Morocco My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 3 / 25

8 Aim of the talk is to discover : Rational Homotopy Theory Conjecture H The symplectic and co-symplectic manifold cases Join work in 2008 with M.R. Hilali Interaction between RHT and Geometry And finally to invite you to Morocco My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 3 / 25

9 Aim of the talk is to discover : Rational Homotopy Theory Conjecture H The symplectic and co-symplectic manifold cases Join work in 2008 with M.R. Hilali Interaction between RHT and Geometry And finally to invite you to Morocco My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 3 / 25

10 Aim of the talk is to discover : Rational Homotopy Theory Conjecture H The symplectic and co-symplectic manifold cases Join work in 2008 with M.R. Hilali Interaction between RHT and Geometry And finally to invite you to Morocco My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 3 / 25

11 Prerequisites Algebraic Topology basic knowledge of what is : Homology Homotopy My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 4 / 25

12 Prerequisites Graded & differential (A, d) that means : A, module, vector space, algebra,... A := k A k d k : A k A k+1 such that d k+1 d k = 0 We write d : A A, where d k = d Ak and d 2 = 0 My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 4 / 25

13 Prerequisites Cohomology (A, d) that means : H k (A, d) := ker d k+1 Imd k H (A, d) := k H k (A, d) My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 4 / 25

14 Prerequisites Homotopy. Two continuous maps f, g : S k X are called homotopic when there is continuous deformation H : S k [0, 1] X, such that H(., 0) = f, H(., 1) = g. We obtain an equivalence relation The homotopy groups : π k (X) := C(S n, X)/ and π (X) := π k (X). k Type of homotopy : X and Y are called with the same type of homotopy when π (X) = π (Y). My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 4 / 25

15 Glossary Rational Space A simply connected space X is called rational if the following is satisfied. π (X) is a Q-vector space. N.B : π (X) Q is a Q-vector space My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 5 / 25

16 Glossary Rationalization Let X be a simply connected space. A rationalization of X is simply connected and rational space Y, such that : π (X) Q = π (Y) H (X;Q) = H (Y;Q) My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 5 / 25

17 Glossary Theorem 9.7, [FHT] Any simply connected space X admits an unique (up to homotopy) CW-complex rationalization My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 5 / 25

18 Glossary Definition The rational homotopy type of a simply connected space X is the homotopy type of its rationalization. My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 5 / 25

19 Rational Homotopy Theory What it is it Rational homotopy theory is the study of rational homotopy types of spaces and of the properties of spaces and maps that are invariant under rational homotopy equivalence. My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 6 / 25

20 Founders in 1967 Denis Sullivan (1941- ), CUNY-SUNY, USA work in topology, both algebraic and geometric, and on dynamical systems Doctoral advisor : William Browder Wolf Prize in Mathematics (2010) Leroy P. Steele Prize (2006) National Medal of Science (2004) My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 7 / 25

21 Founders in 1967 Daniel Quillen ( ), Oxford the "prime architect" of higher algebraic K- theory Doctoral advisor : Raoul Bott Fields Medal (1978) Cole Prize (1975) Putnam Fellow (1959) My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 7 / 25

22 Model of Sullivan CGDA A commutative graded differential algebra over the rational numbers is a graded Q-algebra (A, d) such that { ab = ( 1) a b ba ( ) d(ab) = (da).b +( 1) a b.da for all a, b A My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 8 / 25

23 Model of Sullivan How to build it From any differential and graded Q-vector space V, we define the cgda ΛV denotes defined by ΛV = TV v w ( 1) v w w v where TV denotes the tensor algebra over V. The differential on ΛV is naturally extended from that of V with respecting the condition (*) called of nilpotence or of Leibniz My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 8 / 25

24 Model of Sullivan Model of Sullivan Our cgda is called a model of Sullivan when there exists some well ordered basis (v α ) α I of V such that dv α Λ{v β, β < α} My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 8 / 25

25 Model of Sullivan Minimal model The model of Sullivan is called minimal when α < β = v α v β My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 8 / 25

26 Model of Sullivan Elliptic model The minimal model is called elliptic when both V and H (ΛV, d) are finite dimensional, in this case (ΛV, d) = (Λ{x 1,..., x n }, d) with x 1... x n dx 1 = 0 and dx j Λ(x 1,...,x j 1 ) for j 2 My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 8 / 25

27 Model of Sullivan D. Sullivan, [Su] Any simply connected space have a minimal model of Sullivan, (ΛV, d) (unique up to isomorphism of cgda), who models its cohomology and homotopy as follows : H k (X;Q) = H k (ΛV, d) π k (X) Q = V k My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 8 / 25

28 Model of Sullivan Basic Examples For the odd sphere : S 2k+1, the model is the form (Λ{x}, 0) with x = 2k + 1. So π n (S 2k+1 ) = Z if n = 2k + 1 = 0 if not My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 8 / 25

29 Model of Sullivan Basic Examples For the even sphere :S 2k, the model is the form(λ{x, y}, d) with x = 2k, y = 4k 1, dy = x 2. So π n (S 2k ) = Z if n = 2k = Z if n = 4k 1 = 0 if not My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 8 / 25

30 Conjecture H Hilali Conjecture (1990) For any elliptic and simply connected topological space X, we have dim(π (X) Q) dim H (X;Q) My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 9 / 25

31 Conjecture H Algebraic version For any elliptic model of Sullivan, (ΛV, d) we have dim V dim H (ΛV, d) My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 9 / 25

32 Conjecture H Simple example in which it holds For the sphere S n we have seen that dim V = 1 or 2, and its well known that for all other i. H 0 (S n ;Q) = H n (S n ;Q) = Q and H i (X;Q) = 0 My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 9 / 25

33 Euler-Poincaré characteristic Definition For any 1-connected elliptic model (ΛV, d) we define two invariants. One cohomological : χ c := k 0( 1) k dim H k (ΛV, d) and another homotopic : χ π := k 0( 1) k dim(v k ) My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 10 / 25

34 Euler-Poincaré characteristic S. Halperin, [Ha83] we have the following : Morever, χ c 0 and χ π 0 χ c > 0 χ π = 0 My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 10 / 25

35 Euler-Poincaré characteristic Util Remark The conjecture H was resolved in the so called pure case where χ π = 0 (see [HM08]), so the opposite and remainder case is that when χ c = 0, i.e., dim H (ΛV, d) = 2 dim H even (ΛV, d) My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 10 / 25

36 Motivation Main Motivation Intermediate way to resolve the famous Toral Rank Conjecture My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 11 / 25

37 Motivation Toral Rank rk 0 (X) := The largest integer n 1 for which X admits an almost-free n-torus action My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 11 / 25

38 Motivation Toral Rank Conjecture (TRC), S.Halperin(86) If X is an elliptic and simply connected space, then dim H (X;Q) 2 rk 0(X) My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 11 / 25

39 Motivation The link As χ π = dim V even dim V odd 0, then write dim V even = n, dim V odd = n+ p, dim V = 2n+p. We know by [AH78] that rk 0 (X) χ π = p, the write TRC : Conj. H : dim H (X;Q) 2 p ε dim H (X;Q) 2n+p 2n+p 2 p ε + Conj H = CRT My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 11 / 25

40 Formal dimension Definition For an elliptic space X, we put fd(x) := max{k, H k (X,Q) 0} My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 12 / 25

41 Formal dimension J. Friedlander and S. Halperin, [FH79] If X is a 1-connected and elliptic space of minimal Sullivan model (ΛV, d), then fd(x) dim V Best known result, losed source If X is a 1-connected and elliptic manifold, then fd(x) = dim X My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 13 / 25

42 Symplectic case Definition A symplectic manifold is a smooth manifold X = M 2m, endowed with a closed and nondegenerate 2-form, called the symplectic form. My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 14 / 25

43 Symplectic case Sketch of proof In the compact case, this implies that there is a nonnull cohomolgical class ω in H 2 (X,Q). Morever symplectic manifolds verify a particular Poincaré duality : The "multiplication" ω k : H m k (X,Q) H m+k (X,Q) is an isomorphism for any k {0,..., m}. Thus each cohomolgical class ω k is non-null in H 2k (X,Q), what means that dim H (ΛV, d) = 2 dim H even (ΛV, d) 2m+2 > 2m = fd(x) dim V. My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 15 / 25

44 Cosymplectic manifolds Definition A cosymplectic structure on a 2m + 1-dimension manifold X = M 2n+1, is the data of a closed 1-form θ and a closed 2-form ω such that θ ω k is a volume form on X, where ω k denotes the product of k copies of ω. My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 16 / 25

45 Cosymplectic manifolds Sketch of proof From [BG67], all dim H k (X;Q) non zero. Thus dim H (ΛV, d) 2m+2 > 2m+1 = fd(x) dim V. (ΛV, d) is not necessary 1-connected because all the known examples of cosymplectic manifolds are non 1-connected, but π 1 is that of a torus of dimension 1 or a nilpotent group. However they have minimal models which satisfy the property that fd(x) dimv. My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 17 / 25

46 Cocktail RHT-Geometry Bott Conjecture A closed 1-connected manifold M, which admits a metric of nonnegative sectional curvature, is rationally elliptic. Hopf Conjecture If closed 1-connected manifold M n admits a metric of nonnegative sectional curvature, then χ c 0. My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 18 / 25

47 Cocktail RHT-Geometry Theorem, J-C. Thomas & Y.Félix On a rationally hyperbolic Riemannian manifold M any isometry A has infinitely many geometrically distinct invariant geodesics. Theorem, D. Sullivan & M. Vigué Suppose M is a rationally elliptic Riemannian n-manifold which has an isometry with no invariant geodesics. Then n is even. Theorem, K. Grove, S. Haplerin & M. Vigué A 1-connected closed Riemannian manifold of odd dimension every isometry has an invariant geodesic. My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 18 / 25

48 Main References of RHT C. Allday & S. Halperin, Lie group actions on spaces of finite rank, Quar. J. Math. Oxford 28 (1978), D.E. Blair and S.I. Goldberg, Topology of almost contact manifolds, Journal of Differential Geometry Vol. 1 (1967), Intelpress, J. Friedlander and S. Halperin, An arithmetic characterization of the rational homotopy groups of certain spaces, Invent. Math. 53 (1979), Y. Félix, S. Halperin & J.-C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, S. Halperin, Finitness in the minimal models of Sullivan, Transc. AMS 230 (1983), D. Sullivan, Infinitesimal computations in topology, Publications Mathématiques de l IHÉS, 47 (1977), My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 19 / 25

49 References of RHT-Geometry interaction D. Sullivan & M. Vigué-Poirrier, The homology theory of the closed geodesic problem, J. Differential Geom. Volume 11, Number 4 (1976), K. Grove, S. Halperin & M. Vigué-Poirrier, The rational homotopy theory of certain path-spaces with applications to geodesics, Acta math., 14,0 (1978), K. Grove, S. Halperin, Contributions of rational homotopy theory to global problems in geometry, Publications Mathématiques de L IHÉS, Volume 56, Number 1, My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 20 / 25

50 Our papers M.R Hilali et M.I Mamouni, A lower bound of cohomologic dimension for an elliptic space, Topology and its Applications, Vol. 156, Issue 2 (2008), M.R Hilali et M.I Mamouni, A conjectured lower bound for the cohomological dimension of elliptic spaces, Journal of Homotopy and Related Structures, Vol. 3, No. 1 (2008), M.R. Hilal1, H. Lamane and M. I. Mamouni, Classification of Rational Homotopy Type for 8-Cohomological Dimension Elliptic Spaces, Advances in Pure Mathematics, Vol. 2 No. 1 (2012), My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 21 / 25

51 See You in Morocco My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 22 / 25

52 See You in Morocco CIMPA School on symplectic geometry Meknes, 21 may - 1 june 2012 My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 23 / 25

53 See You in Morocco 2nd physicists and mathematicians meeting Rabat, 2-3 june 2012 algtop.legtux.org My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 24 / 25

54 My Ismail Mamouni (CPGE-CPR Rabat) CIMPA 2012, Beyrouth A conjecture in RHT 25 / 25