3 MARCH 2012, CPR, RABAT

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1 RATIONAL HOMOTOPY THEORY SEMINAR Sullivan s Minimal Models 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 3 MARCH 2012, CPR, RABAT My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 1 / 14

2 Aim of the talk Rational Homotopy Theory : Brief description Modèle minimal de Sullivan : Definition and main proprieties My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 2 / 14

3 Aim of the talk Rational Homotopy Theory : Brief description Modèle minimal de Sullivan : Definition and main proprieties My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 2 / 14

4 Prerequisites Algebraic Topology basic knowledge of what is : Homology Homotopy My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 3 / 14

5 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 3 / 14

6 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 3 / 14

7 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 3 / 14

8 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 4 / 14

9 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 4 / 14

10 Glossary Theorem, [FHT] Any simply connected space X admits an unique (up to homotopy) CW-complex rationalization My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 4 / 14

11 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 4 / 14

12 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 5 / 14

13 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 6 / 14

14 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 6 / 14

15 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 In particular : y 2 = 0 when y odd xy = yx when x even My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 7 / 14

16 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 7 / 14

17 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 7 / 14

18 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 7 / 14

19 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 7 / 14

20 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 7 / 14

21 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 7 / 14

22 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 7 / 14

23 Special Denotations In general for any x ΛV, we have dx = k 0β k lenght=k {}}{ y 1...y }{{ k } y i odd x i even x α i i k Λ k V odd ΛV even Hence ΛV is bi-graded as follows ΛV = p,q (Λp V odd ΛV even ) q. p word-length graduation and q : degree graduation. Λ k V := p k Λp V and Λ + V := Λ 1 V When (ΛV, d) is a simply elliptic minimal model, we have dv Λ 2 V = Λ + V.Λ + V My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 8 / 14

24 Special Denotations Simple Conclusions Λ(V W) = ΛV ΛW When y odd, Λy = {ay + b; a, b Q} = Q 1 [y] When x even, Λx = { k a k x k ; a, b Q} = Q[x] My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 8 / 14

25 Special Denotations Pure Model When dv even = 0 dv odd ΛV even My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 8 / 14

26 Special Denotations Hyperelliptic Model When dv even = 0 My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 8 / 14

27 Special Denotations Two Stage Model When V = U W du = 0 dw ΛU My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 8 / 14

28 Example of Algebraization 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 9 / 14

29 Example of Algebraization Algebraic version For any elliptic model of Sullivan, (ΛV, d) we have dim V dim H (ΛV, d) My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 9 / 14

30 Example of Algebraization 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 9 / 14

31 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 10 / 14

32 Euler-Poincaré characteristic S. Halperin, [Ha83] we have the following : χ c 0 and χ π 0 Morever, In this case χ c > 0 χ π = 0 H (ΛV, d) = H even (ΛV, d) My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 10 / 14

33 Euler-Poincaré characteristic Generalisation For any graded vector space A, the Euler-Poincaré characteristic is defined as follows χ(a) := k 0 ( 1) k dim A k So, N.B χ c = χ(h (ΛV, d)), χ π = χ(v) χ(h (A, d)) = χ(a) My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 10 / 14

34 Euler-Poincaré characteristic Util Remark As χ π = dim V even dim V odd, we put dim V even = p and dim V odd = n+p, so χ π = p and dim V = 2n+p p = 0 H (ΛV, d) = H even (ΛV, d) p 0 dim H (ΛV, d) = 2 dim H even (ΛV, d) My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 10 / 14

35 Toral Rank Definition rk 0 (X) := The largest integer n 1 for which X admits an almost-free n-torus action C. Allday & Halperin, [AH78] rk 0 (X) χ π The equality holds when X is pure My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 11 / 14

36 Toral Rank My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 11 / 14

37 Toral Rank Toral Rank Conjecture (TRC), S.Halperin (1986) For any elliptic and simply connected topological space X, we have dim H (X;Q) 2 rk 0(X) My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 11 / 14

38 Toral Rank The link between ConjH & TRC 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 Conj H+2n+p 2 p ε = CRT My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 11 / 14

39 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 12 / 14

40 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 My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 12 / 14

41 Formal dimension 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 12 / 14

42 Formal dimension J. Friedlander and S. Halperin, [FH79] There exists a special homogeneous basis x 1,...,x n of V even and a basis y 1,... y n+p of V odd such that : x 1 x n 2 x i 1 y i for i = 1..n n x i fd(x) i=1 n+p y i 2fd(X) 1 i=1 n+p y i i=1 n ( x i 1) = fd(x) i My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 12 / 14

43 Main References of RHT C. Allday & S. Halperin, Lie group actions on espace s 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 espace s, 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) RHT Seminar, CPR Rabat Sullivan s Minimal Models 13 / 14

44 My Ismail Mamouni (CPGE-CPR Rabat) RHT Seminar, CPR Rabat Sullivan s Minimal Models 14 / 14