Rational Homotopy Theory II

Transcription

1 Rational Homotopy Theory II Yves Félix, Steve Halperin and Jean-Claude Thomas World Scientic Book, 412 pages, to appear in March Abstract Sullivan s seminal paper, Infinitesimal Computations in Topology, includes the application of his techniques to non-simply connected spaces, and these ideas have been used frequently by other authors. Our objective in this sequel to our Rational Homotopy Theory I, published by Springer-Verlag in 2001, is to provide a complete description with detailed proofs of this material. This then provides the basis for new results, also included, and which we complement with recent advances for simply-connected spaces. There do remain many interesting unanswered questions in the field, which hopefully this text will make it easier for others to resolve. 1 Introduction Rational homotopy theory assigns to topological spaces invariants which are preserved by continuous maps f for which H (f; Q) is an isomorphism. The two standard approaches of the theory are due respectively to Quillen [58] and Sullivan [61], and [62]. Each constructs from a class of CW complexes X an algebraic model M X, and then constructs from M X a CW complex X Q, together with a map ϕ X : X X Q. Both H (X Q ; Z) and π n (X Q ) are rational vector spaces, and with appropriate hypotheses H (ϕ X ) : H (X) Q H (X Q ; Z), and π n (ϕ X ) : π n (X) Q π n (X Q ), n 2, are isomorphisms. In each case the model M X belongs to an algebraic homotopy category, and a homotopy class of maps f : X Y induces a homotopy class of morphisms M f : M X M Y (in Quillen approach) and a homotopy class of morphisms M f : M Y M X (in Sullivan s approach). These are referred to as representatives of f. In Quillen s approach, X is required to be simply connected and M X is a rational differential graded Lie algebra which is free as a graded Lie algebra. In this case H (ϕ X ) and π 2 (ϕ X ) are always isomorphisms. Here, as in [18], we adopt Sullivan s approach, and in this Introduction provide an overview of the material in the monograph, together with brief summaries of the individual Chapters. Sullivan s approach associates to each path connected space X a cochain algebra M X of the form ( V, d) in which the free commutative graded algebra V is generated by V = V 1, and V = m m V with m V = V V (m factors). Additionally, each V k is preserved by d, and d also satisfies a nilpotence condition: ( V, d) is called a minimal Sullivan algebra. A minimal Sullivan algebra determines a simplicial set V, d 1

2 with spatial realization V, d, and when ( V, d) is the model of the CW complex X then this determines (up to homotopy) the map ϕ X : X X Q = V, d. This approach makes non-simply connected spaces accessible to rational homotopy theory. For example, if H 1 (X; Q) is finite dimensional then π 1 (X Q ) is the Malcev completion of π 1 (X): π 1 (ϕ X ) induces an isomorphism lim n π 1(X)/π n 1 (X) Q = π 1 (X Q ), where (π1 n(x)) denotes the lower central series of π 1(X). On the other hand, this approach also comes at the cost of a finiteness condition: If X is simply connected then H (ϕ X ) and π 2 (ϕ X ) are isomorphisms if and only if H (X; Q) is a graded vector space of finite type. In the case of non-simply connected CW complexes X, two additional ingredients are required for rational homotopy theory: first, the action by covering transformations of π 1 (X) on the cohomology H ( X; Q) of the universal covering space of X; second, a Sullivan representative ψ for a classifying map mapping X to the classifying space for π 1 (X) and inducing an isomorphism of fundamental groups. If H ( X; Q) has finite type, then the groups π 2 (X) Q can be computed from a minimal Sullivan model of X, and if the action of π 1 (X) on H ( X; Q) is nilpotent, then this Sullivan model can be computed from ψ. Minimal Sullivan algebras ( V, d) are equipped with a homotopy theory and a range of invariants analogous to those which arise in topology. Key among these are the graded homotopy Lie algebra L = {L p } p 0 and, when dim H 1 ( V, d) <, the group G L. Here L p = Hom(V p+1, Q), the Lie bracket is dual to the component d 1 : V 2 V of d, and an exponential map converts L 0 to G L. The group G L acts by conjugation in L and also in H( V 2, d) where ( V 2, d) is obtained by dividing by V 1 V. A third key invariant is the Lusternik-Schnirelmann category, cat ( V, d), defined as the least m for which ( V, d) is a homotopy retract of ( V/ >m V, d). When ( V, d) is the minimal Sullivan model of a CW complex X such that dim H 1 (X; Q) <, then G L = π 1 (X Q ), L 1 = π 1 (ΩX Q ), and there is a natural map H( V 2, d) H( X; Q) equivariant via π 1 (ϕ X ) with respect to the actions of G L and the action by covering transformations of π 1 (X). Finally, as in the simply connected case ([18]) cat ( V, d) cat X. Sullivan models ( V, d) are constructed via a functor, A P L (inspired by the differential forms on a manifold) from spaces to rational cochain algebras, for which H(A P L (X)) and H (X; Q) are naturally isomorphic algebras. Then ( V, d) is the unique (up to isomorphism) minimal Sullivan algebra admitting a morphism m : ( V, d) A P L (X) for which H(m) is an isomorphism. Moreover any morphism ϕ : ( W, d) A P L (X) from an arbitrary minimal Sullivan algebra determines by adjunction a homotopy class of maps ϕ : X V, d ; in particular, m is homotopic to the map ϕ X above. This second step can be applied to construct a minimal Sullivan model ( V, d) (A, d) for any commutative cochain algebra satisfying H 0 (A, d) = lk. While these may not be Sullivan models of a topological space, the homotopy machinery of minimal Sullivan algebras is established independently of topology, and so can be applied in this more general context. 2

3 In particular, minimal Sullivan algebras become a valuable tool in the study of graded Lie algebras E = E 0 with lower central series denoted by (E n ), provided that E 0 acts nilpotently by the adjoint representation in each E i, i 1, and that dim E 0 /[E 0, E 0 ] <, dim E i <, i 1, and n E n 0 = 0. Such Lie algebras are called Sullivan Lie algebras. Note that we may have dim E 0 =. For a Sullivan Lie algebra, E, lim n C (E/E n ) is a minimal Sullivan algebra, called the associated Sullivan algebra for E, and lim n E/E n is its homotopy Lie algebra. (Here C ( ) is the classical Cartan-Chevalley-Eilenberg cochain algebra construction.) In the reverse direction, if ( V, d) is any minimal Sullivan algebra for which dim H 1 ( V, d) < then its homotopy Lie algebra is a Sullivan Lie algebra whose associated Sullivan algebra is ( V, d 1 ) with d 1 the component of d mapping V to 2 V. In summary, the interplay between spaces, minimal Sullivan algebras and graded Lie algebras is illustrated by the diagram Graded Lie algebras lim C (L/L n ) n Spaces A P L Minimal Sullivan algebras L. Spaces A crucial technical tool in the theory of minimal Sullivan algebras is the conversion of cochain algebra morphisms to Λ-extensions ( V, d) ( V Z, d) in which ( V, d) is a minimal Sullivan algebra and d : Z p V Z p satisfies a nilpotence condition; when V 1 0 it may happen that Z 0 0. Division by + V Z gives a quotient cochain algebra ( Z, d) and the Λ-extension determines holonomy representations of L and, if dim H 1 ( V, d) <, of G L in H( Z, d). These Λ-extensions are the Sullivan analogues of fibrations, and ( Z, d) is the Sullivan analogue of the fibre. The analogy is not merely abstract: suppose ( V, d) α A P L (Y ) A P L (p) ( V Z, d) β A P L (X) is a commutative diagram in which Y is a CW complex and p is the projection of a fibration with fibre F. Then β factors to give a morphism γ : ( Z, d) A P L (F ), and, in this setting H(γ) : H( Z, d) H (F ; Q) is equivariant via π 1 ( ψ ) with respect to the homolony representation of G L in H( Z, d) and π 1 (Y ) in H(F ; Q). There are two important examples of Λ-extensions. First, if ( V, d) is a minimal Sullivan algebra then ( V 1, d) ( V, d) is a Λ-extension, the Sullivan analogue of a classifying map for a CW complex X. If this morphism is a Sullivan representative for the classifying map, and if H (X; Q) has finite type and the covering space action of π 1 (X) is nilpotent, then X is a Sullivan space and the quotient ( V 2, d) is a minimal Sullivan model for X. Many classical examples are Sullivan spaces, including all closed orientable Riemann surfaces. 3

4 Second, if ( V, d) is any minimal Sullivan algebra, converting the augmentation ( V, d) lk yields a Λ-extension ( V, d) ( V U, d) with H( V U, d) = lk; this is the acyclic closure of ( V, d). Here the differential in the quotient U is zero and so the holonomy representation is a representation of the homotopy Lie algebra L of V in U. The Λ-extension also determines a diagonal : U U U which makes U into a commutative graded Hopf algebra, and there is a natural homomorphism η L : UL Hom ( U, lk) of graded algebras which converts right multiplication by L to the dual of the holonomy representation. In particular, in the case of the acyclic closure of the associated Sullivan algebra of a Sullivan Lie algebra, E, this yields a morphism UE Hom( U, lk) which identifies U as a sort of predual of UE. In this setting we define depth E = least p (or ) such that Tor UE p (lk, U) 0. This generalizes the definition in [18] for Lie algebras E = E 1 of finite type, because in this case UE = ( U) # and Ext UE (lk, UE) is the dual of TorUE (lk, U). The invariant depth E plays an important role in the growth and structure theorems for the homotopy Lie algebra of a simply connected space of finite LS category. These were established after [18] appeared, and so are included here. The extent to which they may be generalized to non-simply connected spaces remains an open question. Although the present volume is a sequel to [18] it can be read independently, since all the definitions, conventions and results are stated here, whether or not they appear in [18], although we do quote proofs from [18] whenever this is possible. As in [18], we work where possible over an arbitrary field lk of characteristic zero, and with rare exceptions, definitions and notation are unchanged from [18]; in particular, V # denotes the dual of a graded vector space V. Also, for simplicity, the cohomology algebra H (X; lk) of a space X is denoted by H(X). That said, by and large the material in this monograph either is a non-trivial extension of, or is in addition to, the content of [18]. In particular, it includes: the extension of Sullivan models from simply connected spaces to path connected spaces with general (not necessarily nilpotent) fundamental group G. an analysis of L 0, the fundamental Lie algebra of ( V, d). a description of the holonomy action of π 1 (B) on H (F ) in terms of Sullivan models. a complete proof that under the most general possible hypotheses the Sullivan fibre associated with a fibration B E is the Sullivan model of the fibre F of p, even when B is not simply connected. an analysis of the minimal Sullivan model of a classifying space and the introduction of Sullivan spaces. the definition of the depth of L for any Sullivan algebra and a homological analysis of its properties extending those provided in [18] when L 0 = 0. complete proofs of the growth and structure theorems for the higher rational homotopy groups of a connected CW complex. 4

5 2 Contents 1. Basic definitions and constructions 1.1 Graded algebra 1.2 Differential graded algebra 1.3 Simplicial sets 1.4 Polynomial differential forms 1.5 Sullivan algebras 1.6 The simplicial and spacial realizations of a Λ-algebra 1.7 Homotopy and based homotopy 1.8 The homotopy groups of a minimal algebra The homotopy Lie algebra of a minimal Sullivan algebra 2.2 The fundamental Lie algebra of a Sullivan 1-algebra 2.3 Sullivan Lie algebras 2.4 Primitive Lie algebras and exponential groups 2.5 The lower central series of a group 2.6 The linear isomorphism ( sv ) # = ÛL 2.7 The fundamental group of a 1-finite minimal Sullivan algebra 2.8 The homology Hopf algebra of a 1-finite minimal Sullivan algebra 2.9 The action of G L on π n ( V, d ) 2.10 Formal Sullivan 1-algebras 3. Fibrations et Λ-extensions 3.1 Fibrations, Serre fibrations and homotopy fibrations 3.2 The classifying space fibration and Postnikov decompsition of a CW complex 3.3 λ-extensions 3.4 Existence of minimal Sullivan models 3.5 Uniqueness of the minimal model 3.6 The acyclic closure of a minimal Sullivan algebra 3.7 Sullivan extension and fibrations 4. Holonomy 4.1 Holonomy of a fibration 4.2 Holonomy of a λ-extension 4.3 Holonomy representation for a λ-extension 4.4 Nilpotent and locally nilpotent representations 4.5 Connecting topology and Sullivan homotopy 4.6 The holonomy action on the homotopy groups of the fibre 5

6 5. The model of the fibre is the fibre of the model 5.1 The main theorem 5.2 The holonomy action of π 1 (Y, ) on π (F ) 5.3 The Sullivan model of a universal covering space 5.4 The Sullivan model of a spacial realization 6. Loop spaces and loop spaces actions 6.1 The loop cohomology coalgebra of (λv, d) 6.2 The transformation map η L 6.3 The graded Hopf algebra, H ( λu ; Q) 6.4 Connecting Sullivan algebras with topological spaces 7. Sullivan spaces 7.1 Sullivan spaces 7.2 The classifying space BG 7.3 The Sullivan 1-model of BG 7.4 Malcev completions 7.5 The morphism m λv,d : (λv, d) A P L (λv, d ) 7.6 When BG is a Sullivan space 8. Examples 8.1 Nilpotent and rationally nilpotent groups 8.2 Nilpotent and rationally nilpotent spaces 8.3 The group Z# #Zs 8.4 Semidirect product 8.5 Orientable Riemann surfaces 8.6 The classifying space of the pure braid group is a Sullivan space 8.7 The Heisenberg group 8.8 Siefert manifolds 8.9 Arrangement of hyperplanes 8.10 Connected sum of real projective spaces 8.11 A final example 9. Lusternik-Schnirelmann category 9.1 The LS category of topological spaces and commutative cochain algebras 9.2 The mapping theorem 9.3 Module category and the Toomer invariant 9.4 cat =mcat 9.5 cat = e( ) 6

7 9.7 Jessup s Theorem 9.8 Example 10. Depth of a Sullivan algebra and of a Sullivan Lie algebra 10.1 Ext, Tor and the Hochschild-Serre spectral sequence 10.2 The depth of a minimal Sullivan algebra 10.3 The depth of a Sullivan Lie algebra 10.4 Sub Lie algebras and ideals of Sullivan Lie algebra 10.5 Depth and relative depth 10.6 The radical of a Sullivan Lie algebra 10.7 Sullivan Lie algebras of finite type 11. Depth of a connected Sullivan Lie algebra of finite type 11.1 Summary of previous results 11.2 Modules over an abelian Lie algebra 11.3 Weak depth 12. Trichotomy 12.1 Overview of the results 12.2 The rationally elliptic case 12.3 The rationally hyperbolic case 12.4 The gap theorem 12.5 Rationally infinite spaces of finite category 12.6 Rationally infinite spaces of finite dimension 13. Exponential growth 13.1 The invariant lod index 13.2 Growth of graded Lie algebra 13.3 Weak exponential growth and critical degree 13.4 Approximation of log index 13.5 Moderate exponential growth 13.6 Exponentiam growth 14. Structure of graded Lie algebras of finite depth 14.1 Introduction 14.2 The spectrum 14.3 Minimal sub Lie algebras 14.5 L-equivalence 14.6 The odd part of a graded Lie algebra 15. Weight decomposition of a Sullivan algebra 7

8 15.1 Weight decomposition 15.2 Exponential growth of L 15.3 The fundamental Lie algebra of a 1-formal Sullivan algebra 16. Problems AMS Classification : 55P35, 55P62, 17B70 Key words : Homotopy groups, graded Lie algebra, exponential growth, LS category. 3 Introduction References [1] J.F. Adams and P.J. Hilton, On the chain algebra of a loop space, Comment. Math. Helvetici 30 (1956), [2] J. Amoros, On the Malcev completion of Kahler groups, Comment. Math. Helv. 71 (1996), [3] I.K. Babenko, Analytical properties of Poincaré series of a loop space, Mat. Zametski 27 (1980) [4] H. Baues and J.M. Lemaire, Minimal models in homotopy theory, Math. Ann. 225 (1977), [5] J. Birman and T. Brendle, Handbook of Geometric Topology, Elsevier 2005 [6] A.K. Bousfiled and V.K. Gugenheim, On PL de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 179 (1976) [7] A.K. Bousfield and D.M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer (1972). [8] E. Brieskorn, Sur les groupes de tresses (d après Arnold), Séminaire Bourbaki 1971/72, Lecture Notes in Mathematics, 317, Springer-Verlag, [9] H. Cartan and S.Eilenberg, Homological algebra, Princeton University Press, 1956 [10] B. Cenkl and R. Porter, Malcev s completion of a group and differential forms, J. Differential Geometry 15 (1980), [11] K.T. Chen, Extension of C -function algebra by integrals and Malcev completion of π 1, Advances in Math. 23 (1977), [12] O. Cornea, G. Lupton, J. Oprea and D. Tanré, Lusternik-Schnirelmann Category, Mathematical Surveys and Monographs 103, AMS, [13] P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975),

9 [14] G. Denham and A. Suciu, On the homotopy Lie algebra of an arrangement, [15] M. Falk, The minimal model of the complement of an arrangement of hyperplanes, Trans. Amer. Math. Soc. 309 (1988), [16] M. Falk and R. Randell, The lower central series of a fiber-type arrangement, Invent. Math. 82 (1985), [17] Y. Félix, La dichotomie elliptique-hyperbolique en homotopie rationnelle, Astérisque 176, SMF, [18] Y. Félix, S. Halperin and J.-C. Thomas, 2001, Rational Homotopy Theory, Graduate Texts in Mathematics 205, Springer-Verlag. [19] Y. Félix, S. Halperin and J.-C. Thomas, An asymptotic formula for the ranks of the homotopy groups of finite complex, Expositiones Math. 25 (2007), [20] Y. Félix, S. Halperin and J.-C. Thomas, The ranks of the homotopy groups of a finite complex, Can. J. of Math. 65 (2013), [21] Y. Félix, J. Oprea and D. Tanré, Algebraic models in geometry, Oxford Graduate Texts in Math. 17, 2008 [22] W. Fulton, Algebraic curves, Addison-Wesley, 1969 [23] T. Ganea, Lusternik-Schnirelmann category and strong category, Illinois J. of Math.11 (1967), [24] D.H. Gotllieb, Evaluation subgroups of homotopy groups, Amer. J. Math , [25] G. Grätzer, General Lattice Theory, Birkhäuser [26] P. Griffiths and J. Morgan, Rational Homotopy Theory and Differential Forms, Second Edition, Progress in Mathematics 19, Birkhauser, 2013 [27] P.-P. Grivel, Formes différentielles et suites spectrales, Ann. Inst. Fourier 29 (1979), [28] S. Halperin, Lectures on Minimal Models, Mémoires de la Société Mathématique de France, 230 (1983). [29] S. Halperin and J.-M. Lemaire, Suites inertes dans les algèbres de Lie graduées, Math. Scand. 61 (1987), [30] S. Halperin and J. Stasheff, Obstructions to homotopy equivalences, Adv. in Math. 32 (1979), [31] K. Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), [32] A. Hatcher, Algebraic Topology, 2002, Cambridge University Press. [33] K. Hess, A proof of Ganea conjecture for rational spaces, Topology 30 (1991),

10 [34] K. Hess, Rational homotopy theory: a brief introduction, Interactions between homotopy theory and algebra, , Contemp. Math. 436, Amer. Math. Soc [35] P. Hilton, G. Mislin and J. Roitberg, Localization of nilpotent groups and spaces, North-Holland, [36] I. James, On category in the sense of Lusternik-Schnirelmann, Topology 17 (1978), 331)348. [37] T. Kohno, On the holonomy Lie algebra and the nilpotent completion of the fundamental group of the complement of hypersurfaces, Nagoya Math. J. 92 (1983), [38] T. Kohno, Série des Poincaré-Koszul associée aux groupes de tresses pures, Invent. Math. 82 (1985), [39] J.-L. Koszul, Homologie et cohomologie des algèbres de Lie, Bull. Soc. Math. France 78 (1950), [40] A.L. Kurosh, Theory of groups, II, Chelsea Publishing, [41] P. Lambrechts, Analytic properties of Poincaré series of spaces Topology 37 (1998), [42] M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Ec. Norm. Sup. 71 (1954), [43] J.M. Lemaire and F. Sigrist, Sur les invariants d homotopie rationnelle liés à la LS catégorie, Comment. Math. Helv. 56 (1981), [44] MacLane, Homology, Springer-Verlag, Berlin and New-York, [45] W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Interscience Publishers [46] A.L. Malcev, Nilpotent groups without torsion, Jzv. Akad. Nauk. SSSR, Math. 13 (1949), [47] P. May, Simplicial objects in algebraic topology, Chicago Univ. Press, [48] P. May, Classifying spaces and fibrations, Memoirs of the Amercian Math. Soc. 155, 1975 [49] J. Milnor and J.C. Moore, On the structure of Hopf algebras, Annals of Math. 81 (1965), [50] J. Morgan,The algebraic topology of smooth algebraic varieties, Inst. Hautes Etudes Sci Publ. Math. 148 (1970), [51] J. Neisendorfer and L. Taylor, Lie algebras, coalgebras and rational homotopy theory for nilpotent spaces, Pacific J. of Math. 74 (1978), [52] P. Orlik, Seifert Manifold, Lecture Notes in Mathematics, 231 Springer-Verlag [53] P.Orlik and L.Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980),

11 [54] S. Papadima and A. Suciu, Geometric and algebraic aspects of 1-formality, Bull. Math. Soc. Sci. Math. Roumanie 52 (2009), [55] S. Papadima and A. Suciu, Chen Lie algebras, Internat. Math. Res. Notices 21 (2004), [56] S. Papadima and A. Suciu, Homotopy Lie algebras, lower central series and the Koszul property, Geometry and Topology 8 (2004), [57] S. Papadima and S. Yuzvinsky, On rational K(π, 1) spaces and Koszul algebras, Journal of Pure and Applied Algebra 144 (1999), [58] D. Quillen, Rational homotopy theory, Ann. Math. 90 (1969), [59] H.K. Schenck and A. Suciu, Lower central series and free resolutions of hyperplane arrangements, Trans. Amer. Math. Soc. 354 (2002), [60] J.P. Serre, Lie algebras and Lie groups, Benjamin Inc., [61] D. Sullivan, Differential forms and the topology of manifolds, Proceedings of the International Conference on Manifolds and Related Topics in Topology, Tokyo, 1973, University of Tokyo Press, Tokyo (1975), [62] D. Sullivan, Infinitesimal computations in topology, Publ. IHES , [63] G.H Toomer, Lusternik-Schnirelmann category and the Moore spectral sequence, Math. Z. 138 (1974), [64] J.H. van Lint and R.M. Wilson, A course in combinatorics, Cambridge University Press, second edition, [65] Weibel, An introduction to homological algebra, Cambridge University Press, [66] G. Whitehead, Elements of homotopy theory, Graduate Text in Math. 61, Springer- Verlag, [67] J.H.C. Whitehead, A certain exact sequence, Ann. of Math. 52 (1951),