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Rational Homotopy Theory II Rational Homotopy Theory II Yves F´elix,Steve Halperin and Jean-Claude Thomas World Scientic Book, 412 pages, to appear in March 2012. Abstract Sullivan's seminal paper, Infinitesimal Computations in Topology, includes the ap- plication 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 Homo- topy Theory I, published by Springer-Verlag in 2001, is to provide a complete descrip- tion 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 MX , and then constructs from MX a CW complex XQ, together with a map 'X : X ! XQ. Both H∗(XQ; Z) and πn(XQ) are rational vector spaces, and with appropriate hypotheses H∗('X ): H∗(X) ⊗ Q ! H∗(XQ; Z) ; and πn('X ): πn(X) ⊗ Q ! πn(XQ); n ≥ 2; are isomorphisms. In each case the model MX belongs to an algebraic homotopy category, and a homo- topy class of maps f : X ! Y induces a homotopy class of morphisms Mf : MX !MY (in Quillen approach) and a homotopy class of morphisms Mf : MY !MX (in Sullivan's approach). These are referred to as representatives of f. In Quillen's approach, X is required to be simply connected and MX is a rational differential graded Lie algebra which is free as a graded Lie algebra. In this case H∗('X ) and π≥2('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 MX of the form (^V; d) in which the free commutative graded algebra ^V is generated by ≥1 m m V = V , and V = ⊕m ^ V with ^ 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 h^V; di 1 with spatial realization j^V; dj, and when (^V; d) is the model of the CW complex X then this determines (up to homotopy) the map 'X : X ! XQ = j ^ V; dj : This approach makes non-simply connected spaces accessible to rational homotopy the- ory. For example, if H1(X; Q) is finite dimensional then π1(XQ) is the Malcev completion of π1(X): π1('X ) induces an isomorphism ∼ n = lim π1(X)/π (X) ⊗ / π1(X ) ; − n 1 Q Q n where (π1 (X)) denotes the lower central series of π1(X). On the other hand, this approach also comes at the cost of a finiteness condition: If X is simply connected then H∗('X ) and π≥2('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 π1(X) on the cohomology H (Xe; Q) of the universal covering space of X; second, a Sullivan representative for a classifying map mapping X to the classifying space for ∗ π1(X) and inducing an isomorphism of fundamental groups. If H (Xe; Q) has finite type, then the groups π≥2(X) ⊗ Q can be computed from a minimal Sullivan model of Xe, and if ∗ the action of π1(X) on H (Xe; Q) is nilpotent, then this Sullivan model can be computed from . 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 1 homotopy Lie algebra L = fLpgp≥0 and, when dim H (^V; d) < 1, the group GL. Here p+1 2 Lp = Hom(V ; Q), the Lie bracket is dual to the component d1 : V ! ^ V of d, and an exponential map converts L0 to GL. The group GL 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). 1 When (^V; d) is the minimal Sullivan model of a CW complex X such that dim H (X; Q) < ≥2 1, then GL = π1(XQ), L≥1 = π≥1(ΩXQ), and there is a natural map H(^V ; d) ! H(Xe; Q) equivariant via π1('X ) with respect to the actions of GL and the action by cover- ing transformations of π1(X). Finally, as in the simply connected case ([18]) cat (^V; d) ≤ cat X. Sullivan models (^V; d) are constructed via a functor, APL (inspired by the differen- tial forms on a manifold) from spaces to rational cochain algebras, for which H(APL(X)) ∗ and H (X; Q) are naturally isomorphic algebras. Then (^V; d) is the unique (up to iso- morphism) minimal Sullivan algebra admitting a morphism m :(^V; d) ! APL(X) for which H(m) is an isomorphism. Moreover any morphism ' :(^W; d) ! APL(X) from an arbitrary minimal Sullivan algebra determines by adjunction a homotopy class of maps j'j : X ! j ^ V; dj; in particular, jmj is homotopic to the map 'X above. This second step can be applied to construct a minimal Sullivan model (^V; d) ! (A; d) for any commutative cochain algebra satisfying H0(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 In particular, minimal Sullivan algebras become a valuable tool in the study of graded n Lie algebras E = E≥0 with lower central series denoted by (E ), provided that E0 acts nilpotently by the adjoint representation in each Ei, i ≥ 1, and that n dim E0=[E0;E0] < 1 ; dim Ei < 1 ; i ≥ 1 ; and \n E0 = 0 : Such Lie algebras are called Sullivan Lie algebras. Note that we may have dim E0 = 1. For a Sullivan Lie algebra, E, lim C∗(E=En) is a minimal Sullivan algebra, called −! n the associated Sullivan algebra for E, and lim E=En is its homotopy Lie algebra. (Here − n 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 H1(^V; d) < 1 then its homotopy Lie algebra is a Sullivan Lie algebra whose associated Sullivan algebra 2 is (^V; d1) with d1 the component of d mapping V to ^ V . In summary, the interplay between spaces, minimal Sullivan algebras and graded Lie algebras is illustrated by the diagram Graded Lie algebras O L lim C∗(L=Ln) −! n APL Minimal Sullivan j : j Spaces / / Spaces algebras 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 : Zp ! ^V ⊗ ^Z≤p satisfies a "nilpotence" condition; when V 1 6= 0 it may happen that Z0 6= 0. Division by ^+V ⊗ ^Z gives a quotient cochain algebra (^Z; d) and the Λ-extension determines holonomy representations of L and, if 1 dim H (^V; d) < 1, of GL 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) / (^V ⊗ ^Z; d) α β APL(p) APL(Y ) / APL(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) ! APL(F ), and, ∗ in this setting H(γ): H(^Z; d) ! H (F ; Q) is equivariant via π1(j j) with respect to the homolony representation of GL in H(^Z; d) and π1(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 π1(X) is nilpotent, then X is a Sullivan space and the quotient (^V ≥2; d) is a minimal Sullivan model for Xe. Many classical examples are Sullivan spaces, including all closed orientable Riemann surfaces. 3 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.
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