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Rational Homotopy Theory Author(S): Daniel Quillen Source: Annals of Mathematics, Second Series, Vol Annals of Mathematics Rational Homotopy Theory Author(s): Daniel Quillen Source: Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/1970725 . Accessed: 01/04/2014 10:46 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics. http://www.jstor.org This content downloaded from 129.105.215.146 on Tue, 1 Apr 2014 10:46:25 AM All use subject to JSTOR Terms and Conditions Rationalhomotopy theory* By DANIEL QUILLEN CONTENTS INTRODUCTION PART I. 1. Statement of Theorem I 2. Outline of the proof of Theorem I 3. Application of Curtis' convergence theorems 4. DG and simplicial Lie algebras 5. The Whitehead product 6. The coproduct on homology PART II. 1. Closed model categories and statement of Theorem II 2. Serre theory for simplicial sets 3. Serre theory for simplicial groups 4. (se), as a closed model category 5. (DGL), and (DGC)r,+ as closed model categories 6. Applications APPENDIX A. CompleteHopf algebras. 1. Complete augmented algebras 2. Complete Hopf algebras 3. Relation with Malcev groups APPENDIX B. DG Lie algebras and coalgebras. REFERENCES Rational homotopytheory is the study of the rational homotopycategory, that is the category obtained from the category of 1-connected pointed spaces by localizing with respect to the family of those maps which are isomorphisms modulo the class in the sense of Serre of torsion abelian groups. As the homotopy groups of spheres modulo torsion are so simple, it is reasonable to expect that there is an algebraic model for rational homotopy theory which is much simpler than either of Kan's models of simplicial sets or simplicial groups. This is what is constructed in the present paper. We prove that rational homotopy theory is equivalent to the homotopy theory of reduced differentialgraded Lie algebras over Q and also to the homotopy theory of 2-reduced differentialgraded cocommutative coalgebras over Q. In Part I we exhibit a chain of several categories connected by pairs of adjoint functors joining the category i2 of 1-connected pointed spaces with * This research was supported by NSF GP-6959 and the Alfred P. Sloan Foundation. This content downloaded from 129.105.215.146 on Tue, 1 Apr 2014 10:46:25 AM All use subject to JSTOR Terms and Conditions 206 DANIEL QUILLEN the categories(DGL), and (DGc)2 of reduced differentialgraded Lie algebras and 2-reduceddifferential graded cocommutativecoalgebras over Q respec- tively. We provethat these functorsinduce an equivalence of the rational homotopycategory HoQIT2 with both of the categoriesHO(DGL)1 and Ho(DGC)2 obtainedby localizingwith respectto the maps which induce isomorphisms on homology. Moreoverthese equivalenceshave the propertythat the graded Lie algebra wr_1(X)?& Q under Whitehead productand the homologyco- algebra H. (X, Q) of a space X are canonicallyisomorphic to the homologyof the correspondingdifferential graded Lie algebra and coalgebra respectively. An immediatecorollary is that every reduced graded Lie algebra (resp. 2- reducedgraded coalgebra) over Q occursas the rationalhomotopy Lie algebra (resp. homologycoalgebra) of some simply-connectedspace. This answers a questionwhich is due, we believe, to Hopf. Part I raises someinteresting questions such as how to calculate the maps in the categoryHO(DGL), say fromone DG Lie algebra to another, and also whether or not there is any relationbetween fibrations of spaces and exact sequence of DG Lie algebras. In order to answer these questions, we intro- duced in [21] an axiomatizationof homotopytheory based on the notionof a model category,which is short for a "category of models for a homotopy theory". A modelcategory is a categoryendowed with three families of maps called fibrations,cofibrations, and weak equivalences satisfying certain axioms. To a modelcategory C is associated a homotopycategory Ho C, ob- tained by localizing with respect to the familyof weak equivalences,and extra structureon Ho C such as the suspension and loop functorsand the families of fibrationand cofibrationsequences. The homotopycategory to- getherwith thisstructure is called the homotopytheory of the modelcategory e. In Part II we show that rationalhomotopy theory occurs as the homotopy theoryof a closed modelcategory, that all of the algebraiccategories such as (DGL), and (DGc)2occurring in the proofof TheoremI are closed modelcate- gories in a natural way, and that the various adjoint functorsinduce equiv- alences of homotopytheories. Combiningthis result with TheoremI, we obtain a solutionto the problemraised by Thom [29] of constructinga com- mutative cochain functorfrom the category of simply-connectedpointed spaces to the categoryof (anti-) commutativeDG algebras over Q, givingthe rationalcohomology algebra and having the rightproperties with respect to fibrations. Part II containsa numberof resultsof independentinterest. In ? 2 we show how the Serre modC homotopytheory [27], where C is the class of S- torsionabelian groupsand S is a multiplicativesystem in Z, can be realized This content downloaded from 129.105.215.146 on Tue, 1 Apr 2014 10:46:25 AM All use subject to JSTOR Terms and Conditions RATIONAL HOMOTOPY THEORY 207 as the homotopytheory of a suitable closed model category of simplicial sets. In ? 3 we construct another model category for the same homotopytheory out of simplicial groups. In proving the axioms it was necessary to prove the excision property for the homology functor on the category of simplicial groups (II, 3.12). The category of reduced simplicial sets, with cofibrations defined to be infective maps and with weak equivalences definedto be maps which become homotopy equivalences after the geometric realization functor is applied, turned out to be a closed model category in which it is not true that the base extension of a weak equivalence by a fibrationis a weak equivalence. This is reflected in the fact that there exist fibrations with the property that the fiber is not equivalent to the fiberof any weakly equivalent fibrationof Kan complexes (II, 2.9). Since the base of such a fibrationis never a Kan complex, it does not contribute fibrationsequences to the associated homotopy theory. Thus these pathological fibrationsare a curiosity forced upon us by the model category axioms. The same phenomenonoccurs with DG coalgebras, but not with any of the group-like categories considered here. In Part II, ? 6, we give some applications of the theorems of this paper. In particular we use the DG Lie algebra and DG coalgebra models to derive certain spectral sequences (II, 6.6-6.9) for rational homotopy theory. Of special interest is an unstable rational version (II, 6.9) of the reverse Adams spectral sequence studied in [5]. This raises the question of whether such a spectral sequence holds in general. In addition to Part I and II, the paper contains two appendices. Appendix A contains the theory of complete Hopf algebras, which is the natural Hopf algebra framework for treating the Malcev completion [18] as well as groups defined by means of the Campbell-Hausdorffformula [17]. Appendix B con- tains an exposition of some results of DG mathematics in a form particularly suited for our purposes. The main result is that the generalization to DG Lie algebras of the procedure for calculating the homology of a Lie algebra pro- vides a functor C from DG Lie algebras to DG coalgebras whose adjoint 2 is the primitive Lie algebra of the cobar construction, and that the pair 2, C have the same properties of the functors G, W of Kan. Finally we would like to acknowledge the influenceon this work of many conversations with Daniel Kan and E.B. Curtis; our debt to their work will be abundantly clear to anyone who reads the proof of Theorem I. This content downloaded from 129.105.215.146 on Tue, 1 Apr 2014 10:46:25 AM All use subject to JSTOR Terms and Conditions 208 DANIEL QUILLEN PART I 1. Statement of Theorem I If C is a categoryand S is a familyof morphismsof C, thenthe localiza- tion [9, Ch. I]; [21, Ch. I, 1.11] of C with respectto S is a pair consistingof a categoryS-' and a functor7:C'S-'C whichcarries the maps in S into iso- morphismsin S-' and whichis universalwith this property. In generalthere is a minorset-theoretic difficulty with the existenceof S-' whichmay be avoided by use of a suitable set theorywith universes. We shall therefore ignorethis difficultyand assume the existenceof S 'C; forthe cases we need this can be verified(see Part II, 1.3a). Let 2, be the categoryof (r - l)-connected pointedtopological spaces and continuousbasepoint preserving maps. (The reason for the notation2r is to save space in Part II. The subscriptr shouldbe read "begins in dimen- sion r.") We recall the followingtheorem of Serre [27]. PROPOSITION1.1. The following assertions are equivalent for a map f: X Yin 22. ( i) r*(f) ?& Q: w*(X) ?z Q wr*(Y)?& Q is an isomorphism. (ii) H*(f, Q): H*(X, Q) - H*(X, Q) is an isomorphism. A map satisfyingthese conditionswill be called a rational homotopy equivalence. The localizationof 22 with respect to the familyof rational homotopyequivalences will be denotedHOQ 22 and called the rational homo- topy category. The study of this category is what Serre calls homotopy theorymodulo the class of torsionabelian groups.
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