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Annals of Mathematics Annals of Mathematics Genetics of Homotopy Theory and the Adams Conjecture Author(s): Dennis Sullivan Source: Annals of Mathematics, Second Series, Vol. 100, No. 1 (Jul., 1974), pp. 1-79 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/1970841 Accessed: 07-12-2015 16:50 UTC 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 146.96.147.130 on Mon, 07 Dec 2015 16:50:45 UTC All use subject to JSTOR Terms and Conditions Geneticsof homotopytheory and the Adams conjecture By DENNIS SULLIVAN Dedicated to Norman Steenrod This is the firstin a seriesof papersdevoted to the invariantsand classi- ficationof compactmanifolds. One mighthope for a classificationtheory of manifoldshaving the follow- ing form: First, there is an understandablealgebraic descriptionof the homotopytheory of manifolds. Second, the salient geometricproperties of manifoldsbeyond homotopy theory can be isolated and understood. Third, the algebraic structureof these geometricinvariants can be determined. And fourth,this structurecan be successfullyintertwined with the algebraic descriptionof the homotopytheory of manifoldsto completethe theory. We shall tryto outlineand motivatethis paper in termsof this program. First of all, the Adams conjecture concerns the homotopytheory of spherebundles associated to vector bundles. This is importantin the second and thirdparts of the programfor a theoryof smoothmanifolds and follows fromthe fact that the possible tangent vectorbundles of closed manifolds in a homotopytype must have sphere bundles of the same fiberhomotopy type. In fact the work of Browdershows that this spherebundle condition is almost a sufficientcondition for a bundle to be a tangent bundle. The affirmedAdams conjecture then allows us to find the possible tangentbundles of all manifoldsin the homotopytype X by calculatingthe K-groupof the homotopytype K(X) and the Adams' operations{jk} in this group. A more detailed descriptionof Adams' statement is given below. This explains our interestin the Adams conjecture beyondthat inspiredby the natural beauty of the statement. Now it was well-knownfrom the outset that the Adams conjecture concernedonly questions of torsionand divisibilityof a certainsubgroup of K(X) of maximalrank. So the questionwas trivialafter tensoringwith the fieldof rationals. Thus it is natural (and indeed imperative) to discuss the questionin a contextwhere only p-primaryconsiderations are left, where the influence of the rationalsis minimal. This accounts forthe profinitehomotopy theory This content downloaded from 146.96.147.130 on Mon, 07 Dec 2015 16:50:45 UTC All use subject to JSTOR Terms and Conditions 2 DENNIS SULLIVAN of Section 3 of this paper which is, I must say, at firstflush somewhat foreboding. In this profinitesetting the Adams' statementassumes its mostelegant form(see below). Also the homotopyequivalences of the unstable grass- manniansrequired for its proofappear very naturally fromthe fact that the profinitehomotopy type of an algebraic varietycan be constructedusing onlythe abstractalgebraic structureof the variety(etale homotopytheory). This latter pointis in fact the main motive forthe profinitehomotopy theoryof Section3. In orderto derivebenefit from the richGalois symmetry in the homotopytheory of varieties we had to domesticatethe abstract beasts of [51to make themusable in ordinaryalgebraic topology. The Adams conjectures become immediate consequences of general homotopytheoretical discussions. The localization in homotopytheory discussed in Section 2, which is easier to consciencethan the completionof Section 3, plays the following role: First, localizationarises in a naturalway in each of the foursteps of the outline for a classificationtheory of manifolds. Many workershave found in discussionsabout manifoldspersistent disparity between the prime2 and odd primes(see forexample the statementsabout the universalspaces G/PL and G/TOP in Section 2). Analysis of these spaces essentially solves the thirdstep in the programfor a manifoldtheory of the piecewiselinear and topologicalcategories. A detailed solutionof the Adams conjectureleads to an analysis of G/O, the relevant space for this step in smooth manifold theory. Second,localization allows us to constructthe rational homotopytype which complementsthe informationin the profinitehomotopy type. This complementationmakes very clear the relationshipbetween the ordinary (or transcendental)homotopy theory of complex algebraic varieties and theiretale (or algebraic) homotopytheory. Also in beautifulcounterpoint to the relationshipbetween profinite homotopy theory and abstractalgebraic geometryis a new relationshipbetween a rational de Rham theory on polyhedraand the rational homotopytype (see [17]). This new relationship allows a determinationof the "real homotopy"of a smoothmanifold by the de Rham algebra of C--forms. Thus analytical and arithmetical structure' play a role in real and rational homotopytheory akin to that of abstract algebraic structurein profinitehomotopy theory. 1 I.e., C- manifoldsand triangulatedspaces. This content downloaded from 146.96.147.130 on Mon, 07 Dec 2015 16:50:45 UTC All use subject to JSTOR Terms and Conditions GENETICS OF HOMOTOPY THEORY 3 Now we shall outline our structuralanalysis of homotopytheory and discuss the Adams conjecturesin some detail. First of all a remarkon terminology:We are studyingthe structureof homotopytypes to deepen our understandingof more complicatedor richer mathematicalobjects such as manifoldsor algebraic varieties. The relationshipbetween these two types of objects is I think rather strikinglyanalogous to the relationshipin biologybetween the geneticstruc- ture of living substancesand the visible structureof completedorganisms or individuals. The specificationsof the genetic structureof an organismand of the homotopystructure of a manifoldhave similartexture; they are bothdiscrete, combinatorial,rigid, interlocking and sequential.' On the otherhand, a completedindividual has shape, form,geometrical substance. Its visible expression admits continuousvariation. All these attributesare possessed by a manifoldin Euclidean space. So we shall think of our fracturingprocess for homotopytypes as a genetic analysis. We shall refer to the array of irreduciblepieces of in- formationin the homotopytype as the genotype. This is similarto the use of the termgenus in the theoryof quadratic forms. Thus the genotypeof a space is determinedby the rational type and profinitetype. The latter oftensplits completelyinto a collectionof p-adic types,one foreach prime. We cannot combinethese ingredientsarbitrarily to form a space. A certaincompatibility must persistamong the pieces. This coherencein the genotypeis the second piece of structurein our analysis of homotopytheory. Now the detailed description: Let Xdenote an "ordinaryhomotopy type". We shall thinkof Xas arising froma compactmanifold, an algebraicvariety, or one of the universalspaces associated to geometricproblems concerning these spaces. The homology groupsof these spaces are almost always finitelygenerated Abelian groups, and one oftenhas some analogous controlover the fundamentalgroup and the higherhomotopy groups. We shall associate to X a natural array of morebasic homotopytypes whichare not such "ordinaryhomotopy types" but which satisfyanalogous finitenessconditions over the rings - a Z, e Q: (b, 1) = 1) 1 In fact many of our constructionson spaces proceedby a replicationprocess using the Postnikovtower as a template. This content downloaded from 146.96.147.130 on Mon, 07 Dec 2015 16:50:45 UTC All use subject to JSTOR Terms and Conditions 4 DENNIS SULLIVAN or = Z Iimn,,j Z/nZ where {I} is the multiplicativesubset of Z generated by a collection I of primes. The two importantcases for us correspondto the ground rings Q and Z = lim,,Z/n. These more elemental spaces associated to X are constructedby the processesof localizationand completion. These processes mightbe thought of as analogous to laboratorytechniques which break a more complicated substance into simplercomponents. The firstprocess, localization at 1,is a kindof direct limit procedure which stripsaway the p-torsionand p-divisibilitystructure in the algebraictopology of the space for the primesp not in the collection1. The remaininginfor- mationin X is carriedby the associated space Xl to which X maps X ) Xl. This localization procedure is well-understoodif X has a principal Postnikovdecomposition; this happens preciselywhen each w,-moduleSEX, n> 1 has a finitefiltration ec kck( *cFk=r,,Xc with trivialaction on the successivequotients.' In this "nilpotentcase" the homologygroups of Xl are the localizationsof those of X: H*X--H*X ZZ, * > o . Similarly,the homotopygroups of Xl are the localizationsof those of X. More precisely,each r,,X,admits a finitefiltration so that the action of wrX,is trivialon the successivequotients. The map X - Xl inducesa filtra- tion preservingmap
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