Geometric Topology Localization, Periodicity, And Galois .
Geometric TopologyLocalization, Periodicity, and Galois Symmetry(The 1970 MIT notes)byDennis SullivanEdited by Andrew Ranicki
iiDennis SullivanDepartment of MathematicsThe Graduate CenterCity University of New York365 5th AveNew York, NY 10016-4309USAemail: dsullivan@gc.cuny.eduMathematics DepartmentStony Brook UniversityStony Brook, NY 11794-3651USAemail: dennis@math.sunysb.eduAndrew RanickiSchool of MathematicsUniversity of EdinburghKing’s BuildingsMayfield RoadEdinburgh EH9 3JZScotland, UKemail: a.ranicki@ed.ac.uk
ContentsEDITOR’S PREFACEviiPREFACEix1. ALGEBRAIC CONSTRUCTIONS12. HOMOTOPY THEORETICALLOCALIZATION313. COMPLETIONS IN HOMOTOPY THEORY514. SPHERICAL FIBRATIONS895. ALGEBRAIC GEOMETRY1136. THE GALOIS GROUP IN GEOMETRIC TOPOLOGY187REFERENCES241GALOIS SYMMETRY IN MANIFOLD THEORY ATTHE PRIMESReprint from Proc. 1970 Nice ICMPOSTSCRIPT (2004)251261v
Editor’s PrefaceThe seminal ‘MIT notes’ of Dennis Sullivan were issued in June1970 and were widely circulated at the time. The notes had a major influence on the development of both algebraic and geometrictopology, pioneeringthe localization and completion of spaces in homotopy theory,including p-local, profinite and rational homotopy theory, leading to the solution of the Adams conjecture on the relationshipbetween vector bundles and spherical fibrations,the formulation of the ‘Sullivan conjecture’ on the contractibilityof the space of maps from the classifying space of a finite groupto a finite dimensional CW complex,e ofthe action of the Galois group over Q of the algebraic closure QQ on smooth manifold structures in profinite homotopy theory,the K-theory orientation of P L manifolds and bundles.Some of this material has been already published by Sullivan himself: in an article1 in the Proceedings of the 1970 Nice ICM, andin the 1974 Annals of Mathematics papers Genetics of homotopytheory and the Adams conjecture and The transversality characteristic class and linking cycles in surgery theory2 . Many of the ideasoriginating in the notes have been the starting point of subsequent1 reprinted2 jointat the end of this volumewith John Morganvii
viiidevelopments3 . However, the text itself retains a unique flavour ofits time, and of the range of Sullivan’s ideas. As Wall wrote in section 17F Sullivan’s results of his book Surgery on compact manifolds(1971) : Also, it is difficult to summarise Sullivan’s work so briefly:the full philosophical exposition in (the notes) should be read. Thenotes were supposed to be Part I of a larger work; unfortunately,Part II was never written. The volume concludes with a Postscriptwritten by Sullivan in 2004, which sets the notes in the context ofhis entire mathematical oeuvre as well as some of his family life,bringing the story up to date.The notes have had a somewhat underground existence, as a kindof Western samizdat. Paradoxically, a Russian translation was published in the Soviet Union in 19754 , but this has long been out ofprint. As noted in Mathematical Reviews, the translation does notinclude the jokes and other irrelevant material that enlivened theEnglish edition. The current edition is a faithful reproduction ofthe original, except that some minor errors have been corrected.The notes were TeX’ed by Iain Rendall, who also redrew all thediagrams using METAPOST. The 1970 Nice ICM article was TeX’edby Karen Duhart. Pete Bousfield and Guido Mislin helped preparethe bibliography, which lists the most important books and papersin the last 35 years bearing witness to the enduring influence of thenotes. Martin Crossley did some preliminary proofreading, whichwas completed by Greg Brumfiel (“ein Mann der ersten Stunde”5 ).Dennis Sullivan himself has supported the preparation of this editionvia his Albert Einstein Chair in Science at CUNY. I am very gratefulto all the above for their help.Andrew RanickiEdinburgh, October, 20043 For example, my own work on the algebraic L-theory orientations of topological manifoldsand bundles.4 The picture of an infinite mapping telescope on page 34 is a rendering of the picture in theRussian edition.5 A man of the first hour.
PrefaceThis compulsion to localize began with the author’s work on invariants of combinatorial manifolds in 1965-67. It was clear from thebeginning that the prime 2 and the odd primes had to be treateddifferently.This point arises algebraically when one looks at the invariants ofa quadratic form1 . (Actually for manifolds only characteristic 2 andcharacteristic zero invariants are considered.)The point arises geometrically when one tries to see the extent ofthese invariants. In this regard the question of representing cyclesby submanifolds comes up. At 2 every class is representable. At oddprimes there are many obstructions. (Thom).The invariants at odd primes required more investigation becauseof the simple non-representing fact about cycles. The natural invariant is the signature invariant of M – the function which assigns the“signature of the intersection with M ” to every closed submanifoldof a tubular neighborhood of M in Euclidean space.A natural algebraic formulation of this invariant is that of a canonical K-theory orientation4M K-homology of M .1 Whichaccording to Winkelnkemper “. is the basic discretization of a compact manifold.”ix
xIn Chapter 6 we discuss this situation in the dual context of bundles. This (Alexander) duality between manifold theory and bundletheory depends on transversality and the geometric technique ofsurgery. The duality is sharp in the simply connected context.Thus in this work we treat only the dual bundle theory – howevermotivated by questions about manifolds.The bundle theory is homotopy theoretical and amenable to thearithmetic discussions in the first Chapters. This discussion concerns the problem of “tensoring homotopy theory” with variousrings. Most notable are the cases when Z is replaced by the rationals Q or the p-adic integers Ẑp .These localization processes are motivated in part by the ‘invariants discussion’ above. The geometric questions do not howevermotivate going as far as the p-adic integers.2One is led here by Adams’ work on fibre homotopy equivalencesbetween vector bundles – which is certainly germane to the manifoldquestions above. Adams finds that a certain basic homotopy relationshould hold between vector bundles related by his famous operationsψk .Adams proves that this relation is universal (if it holds at all) –a very provocative state of affairs.Actually Adams states infinitely many relations – one for eachprime p. Each relation has information at every prime not equal top.At this point Quillen noticed that the Adams conjecture has ananalogue in characteristic p which is immediately provable. He suggested that the etale homotopy of mod p algebraic varieties be usedto decide the topological Adams conjecture.Meanwhile, the Adams conjecture for vector bundles was seen toinfluence the structure of piecewise linear and topological theories.The author tried to find some topological or geometric understanding of Adams’ phenomenon. What resulted was a reformulation which can be proved just using the existence of an algebraic2 Althoughthe Hasse-Minkowski theorem on quadratic forms should do this.
xiconstruction of the finite cohomology of an algebraic variety (etaletheory).This picture which can only be described in the context of thep-adic integers is the following – in the p-adic context the theory ofvector bundles in each dimension has a natural group of symmetries.These symmetries in the (n 1) dimensional theory provide canonical fibre homotopy equivalence in the n dimensional theory whichmore than prove the assertion of Adams. In fact each orbit of theaction has a well defined (unstable) fibre homotopy type.The symmetry in these vector bundle theories is the Galois symmetry of the roots of unity homotopy theoretically realized in the‘Čech nerves’ of algebraic coverings of Grassmannians.The symmetry extends to K-theory and a dense subset of the symmetries may be identified with the “isomorphic part of the Adamsoperations”. We note however that this identification is not essentialin the development of consequences of the Galois phenomena. Thefact that certain complicated expressions in exterior powers of vector bundles give good operations in K-theory is more a testament toAdams’ ingenuity than to the ultimate naturality of this viewpoint.The Galois symmetry (because of the K-theory formulation ofthe signature invariant) extends to combinatorial theory and eventopological theory (because of the triangulation theorems of KirbySiebenmann). This symmetry can be combined with the periodicityof geometric topology to extend Adams’ program in several ways –i) the homotopy relation implied by conjugacy under the actionof the Galois group holds in the topological theory and is alsouniversal there.ii) an explicit calculation of the effect of the Galois group on thetopology can be made –for vector bundles E the signature invariant has an analyticaldescription,4E in KC (E) ,and the topological type of E is measured by the effect of theGalois group on this invariant.
xiiOne consequence is that two different vector bundles which arefixed by elements of finite order in the Galois group are also topologically distinct. For example, at the prime 3 the torsion subgroup isgenerated by complex conjugation – thus any pair of non isomorphicvector bundles are topologically distinct at 3.The periodicity alluded to is that in the theory of fibre homotopyequivalences between PL or topological bundles (see Chapter 6 Normal Invariants).For odd primes this theory is isomorphic to K-theory, and geometric periodicity becomes Bott periodicity. (For non-simply connectedmanifolds the periodicity finds beautiful algebraic expression in thesurgery groups of C. T. C. Wall.)To carry out the discussion of Chapter 6 we need the works of thefirst five chapters.The main points are contained in chapters 3 and 5.In chapter 3 a description of the p-adic completion of a homotopytype is given. The resulting object is a homotopy type with theextra structure3 of a compact topology on the contravariant functorit determines.The p-adic types one for each p can be combined with a rationalhomotopy type (Chapter 2) to build a classical homotopy type.One point about these p-adic types is that they often have symmetry which is not apparent or does not exist in the classical context. For example in Chapter 4 where p-adic spherical fibrations arediscussed, we find from the extra symmetry in C P , p-adically completed, one can construct a theory of principal spherical fibrations(one for each divisor of p 1).Another point about p-adic homotopy types is that they can benaturally constructed from the Grothendieck theory of etale cohomology in algebraic geometry. The long chapter 5 concerns thisetale theory which we explicate using the Čech like construction ofLubkin. This construction has geometric appeal and content andshould yield many applications in geometric homotopy theory.43 whichis “intrinsic” to the homotopy type in the sense of interest here.study of homotopy theory that has geometric significance by geometrical qua homotopytheoretical methods.4 The
xiiiTo form these p-adic homotopy types we use the inverse limittechnique of Chapter 3. The arithmetic square of Chapter 3 showswhat has to be added to the etale homotopy type to give the classicalhomotopy type.5We consider the Galois symmetry in vector bundle theory in somedetail and end with an attempt to analyze “real varieties”. Theattempt leads to an interesting topological conjecture.Chapter 1 gives some algebraic background and preparation forthe later Chapters. It contains the examples of profinite groups intopology and algebra that concern us here.In part II6 we study the prime 2 and try to interpret geometricallythe structure in Chapter 6 on the manifold level. We will also pursuethe idea of a localized manifold – a concept which has interestingexamples from algebra and geometry.Finally, we acknowledge our debt to John Morgan of PrincetonUniversity – who mastered the lion’s share of material in a few shortmonths with one lecture of suggestions. He prepared an earlier manuscript on the beginning Chapters and I am certain this manuscriptwould not have appeared now (or in the recent future) without hisconsiderable efforts.Also, the calculations of Greg Brumfiel were psychologically invaluable in the beginning of this work. I greatly enjoyed and benefited from our conversations at Princeton in 1967 and later.5 Actually6 whichit is a beginning.was never written (AAR).
Chapter 1ALGEBRAIC CONSTRUCTIONSWe will discuss some algebraic constructions. These are localization and completion of rings and groups. We consider properties ofeach and some connections between them.LocalizationUnless otherwise stated rings will have units and be integral domains.Let R be a ring. S R {0} is a multiplicative subset if 1 Sand a, b S implies a · b S.Definition 1.1 If S R {0} is a multiplicative subset thenS 1 R , “R localized away from S”is defined as equivalence classes{r/s r R, s S}wherer/s r0 /s0 iff rs0 r0 s .1
2S 1 R is made into a ring by defining[r/s] · [r0 /s0 ] [rr0 /ss0 ] and· 0 rs sr000[r/s] [r /s ] .ssThe localization homomorphismR S 1 Rsends r into [r/1].Example 1 If p R is a prime ideal, R p is a multiplicative subset.DefineRp , “R localized at p”as (R p) 1 R.In Rp every element outside pRp is invertible. The localizationmap R Rp sends p into the unique maximal ideal of non-units inRp .If R is an integral domain 0 is a prime ideal, and R localized atzero is the field of quotients of R.The localization of the ring R extends to the theory of modulesover R. If M is an R-module, define the localized S 1 R-module,S 1 M byS 1 M M R S 1 R .Intuitively S 1 M is obtained by making all the operations on Mby elements of S into isomorphisms.Interesting examples occur in topology.Example 2 (P. A. Smith, A. Borel, G. Segal) Let X be a locallycompact polyhedron with a symmetry of order 2 (involution), T .What is the relation between the homology of the subcomplex offixed points F and the “homology of the pair (X, T )”?Let S denote the (contractible) infinite dimensional sphere withits antipodal involution. Then X S has the diagonal fixed pointfree involution and there is an equivariant homotopy class of mapsX S S
3Algebraic Constructions(which is unique up to equivariant homotopy). This gives a mapXT (X S)/T S/T R P and makes the “equivariant cohomology of (X, T )”H (XT ; Z/2)into an R-module, whereR Z2 [x] H (R P ; Z/2) .In R we have the multiplicative set S generated by x, and the cohomology of the fixed points with coefficients in the ring S 1 R Rx R[x 1 ] is just the localized equivariant cohomology,H (F ; Rx ) H (XT ; Z/2) with x inverted H (XT ; Z/2) R Rx .For most of our work we do not need this general situation oflocalization. We will consider most often the case where R is thering of integers and the R-modules are arbitrary Abelian groups.Let be a set of primes in Z. We will write “Z localized at ”Z S 1 Zwhere S is the multiplicative set generated by the primes not in .When contains only one prime {p}, we can writeZ Zpsince Z is just the localization of the integers at the prime ideal p.Other examples areZ{all primes} Z and Z Q Z0 .In general, it is easy to see that the collection of Z ’s{Z }is just the collection of subrings of Q with unit. We will see belowthat the tensor product over Z,Z Z Z 0 Z 0
4and the fibre product over QZ Q Z 0 Z 0 .We localize Abelian groups at as indicated above.Definition 1.2 If G is an Abelian group then the localization of Gwith respect to a set of primes , G is the Z -moduleG Z .The natural inclusion Z Z induces the “localization homomorphism”G G .We can describe localization as a direct limit procedure.Order the multiplicative set {s} of products of primes not in bydivisibility. Form a directed system of groups and homomorphismsindexed by the directed set {s} withmultiplication by s0 /sGs G Gs0 if s 6 s0 .Proposition 1.1lim Gs G Z G . sProof: Define compatible mapsGs G Z by g 7 g 1/s. These determinelim Gs G Z . sIn case G Z this map is clearly an isomorphism. (Each mapZ Z is an injection thus the direct limit injects. Also a/s in Z isin the image of Z Gs Z .)The general case follows since taking direct limits commutes withtensor products.Lemma 1.2 If and 0 are two sets of primes, then Z Z 0 is isomorphic to Z 0 as rings.
5Algebraic ConstructionsProof: Define a map on generatorsρZ Z 0 Z 0by ρ(a/b a0 /b0 ) aa0 /bb0 . Since b is a product of primes outside and b0 is a product of primes outside 0 , bb0 is a product of primesoutside 0 and ρ is well defined.To see that ρ is onto, take r/s in Z 0 and factor s s1 s2 so that“s1 is outside ” and “s2 is outside 0 .” Then ρ(1/s1 r/s2 ) r/s.To see that ρ is an embedding assumeXρai /bi ci /di 0.iThenXai ci /bi di 0, oriXai ciiYbj dj 0 .i6 jThis means thatXXai /bi ci /di ai ci (1/bi 1/di )ii X¡YiY ¡ Ydhbh 1/bj dj ai ci 1/i6 jhh 0so ρ has kernel {0}.Lemma 1.3 The Z-module structure on an Abelian group G extendsto a Z -module structure if and only if G is isomorphic to its localizations at every set of primes containing .Proof: This follows from Proposition 1.1.Example 3½0p / (Z/p ) Z/p Z nZ/p p nn³ finitely generated Z · · · Z Z -torsion GAbelian group G {z } rank G factors
6Proposition 1.4 Localization takes exact sequences of Abelian groupsinto exact sequences of Abelian groups.Proof: This also follows from Proposition 1.1 since passage to adirect limit preserves exactness.Corollary 1.5 If 0 A B C 0 is an exact sequence ofAbelian groups and two of the three groups are Z -modules then sois the third.Proof: Consider the localization diagram00/A²/B Z ² Z / B / A /C²/0 Z / C /0The lower sequence is exact by Proposition 1.4. By hypothesis andLemma 1.2 two of the maps are isomorphisms. By the Five Lemmathe third is also.Corollary 1.6 If in the long exact sequence· · · An Bn Cn An 1 Bn 1 . . .two of the three sets of groups{An }, {Bn }, {Cn }are Z -modules, then so is the third.Proof: Apply the Five Lemma as above.Corollary 1.7 Let F E B be a Serre fibration of connectedspaces with Abelian fundamental groups. Then if two ofπ F, π E, π Bare Z -modules the third is also.Proof: This follows from the exact homotopy sequence· · · πi F πi E πi B . . . .This situation extends easily to homology.Proposition 1.8 Let F E B be a Serre fibration in which π1 B
7Algebraic Constructionse (F ; Z/p) for primes p not in . Then if two ofacts trivially on Hthe integrale F, He E, He BHare Z -modules, the third is also.e X is a Z -module iff He (X; Z/p) vanishes for p not in .Proof: HThis follows from the exact sequence of coefficientspe i (X) e i (X) He i (X; Z/p) . . . .··· H HBut from the Serre spectral sequence with Z/p coefficients we canconclude that if two ofe (F ; Z/p), He (E; Z/p), He (B; Z/p)Hvanish the third does also.Note: We are indebted to D. W. Anderson for this very simpleproof of Proposition 1.8.Let us say that a square of Abelian groupsAi/Bj²C²kl/Dis a fibre square if the sequencei jl k0 A B C D 0is exact.Lemma 1.9 The direct limit of fibre squares is a fibre square.Proof: The direct limit of exact sequences is an exact sequence.Proposition 1.10 If G is any Abelian group and and 0 are twosets of primes such that 0 , 0 all primesthenG/ G Z ²²/G QG Z 0
8is a fibre square.Proof:Case 1: G Z: an easy argument shows0 Z Z Z 0 Q 0is exact.Case 2: G Z/pα , the square reduces toZ/pα/0²²/0Z/pαorZ/pα ²/ Z/pα²0/ .0Case 3: G is a finitely generated group: this is a finite direct sum ofthe first two cases.Case 4: G any Abelian group: this follows from case 3 and Lemma1.9.We can paraphrase the proposition “G is the fibre product of itslocalizations G and G 0 over G0 ,”More generally, we haveMeta Proposition 1.12 Form the infinite diagramG2 BG3G5 . . .BBxxBBxxBBxB! ² {xxxG0Then G is the infinite fibre product of its localizations G2 , G3 ,. . . over G0 .Proof: The previous proposition shows G(2,3) is the fibre productof G(2) and G(3) over G(0) . Then G(2,3,5) is the fibre product of G(2,3)and G(5) over G(0) , etc. This description depends on ordering theprimes; however since the particular ordering used is immaterial thestatement should be regarded symmetrically.
9Algebraic ConstructionsCompletionsWe turn now to completion of rings and groups. As for ringswe are again concerned mostly with the ring of integers for whichwe discuss the “arithmetic completions”. In the case of groups weconsider profinite completions and for Abelian groups related formalcompletions.At the end of the Chapter we consider some examples of profinitegroups in topology and algebra and discuss the structure of the padic units.Finally we consider connections between localizations and completions, deriving certain fibre squares which occur later on the CWcomplex level.Completion of Rings – the p-adic IntegersLet R be a ring with unit. LetI1 I2 . . .be a decreasing sequence of ideals in R with \Ij {0} .j 1We can use these ideals to define a metric on R, namelyd(x, y) e k , e 1where x y Ik but x y 6 Ik 1 , (I0 R). If x y Ik and y z Ilthen x z Imin(k, l) . Thus¡ d(x, z) 6 max d(x, y), d(y, z) ,a strong form of triangle inequality. Also, d(x, y) 0 meansx y \Ij {0} .j 0This means that d defines a distance function on the ring R.
10Definition 1.3 Given a ring with metric d, define the completionbd , by the Cauchy sequence procedure. Thatof R with respect to d, Ris, form all sequences in R, {xn }, so that1lim d(xn , xm ) 0 .n,m Make {xn } equivalent to {yn } if d(xn , yn ) 0. Then the set ofbd is made into a topological ring by definingequivalence classes R[{xn }] [{yn }] [{xn yn }] ,[{xn }] · [{yn }] [{xn yn }] .There is a natural completion homomorphismc bR Rdsending r into [{r, r, . . . }]. c is universal with respect to continuousring maps into complete topological rings.Example 1 Let Ij (pj ) Z. The induced topology is the p-adicbp.topology on Z, and the completion is the ring of p-adic integers, Zb p was constructed by Hensel to study Diophantine equaThe ring Zb p corresponds to solving the associated Diotions. A solution in Zphantine congruence modulo arbitrarily high powers of p.Solving such congruences for all moduli becomes equivalent to aninfinite number of independent problems over the various rings ofp-adic numbers.bp,Certain non-trivial polynomials can be completely factored in Zfor examplexp 1 1(see the proof of Proposition 1.16.)Thus here and in other situations we are faced with the pleasant possibility of studying independent p-adic projections of familiarproblems over Z armed with such additional tools as (p 1)st rootsof unity.1 Inthis context it is sufficient to assume that d(xn , xn 1 ) 0 to have a Cauchy sequence.
11Algebraic ConstructionsExample 2 Let be a non-void subset of the primes (p1 , p2 , . . . ) (2, 3, . . . ). DefineYIj ( pj )p ,p6pj .The resulting topology on Z is the -adic topology and the compleb .tion is denoted Z0If 0 then Ij Ij and any Cauchy sequence in the -adictopology is Cauchy in the 0 -adic topology. This gives a mapb Zb 0 .ZProposition 1.13 Form the inverse system of rings {Z/pn }, whereZ/pn Z/pm is a reduction mod pm whenever n m. Then thereis a natural ring isomorphismρbpbp Z lim{Z/pn } . Proof: First define a ring homomorphismb p Z/pn .ZρnIf {xi } is a Cauchy sequence in Z, the pn residue of xi is constantfor large i so defineρbn [{xi }] stable residue xi .If {xi } is equivalent to {yi }, pn eventually divides every xi yi , soρn is well defined.The collection of homomorphisms ρn are clearly onto and compatible with the maps in the inverse system. Thus they defineb p lim Z/pn .Zρbp ρbp is injective. For ρbp {xi } 0, means pn eventually divides xi forall n. Thus {xi } is eventually in In for every n. This is exactly thecondition that {xi } is equivalent to {0, 0, 0, . . . }.ρbp is surjective. If (ri ) is a compatible sequence of residues inlim Z/pn , let {eri } be a sequence of integers in this sequence of residue classes. {eri } is clearly a Cauchy sequence andρbp {eri } (ri ) lim Z/pn
12b p is compact.Corollary ZProof: The isomorphism ρbp is a homeomorphism with respect tothe inverse limit topology on lim Z/pn . b Zbp, p Proposition 1.14 The product of the natural maps Zyields an isomorphism of ringsYb bp .Z Z p b is an inProof: The argument of Proposition 1.13 shows that Zverse limit of finite -ringsZ/Ij .Butsincelim Z/Ij jZ/Ij YbZ/pp YZ/pj .p ,p6pjb is a ring with unit, but unlike Zb p it is not an integralNote: Zbp, Zb is compactdomain if contains more than one prime. Like Zand topologically cyclic – the multiples of one element can form adense set.¡ Example 3 K(R P ), AtiyahLet R be the ring of virtual complex representations of Z/2,R Z [x]/(x2 1) .n mx corresponds to the representation 11 11 n 1. . 1{1, 0} , 1 m . . 1 . .of Z/2 on Cn m . Let Ij be the ideal generated by (x 1)j . Thecompletion of the representation ring R with respect to this topology1
Algebraic Constructions13is naturally isomorphic to the complex K-theory of R P ,b R K(R P ) [R P , Z BU ] .It is easy to see that additively this completion of R is isomorphicto the integers direct sum the 2-adic integers.Example 4 (K(fixed point set), Atiyah and Segal) Consider againthe compact space X with involution T , fixed point set F , and ‘homotopy theoretical orbit space’, XT X S /((x, s) (T x, s)).We have the Grothendieck ring of equivariant vector bundles overX, KG (X) – a ring over the representation ring R. KG (X) is arather subtle invariant of the geometry of (X, T ). However, Atiyahand Segal show thati) the completion of KG (X) with respect to the ideals (x 1)j KG (X)is the K-theory of XT .ii) the completion of KG (X) with respect to the ideals (x 1)j KG (X)is related to K(F ).If we complete KG (X) with respect to the ideals (x 1, x 1)j KG (X)(which is equivalent to 2-adic completion)2 we obtain the isomorphismb 2 [x]/(x2 1) K(F ) Z K(XT )ˆ2 .We will use this relation in Chapter 5 to give an ‘algebraic description’ of the K-theory3 of the real points on a real algebraic variety.Completions of GroupsNow we consider two kinds of completions for groups. First thereare the profinite completions.Let G be any group and a non-void set of primes in Z. Denotethe collection of those normal subgroups of G with index a productof primes in by {H} .2 (x 1, x 1)2 (2) (x 1, x 1).with the group ring of Z/2 over the 2-adic integers.3 Tensored
14Now {H} can be partially ordered byH1 6 H2 iff H1 H2 .Definition 1.4 The -profinite completion of G is the inverse limitof the canonical finite -quotients of G –b lim (G/H) .G {H} b is topologized by the inverse limitThe -profinite completion Gb becomes a totallyof the discrete topologies on the G/H’s. Thus Gdisconnected compact topological group.The natural mapb G Gis clearly universal4 for maps of G into finite -groups.This construction is functorial because the diagram/ H0H f 1 H 0²Gf²G/H²/ G0²/ G0 /H 0f shows f induces a map of inverse systems –{H 0 } {f 1 H 0 } {H} f G/H G0 /H 0 , H f 1 H 0and thus we have fbbfb lim G/H b0 .G lim G0 /H 0 G 4 There is a unique continuous map of the completion extending over a given map of G intoa finite group.
15Algebraic ConstructionsExamples1) Let G Z, {p}. Then the only p-quotients are Z/pn . Thusp-profinite completionof the group G lim Z/pn nwhich agrees (additively) with the ring theoretic p-adic completion of Z, the “p-adic integers”.2) Again G Z, {p1 , p2 , . . . }. ThenZ lim Z/pα1 1 . . . pαi i αwhereα {(α1 , α2 , . . . , αi , 0, 0, 0, . . . )}is the set of all non-negative exponents (eventually zero) partiallyordered byα 6 α0 if αi 6 αi0 for all i .The cofinality of the sequenceαk (k, k, . . . , k , 0, 0, 0, . . . ) {z }k placesshows thatkYbZ limZ/pki k i 0Y lim Z/pk p kYbp Zp 3) For any Abelian group Gb G Ybp .Gp 4) The -profinite completion of a finitely generated Abelian groupof rank n and torsion subgroup T is justb b ··· Zb -torsion T .G Z Z Z} {zn summands
165) If G is -divisible, then -profinite completion reduces G to thetrivial group. For example,b 0, (Q / Z) 0 .Q6) The p-profinite completion of the infinite direct sum Ythe infinite direct productZ/p. MZ/p isFrom the examples we see that profinite completion is exact forfinitely generated Abelian groups but is not exact in general, e.g.0 Z Q Q/Z 0becomesb 0 0 0 .0 ZIf we wanted a construction related to profinite completion whichpreserved exactness for non-finitely generated groups, we could simply make theDefinition 1.5 The formal -completion of an Abelian group G, Ḡ is given byb .Ḡ G ZProposition 1.15 The functor G Ḡ is exact. It is the uniquefunctor which agrees with the profinite completion for finitely generated groups and commutes with direct limits.b is torsion free. The secondProof: The first part follows since Zpart follows fromi) any group is the direct limit of its finitely generated subgroupsii) tensoring commutes with direct limits.b is called “the profinite completion” ofIf is {all primes} then Gb Ḡ is the “formal completion” of G and denotedG and denoted G.bḠ. T
The periodicity alluded to is that in the theory of flbre homotopy equivalences between PL or topological bundles (see Chapter 6 - Normal Invariants). For odd primes this theory is isomorphic to K-theory, and geomet-ric periodicity becomes Bott periodicity. (For non-simply connected manifolds the
The formula for the sum of a geometric series can also be written as Sn a 1 1 nr 1 r. A geometric series is the indicated sum of the terms of a geometric sequence. The lists below show some examples of geometric sequences and their corresponding series. Geometric Sequence Geometric Series 3, 9, 27, 81, 243 3 9 27 81 243 16, 4, 1, 1 4, 1 1 6 16 .
Localization processes and best practices will be examined from the perspective of Web developers and translators, and with these considerations in mind, an online localization management tool called Localize1will be evaluated. The process of localization According to Miguel Jiménez-Crespo (2013, 29-31) in his study of Web localization, the
es the major management issues that are key to localization success and serves as a useful reference as you evolve in your role as Localization Manager. We hope that it makes your job easier and furthers your ability to manage complex localization projects. If the Guide to Localization Management enables you to manage localiza-
quires a geometric topology approach to graphs, which is the main topic of this book. The essential point of the geometric topology of graphs is that, if the graph is viewed as a transport medium (transport of packets, heat diffusion, spin & entanglement transport, etc.), its relevant properties are encapsulated in the concept of curvature.
been proven to be stable and effective and could significantly improve the geometric accuracy of optical satellite imagery. 2. Geometric Calibration Model and the Method of Calculation 2.1. Rigorous Geometric Imaging Model Establishment of a rigorous geometric imaging model is the first step of on-orbit geometric calibration
The first term in a geometric sequence is 54, and the 5th term is 2 3. Find an explicit form for the geometric sequence. 19. If 2, , , 54 forms a geometric sequence, find the values of and . 20. Find the explicit form B( J) of a geometric sequence if B(3) B(1) 48 and Ù(3) Ù(1) 9. Lesson 4: Geometric Sequences Unit 7: Sequences S.41
The first term in a geometric sequence is 54, and the 5th term is 2 3. Find an explicit form for the geometric sequence. 19. If 2, , , 54 forms a geometric sequence, find the values of and . 20. Find the explicit form B( J) of a geometric sequence if B(3) B(1) 48 and Ù(3) Ù(1) 9. Lesson 7: Geometric Sequences
3. Wenger-Clemons, J. (2014). Client system assessment tools for social work practice (Canvas). Group Assignment s . 1. Therapeutic Group Work Assignment 2. Therapeutic Group Work Twitter Feed Individual Assignment 1. M2 Quiz . Module 3: Identity-Based Social Action Group Work . Weeks 5 and 6 . Overview . A. This module reviews a number of topics around how we construct self-concepts and self .
