Rational algebraic K-theory of topological K-theory

RA TIONAL ALGEBRAIC K-THEOR Y OF TOPOLOGICAL K-THEOR Y Christian A usoni and John R ognes August 14th 2007 A bs tr act . W e sho w that after rationalization there is a homotop y fib er sequence B B U ⊗ → K ( ku ) → K ( Z ). W e interpret this as a correspondence be t ween the virtual 2-v ector bundles o ver a space X and their asso ciated anomaly bundle s o ver the free lo op space L X . W e also rati onally compute K ( K U ) by using the lo calization sequence, and K ( M U ) b y a metho d tha t applies to all connective S -algebras. Intr oduction W e are in terested in t he algebraic K -theory K ( k u ) of the connective complex K - theory sp ectrum k u . By the calculations o f [AR02, Thm. 0 . 4] and [Au], the “mo d p and v 1 ” homotop y of K ( k u ) is purely v 2 -p erio dic, a distinctive homotopy theoretic prop ert y it shares with the sp ectra represen ting elli ptic cohomolog y [LRS95] and top ological mo dular forms [ Ho02, § 4]. The theory of 2-vector bundles from [BDR04] and [B DRR] therefore ex hi bi t s K ( k u ) as a geometrically defined form of ellipti c cohomology . In Section 5 w e outline how a 2-v ector bund le with connecting d ata, similar to a connection in a vector bundle, is thought to sp ecify a 1 + 1-d imensional conformal field theory . Since t hese 2-v ector bundles are a l so effective cycles for the form of elli pti c cohomology theory represen ted b y K ( k u ) , w e ha ve some justification for referring to them as ell iptic ob jects, as prop osed b y Segal [Se89]. As illustrated by the authors’ calculations referred to ab ov e, the arithmetic and homotop y-theoretic information captured by a l gebraic K -theory b ecomes more ac- cessible after t he i ntro duction o f suita bl e finite co efficien ts. How ev er, for the ex- traction of C -v al ued n umerical i n v arian ts from a conformal field theory , only the rational homotop y type of K ( ku ) will mat t er. W e are g rateful to Ib Madsen and Dennis Sulliv an for insisting that for suc h geometri c applicati o ns, we should first w an t to compute K ( k u ) rationall y . T o this end we can offer the follo wing t heorem. There is a unit i nclusion ma p w : B B U ⊗ → B GL 1 ( k u ) → B GL ∞ ( k u ) + → K ( k u ) . Let π : K ( k u ) → K ( Z ) b e induced by the zero-th P ostniko v section k u → H Z . The comp osite π ◦ w is the constan t map to the base-po in t o f t he 1-comp onen t of K ( Z ). T yp eset by A M S -T E X 1 2 CHRISTIA N AUSONI AND JOHN R OGNES Theorem 0.1. (a) After r ationalization, B B U ⊗ w − → K ( k u ) π − → K ( Z ) is a split homotopy fib er se quenc e. (b) The Poinc ar ´ e series of K ( k u ) is 1 + t 3 1 − t 2 + t 5 1 − t 4 = 1 + t 3 + 2 t 5 1 − t 4 . (c) Ther e is a r ational deter mi na nt map det Q : B GL ∞ ( k u ) + → B GL 1 ( k u ) Q that, in its r e lative form for k u → H Z , r ational ly splits w . By the Poincar ´ e series of a space X of finite type, we mean the formal p o w er series P n ≥ 0 r n t n in Z [[ t ]], where r n is the ra nk of π n ( X ). Theorem 0.1 i s prov ed b y assem bling Theorem 2.5( a) and Theorem 4.8(a). The other parts of t hose theorems pro v e simil ar results f or K ( k o ) and K ( ℓ ), where ℓ = B P h 1 i . The splitting of K ( k u ) Q sho ws that for a (vi rtual) 2-vector bund le ov er X , rep- resen ted by a map E : X → K ( k u ), the rational information split s in to t w o pieces. The less in teresting piece is the decategorified information carried by the dimension bundle dim( E ) = π ◦ E : X → K ( Z ). The more in teresting piece is the determinant bundle |E | = det Q ◦ E : X → ( B B U ⊗ ) Q . T o sp ecify |E | is eq ui v alen t to sp ecifying a rational virtual v ector bundle H : L X → ( B U ⊗ ) Q o v er t he free lo op space L X = Map( S 1 , X ), called t he “anomaly bun- dle”, sub ject to a coherence condition relating the comp osition of free lo ops, when defined, to the tensor pro duct of vi r t ual vector spaces. See diagram ( 5.3). The conclus ion is that for rational purposes the information in a 2-v ector bu n- dle E ov er X is the same a s that in its anomaly bundle H ov er L X (sub ject to the indicated coherence condition, which we think of as implicit , a nd tog ether with the dimension bundle dim( E ) o v er X , which w e tend to ignore). In ph ysical language, the fib er of the anomaly bundle at a free lo op γ : S 1 → X plays t he role of the state space of γ viewed as a closed string in X . The adv antage of 2-v ector bundles o v er their homot o py-theoretic alternatives, suc h a s represen ting maps to classifying spaces or bundles of k u -mo dules, is t hat they are geometri cally mo deled in t erms o f v ector bundles, rather than virtual vec tor bundles. This seems to b ecome an essen- tial virtue when one wan ts to treat differen tial-geometri c structures li k e connections on these bundles. W e also compute t he rational algebraic K -theory K ( K U ) of the p erio dic complex K -theory sp ectrum K U . T o this end w e ev aluate the (rationali zed) transfer map π ∗ in the lo cali zat ion sequence K ( Z ) π ∗ − → K ( k u ) ρ − → K ( K U ) predicted by the second author and es tablished by Blum b erg and Mandell [BM]. RA TIONAL ALGEBRAIC K-THEOR Y OF TOPOLO GICA L K-THEOR Y 3 Theorem 0.2. (a) Ther e is a r ational ly split homotopy fib er se quenc e of infini te lo op sp ac es K ( k u ) ρ − → K ( K U ) ∂ − → B K ( Z ) wher e ρ is induc e d by the c onne ctive c over map k u → K U , and B K ( Z ) denotes the first c onne cte d delo oping of K ( Z ) . (b) The Poinc ar ´ e series of K ( K U ) i s (1 + t ) + t 3 + 2 t 5 + t 6 1 − t 4 . See Theorem 2.12 for our pro of. As stated, Theorems 0.1 and 0.2 o nl y concern the algebraic K -theory of t opo - logical K -theory , but we dev elop our pro ofs in the greater generality of ar bi t rary connectiv e S -al g ebras. In Sec tion 1 w e observe how the calculation b y Go o dwill ie [Go86] of the relati v e rational algebraic K -theory for a 1-connected map R → π 0 R of simplicial rings (whic h generalizes earlier cal culations by Hsi a ng and Staffel dt [HS82] for simplici a l group rings), also applies t o determine the relative rational algebraic K -theory for a 1-connected map A → H π 0 A of connective S -algebras. The answ er i s given i n terms of negative cyclic homology; see T heorem 1. 5 and Corollary 1.6. When π 0 A is close to Z , a nd A → H π 0 A is a “rational de Rham equiv alence”, w e get a ve ry simple expression for t he relative r a tional algebraic K -theory as the image of Connes’ B -op erator on Ho chs c hild homology ; see P r o posit ion 1.8 and Corollary 1.9. These h yp otheses a pply to a n um b er of interes ting examples of con- nectiv e S -algebras, including the K -theory sp ectra ku , k o and ℓ , and the bordi sm sp ectra M U , M S O and M S p . W e work these ex a mples out i n Theorem 2.5 and Theorem 3.4, resp ectiv ely . In Section 4 we consider the unit inclusion map w : B GL 1 ( A ) → K ( A ). F or comm utative A , the rat ionalization A Q is equiv alen t as a comm utative H Q -algebra to the Eilen b erg–Mac Lane sp ectrum H R of a commu tative simplici a l Q -algebra, so we can use the determinan t GL n ( R ) → GL 1 ( R ) to define a rational determinan t map det Q : B GL ∞ ( A ) + → B GL 1 ( A ) Q . W e sho w in Prop osition 4.7 that the compo site det Q ◦ w is the rati o nal ization map, and apply this in Theorem 4.8 t o sho w that w induc es a rat ional eq uiv alence from B B U ⊗ to the homotop y fib er of π : K ( k u ) → K ( Z ), and similarly for k o and ℓ . This last step is a counting argumen t; it do es not apply for M U or the other b ordism sp ectra. Ac kno wledgments. In an earl ier version of this pap er, w e emphasized a trace map to T H H ( k u ) o v er the determinan t map to ( B B U ⊗ ) Q , in order to detect the image of w i n K ( k u ). W e are grateful to Bjørn Dundas for reminding us of the existence of determinants for comm utative simplicial rings, which is half of the basis for t he existence of the map det Q defined in Lemma 4 .6. W e are also grateful to Mik e Mand ell and Bro oke Shipley for help wi th some of the refe rences concerning comm utative simplici al Q -algebras giv en in Subsection 1.1. 4 CHRISTIA N AUSONI AND JOHN R OGNES § 1. Ra tional algebraic K - theor y of connective S -algebras 1.1. S -algeb ras. W e work in one of the mo dern symmetric monoidal categori es of sp ectra [ E KMM97], [Ly99] , [H SS00] , [MMSS01], whic h w e shall refer to as S - mo dules. The monoids (resp. commutativ e monoids) i n t his categor y are called S -algebras (resp. commutativ e S -algebras), and are equiv al en t to the A ∞ ring sp ec- tra (resp. E ∞ ring sp ectra) considered since the 1970 ’ s. Th e Ei len berg–Mac Lane functor H : R 7→ H R maps the catego ry of simplicial rings (resp. commutativ e simplicial rings) to the category of S -algebras (resp. comm utativ e S -algebras). Sc h w ede pro v ed in [Sc h w99, 4.5] that H is part o f a Quillen equiv alence from the category of simplicial rings to the category of connec tive H Z -algebras. T here is a simi l ar eq uiv alence b et we en the category of comm utative simplicial Q -algebras and the category of connective comm utativ e H Q -algebras. One form of the l atter equiv alence app ears in [KM95, I I.1.3]. In a littl e more detail, the category of connectiv e commutativ e H Q -algebras i s “connectiv e Quillen equiv al en t” [MMSS01, p. 445] to the category of connectiv e E ∞ H Q -ring sp ec- tra [E KMM97, I I.4], and connectiv e E ∞ H Q -ring sp ectra are the E ∞ ob jects in connectiv e H Q -mo dules, whic h are Quillen equiv al en t to E ∞ simplicial Q -algebras [Sc h w99, 4. 4]. The monads defining E ∞ algebras and comm utative al gebras i n sim- plicial Q -mo dules are weakly equiv alent, since for ev ery j ≥ 0 the group homology of Σ j with co efficien ts i n an y Q -mo dule is conc en trated in degree zero. Hence E ∞ simplicial Q -algebras are Quillen equiv alen t to comm utative simplicial Q -algebras [Ma03, 6.7]. The homoto p y categories of comm utative simplicial rings and connectiv e com- m utative H Z -algebras are not equiv alen t. 1.2. Lineariza t ion. Let A b e a connective S -algebra. W e wri te π = π A : A → H π 0 A for its zero-th Postnik o v section, and define the l inearization map λ = λ A : A → H Z ∧ A to b e π ∧ id : A ∼ = S ∧ A → H Z ∧ A . It is a π 0 -isomorphism and a rational equiv alence of connec tive S -algebras. F or eac h (sim pl i cial or top o- logical) mo noi d G let S [ G ] = Σ ∞ G + b e its unreduced susp ension sp ectrum. F or A = S [ G ] , the linearization map λ : S [ G ] → H Z ∧ S [ G ] ∼ = H Z [ G ] agrees with the map considered b y W aldhausen [W a78, p. 43]. In general, H Z ∧ A is a connectiv e H Z -algebra, so by the first Quillen equiv a- lence ab o v e there is a naturally asso ciated simplici al ring R with H Z ∧ A ≃ H R . F or connective comm utative A , t he rationalizati o n A Q = H Q ∧ A i s a connective comm utative H Q -algebra, so b y t he second Quill en equiv alence ab o ve there i s a naturally asso ciated commutativ e simplicial Q -algebra R wit h A Q ≃ H R . 1.3. Algebraic K -theory. F or a general S -algebra A , the algebraic K -theory space K ( A ) can b e defined as Ω | hS • C A | , where C A is the category of finite cell A - mo dule spectra a nd t heir retracts, S • denotes W aldhausen’s S • -construction [W a85, § 1.3], and | h ( − ) | indicates the nerv e of the sub category of weak eq uiv alences. By iterating the S • -construction, we ma y also v i ew K ( A ) as a sp ectrum. F or con- nectiv e S -algebras, K ( A ) can al ternativ ely b e defined in terms of Quillen’s plus- construction as K 0 ( π 0 A ) × B GL ∞ ( A ) + , and then the tw o definitions are equiv alen t, see [EKMM97, VI.7.1]. W e wri te K ( R ) for K ( H R ), a nd similarly for other functors. An y map A → A ′ of connectiv e S -algebras that i s a π 0 -isomorphism and a rational eq ui v alence induces a ra t ional equiv alence K ( A ) → K ( A ′ ), b y [W a78, 2.2]. The pro of go es b y observing t hat B GL n ( A ) → B GL n ( A ′ ) is a r a tional equiv alence RA TIONAL ALGEBRAIC K-THEOR Y OF TOPOLO GICA L K-THEOR Y 5 for eac h n . In particular, for R with H Z ∧ A ≃ H R there is a nat ural rational equiv al ence λ : K ( A ) → K ( H Z ∧ A ) ≃ K ( R ). F or X ≃ B G , W aldhausen writes A ( X ) for K ( S [ G ]) , and λ : A ( X ) → K ( Z [ G ]) is a rat ional equiv alence. In this case, A ( X ) can also b e defined a s the alg ebraic K -theory of a category R f ( X ) of suitably finite retractive spaces o v er X , see [W a85, § 2.1]. 1.4. C yclic homology. There is a natural tra ce map tr : K ( A ) → T H H ( A ) to the top ologi cal Ho c hsc hild homo l ogy of A , see [BHM93, § 3]. The target is a cycl ic ob ject in the sense o f Connes, hence carries a natural S 1 -action. There exi sts a mo del for the trace map t hat factors through the fixed p oin ts of this circle action [Du04], hence it also factors through the homotopy fixed p oin ts T H H ( A ) hS 1 . W e get a natural commu tati ve triangle K ( A ) α / / tr & & L L L L L L L L L L T H H ( A ) hS 1 F   T H H ( A ) where the F rob enius map F forgets ab out S 1 -homotop y in v ariance. F or an y sim- plicial ring R there are natural isomorphisms T H H ∗ ( H R Q ) ∼ = H H ∗ ( R ⊗ Q ) (Ho ch - sc hild homology) and T H H ∗ ( H R Q ) hS 1 ∼ = H C − ∗ ( R ⊗ Q ) (negat i v e cyclic homology) . See e.g. [EKMM 97, IX.1.7 ] and [CJ90, 1.3(3)]. With thes e iden tifications, the tri- angle ab o v e realizes the comm utative dia g ram of [ Go 8 6, I I.3.1]. In [Go86, I I.3.4], Go o dwillie pro v ed: Theorem 1.5. L e t f : R → R ′ b e a map of si mplicial ring s , wi th π 0 R → π 0 R ′ a surje ction with nilp o te nt kernel. Then K ( R ) Q α / / f   H C − ( R ⊗ Q ) f   K ( R ′ ) Q α / / H C − ( R ′ ⊗ Q ) is homotopy Cartesi an, i .e., the map of ve rti c al homotopy fib ers α : K ( f ) Q → H C − ( f ⊗ Q ) is an e quivalenc e. Here we write K ( f ) for t he homoto p y fiber of K ( R ) → K ( R ′ ), so t hat there is a long exact sequence · · · → K ∗ +1 ( R ′ ) → K ∗ ( f ) → K ∗ ( R ) → K ∗ ( R ′ ) → . . . , and similarly for ot her functors from ( S -)al g ebras to spaces. (Go o dwilli e wri tes K ( f ) for a delo oping of our K ( f ) , but w e ne ed to emphasize fibers o v er cofib ers.) W e writ e K ( R ) Q for the rat i onalization of K ( R ), and similarl y for other spaces and S -algebras. 6 CHRISTIA N AUSONI AND JOHN R OGNES Corollary 1.6. L et g : A → A ′ b e a map o f c onne ctive S -algebr as, with π 0 A → π 0 A ′ a surje ction with ni lp otent kernel, Then K ( A ) Q α / / g   T H H ( A Q ) hS 1 g   K ( A ′ ) Q α / / T H H ( A ′ Q ) hS 1 is homotopy Cartesi an, i .e., the map of ve rti c al homotopy fib ers α : K ( g ) Q → T H H ( g Q ) hS 1 is an e quivalenc e. Pr o of. Given g : A → A ′ w e find f : R → R ′ with H Z ∧ A ≃ H R and H Z ∧ A ′ ≃ H R ′ making the diagram A λ / / g   H R H f   A ′ λ / / H R ′ homotop y comm ute. Then λ : K ( A ) → K ( R ) is a rational equi v alence and A Q ≃ H ( R ⊗ Q ), so t he square in the corol l ary is equiv alen t to the square in Go o dwill ie’s theorem.  1.7. De Rham homology . The (sp ectrum level) circle action on T H H ( A ) in- duces a susp ension op erator d : T H H ∗ ( A ) → T H H ∗ +1 ( A ), a nal ogous to Connes’ op erator B : H H ∗ ( R ) → H H ∗ +1 ( R ). When A Q ≃ H ( R ⊗ Q ), t hese op erat ors are compatible under the isomorphism T H H ∗ ( A Q ) ∼ = H H ∗ ( R ⊗ Q ). In general dd = dη is not zero [He97, 1.4.4], where η is the stable Hopf map, but i n the algebraic case B B = 0 , so one can define t he de Rham homology H dR ∗ ( R ) = ke r( B ) / im( B ) of a simplicial ring R as the homology of H H ∗ ( R ) with resp ect to the B -op erator. F or a map g : A → A ′ of S -algebras, the homot op y fib er T H H ( g ) of T H H ( A ) → T H H ( A ′ ) inherits a circle acti on and asso ciated susp ension op erator. Similarly , for a map f : R → R ′ of simpl i cial rings there is a rela t iv e B -op erator a cting on the term H H ∗ ( f ) i n t he long exact sequence · · · → H H ∗ +1 ( R ′ ) → H H ∗ ( f ) → H H ∗ ( R ) → H H ∗ ( R ′ ) → . . . , and w e define H dR ∗ ( f ) to b e the homology of H H ∗ ( f ) with respect t o this B - op erator. W e sa y that f : R → R ′ is a de Rham equiv al ence if H dR ∗ ( f ) = 0, and that f is a rational de Rham equiv alence if H dR ∗ ( f ⊗ Q ) = 0 . If w e assume that H H ∗ ( R ) → H H ∗ ( R ′ ) i s surjective i n eac h degree, t hen there is a lo ng exact sequence · · · → H dR ∗ +1 ( R ′ ) → H dR ∗ ( f ) → H dR ∗ ( R ) → H dR ∗ ( R ′ ) → . . . , in whic h case f is a de Rham equiv alence if and only if H dR ∗ ( R ) → H dR ∗ ( R ′ ) is an isomorphism in every degree. RA TIONAL ALGEBRAIC K-THEOR Y OF TOPOLO GICA L K-THEOR Y 7 Prop osition 1.8. If f : R → R ′ is a de R ham e quivalenc e, then ther e is an exact se quenc e 0 → H C − ∗ ( f ) F − → H H ∗ ( f ) B − → H H ∗ +1 ( f ) that identifies H C − ∗ ( f ) wi th ker( B ) ⊂ H H ∗ ( f ) . Pr o of. By a nalogy with the homotopy fixed p oin t sp ectral sequence for T H H ( g ) hS 1 , there is a second q uadran t homological sp ectral sequence E 2 ∗∗ = Q [ t ] ⊗ H H ∗ ( f ) = ⇒ H C − ∗ ( f ) with t ∈ E 2 − 2 , 0 and d 2 ( t i · x ) = t i +1 · B ( x ) for all x ∈ H H ∗ ( f ), i ≥ 0. So E 3 ∗∗ is the sum of k er( B ) ⊂ H H ∗ ( f ) in the zero-th column and a cop y of H dR ∗ ( f ) i n eac h ev en, negati ve column. B y assumption the latter groups are al l zero, so the sp ectra l sequence collapses to the zero-th column at the E 3 -term. The F rob enius F i s t he edge homomorphism for this sp ectral se quence, and the assertion follo ws.  Corollary 1.9. L et A b e a c onne ctive S -algebr a such that π 0 A is any l o c alizatio n of the i nte gers, and let R b e a simplicial Q -algebr a with A Q ≃ H R . (a) The homotopy fib er se quenc e K ( π A ) → K ( A ) π A − − → K ( π 0 A ) is r ational ly split, whe r e π A : A → H π 0 A is the zer o-th P o s tnikov se ction. (b) Ther e ar e e quiv a lenc es K ( π A ) Q α − → ≃ T H H ( π A Q ) hS 1 ≃ H C − ( π R ) , wher e π R : R → π 0 R = Q is the ze r o-th Postnikov se ction. Supp os e furthermor e that H dR ∗ ( R ) ∼ = Q is triv ial in p os itive de gr e es. (c) The map π R is a de R ham e quiva lenc e, and the F r ob eni us map ide ntifies H C − ∗ ( π R ) with the p ositive-de gr e e p art of k er( B ) ⊂ H H ∗ ( R ) ∼ = T H H ∗ ( A ) ⊗ Q . That p art is a lso e qual to im( B ) ⊂ T H H ∗ ( A ) ⊗ Q . (d) The tr ac e map tr : K ( A ) → T H H ( A ) induc e s the c omp os i te identific ation of K ∗ ( π A ) ⊗ Q with the p ositive-de gr e e p art of ke r( B ) ⊂ T H H ∗ ( A ) ⊗ Q . Pr o of. (a) W ri te π 0 A = Z ( P ) for some (possibly empt y) set of primes P . The unit map i : S → A factors through S ( P ) , and the comp osite map S ( P ) → A → H π 0 A is a π 0 -isomorphism and a rational equiv alence. Hence the comp osite K ( S ( P ) ) → K ( A ) π A − − → K ( π 0 A ) is a rational equiv alence. (b) The map π A : A → H π 0 A induces the i den tit y on π 0 , so α is a n equiv al ence b y Coro l lary 1.6. W e recalled the second i den tification i n Subsection 1.4. It is cl ear that π 0 R = π 0 A Q = π 0 A ⊗ Q = Q . (c) Since H H ∗ ( Q ) = Q i s trivi al in p ositive degrees, t he map H H ∗ ( R ) → H H ∗ ( Q ) is surjectiv e in eac h degree, so π R is a de Rham equiv alence if (a nd only 8 CHRISTIA N AUSONI AND JOHN R OGNES if ) H dR ∗ ( R ) ∼ = H dR ∗ ( Q ) = Q is trivi al in all p osit i v e degrees. The homotop y fib er sequence H H ( π R ) → H H ( R ) → H H ( Q ) iden tifies H H ∗ ( π R ) wi th t he p ositive-degree part of H H ∗ ( R ), so ker( B ) ⊂ H H ∗ ( π R ) is the p ositive -degree part of ke r( B ) ⊂ H H ∗ ( R ). The identification i m( B ) = ker( B ) in p ositive degrees i s of course equiv alen t to the v anishing of H dR ∗ ( R ) in p ositive degrees. (d) The trace map factors a s tr = F ◦ α .  § 2. Examples fr om topological K -theor y 2.1. Connecti ve K -theory sp ectra. Let k u b e the connectiv e complex K - theory sp ectrum, k o the connective real K -theory sp ectrum, and ℓ = B P h 1 i the Adams summand of k u ( p ) , for p an o dd prime. These are all comm utativ e S - algebras. W e write Ω ∞ k u = B U × Z , Ω ∞ k o = B O × Z and Ω ∞ ℓ = W × Z ( p ) for the unde rlyi ng infinite lo op spac es (see [Ma77, V.3-4]). The homotop y u nits form infinite lo op spaces, namely GL 1 ( k u ) = B U ⊗ × {± 1 } , GL 1 ( k o ) = B O ⊗ × {± 1 } and GL 1 ( ℓ ) = W ⊗ × Z × ( p ) . The homotopy algebras are π ∗ k u = Z [ u ] with | u | = 2, π ∗ k o = Z [ η , α, β ] / (2 η , η 3 , η α, α 2 − 4 β ) w i th | η | = 1, | α | = 4, | β | = 8, and π ∗ ℓ = Z ( p ) [ v 1 ] with | v 1 | = 2 p − 2 . T he complexification map k o → k u tak es η to 0, α to 2 u 2 and β to u 4 . The inclusion ℓ → k u ( p ) take s v 1 to u p − 1 . Prop osition 2.2. (a) Ther e ar e π 0 -isomorphis ms and r ational e quiv alenc es κ : S [Ω S 3 ] → S [ K ( Z , 2)] → k u of S - algebr as, s o k u Q ≃ H Q [Ω S 3 ] as homotopy c ommutative H Q -algebr as, wher e Q [Ω S 3 ] is a simplici al Q -algebr a, and k u Q ≃ H Q [ K ( Z , 2)] as c ommutative H Q - algebr as, wher e Q [ K ( Z , 2)] is a c ommutative simplicial Q -algebr a. (b) Ther e ar e π 0 -isomorphis ms a nd r ational e quivalenc es ¯ α : S [Ω S 5 ] → k o and ¯ v 1 : S ( p ) [Ω S 2 p − 1 ] → ℓ , so k o Q ≃ H Q [Ω S 5 ] and ℓ Q ≃ H Q [Ω S 2 p − 1 ] as H Q -algebr a s , wher e Q [Ω S 5 ] and Q [Ω S 2 p − 1 ] ar e simplicial Q -algebr as. (c) In p articular, ther e is a r ational e quivalenc e κ : A ( S 3 ) → K ( k u ) of S - algebr as, a r ational e quivalenc e A ( K ( Z , 3)) → K ( k u ) of c omm utative S -algebr as, and a r ational e quivalenc e ¯ α : A ( S 5 ) → K ( k o ) of S -mo dules. Pr o of. (a) Let B S 3 → K ( Z , 4) represen t a g enerator of H 4 ( B S 3 ). It induces a double lo op ma p Ω S 3 → K ( Z , 2), suc h that the comp osite S 2 → Ω S 3 → K ( Z , 2) represen ts a generator of π 2 K ( Z , 2) . The inclusions K ( Z , 2 ) ≃ B U (1) → B U ⊗ → GL 1 ( k u ) are infinite lo op maps, and the generator of π 2 K ( Z , 2) maps to a gener- ator of π 2 GL 1 ( k u ). By adjunction we hav e an E 2 ring sp ectrum map S [Ω S 3 ] → S [ K ( Z , 2) ] a nd an E ∞ ring sp ectrum map S [ K ( Z , 2)] → k u , with comp osite the E 2 ring sp ectrum map κ : S [Ω S 3 ] → k u . These are ra tional equiv alences, b ecause π ∗ S [Ω S 3 ] ⊗ Q ∼ = H ∗ (Ω S 3 ; Q ) ∼ = Q [ x ], H ∗ ( K ( Z , 2); Q ) ∼ = Q [ b ] and π ∗ k u ⊗ Q = Q [ u ], with κ mapping x via b to u . W e may take the Kan lo op gro up of S 3 (a simpli cial group, see e.g. [W a96 ] ) as our mo del for Ω S 3 , and rigidify κ to a map of S -algebras. F ollowing [FV], there remains an E 1 = A ∞ op erad action on these S -algebras and κ , whic h i n parti cular implies that κ : A ( S 3 ) → K ( k u ) is homot op y comm utativ e. RA TIONAL ALGEBRAIC K-THEOR Y OF TOPOLO GICA L K-THEOR Y 9 (b) F or the real case, l et S 4 → B O ⊗ ⊂ GL 1 ( k o ) represen t a generator of π 4 GL 1 ( k o ). By the lo op structure on the t arget, it ex t ends to a lo op map Ω S 5 → GL 1 ( k o ), with left adjoin t a n A ∞ ring sp ectrum map ¯ α : S [Ω S 5 ] → k o . It is a ratio - nal equiv alence, b ecause π ∗ S [Ω S 5 ] ⊗ Q ∼ = H ∗ (Ω S 5 ; Q ) ∼ = Q [ y ] and π ∗ k o ⊗ Q = Q [ α ], with ¯ α mapping y to α . W e interpret Ω S 5 as t he Kan lo op group, and form the simplicial Q -algebra Q [Ω S 5 ] as it s rat ional group ring. The Adams summand case is en tirely simil a r, start ing with a map S 2 p − 2 → W ⊗ ⊂ GL 1 ( ℓ ). (c) By [FV] and naturality there is an A ∞ op erad acti on on the induced map of sp ectra A ( S 3 ) = K ( S [ Ω S 3 ]) → K ( k u ) (rather than of spaces), which w e can rigidify to a map of S -algebras. The S -alg ebra m ultiplicat ion A ( S 3 ) ∧ A ( S 3 ) → A ( S 3 ) is induced b y the gr o up multiplication S 3 × S 3 → S 3 .  Lemma 2.3. (a) F or any inte ger n ≥ 1 the simplicial Q -algebr a R = Q [Ω S 2 n +1 ] has Ho chschild homolo gy H H ∗ ( R ) ∼ = Q [ x ] ⊗ E ( dx ) with | x | = 2 n , wher e Connes’ B -op er ator satisfies B ( x ) = dx . Her e E ( − ) denotes the exterior algebr a. (b) The de Rham homolo gy H dR ∗ ( R ) ∼ = Q is c onc entr ate d in de gr e e zer o, s o π R : R → Q is a de R ham e quivalenc e. (c) The p ositive-de gr e e p art of k er( B ) ⊂ H H ∗ ( R ) is Q [ x ] { dx } = Q { dx, x dx, x 2 dx, . . . } . Pr o of. (a) The Ho ch sc hild filtra t ion on the bisimplicial Q -algebra H H ( R ) y i elds a sp ectral sequence (2.4) E 2 ∗∗ = H H ∗ ( π ∗ ( R )) = ⇒ H H ∗ ( R ) , and π ∗ ( R ) = Q [ x ] with | x | = 2 n . The H o c hsc hi l d homol ogy of this gr a ded com- m utative ring is Q [ x ] ⊗ E ( dx ), where dx ∈ E 2 1 , 2 n is the image of x under Connes’ B -op erator. The spectral sequen ce coll a pses at that stage, for bidegree reasons. (b,c) The B -op erator is a deriv ati on, hence t ak es x m to mx m − 1 dx for a ll m ≥ 0. It follo ws easily that the de Rham homology is tri vial in p ositive degrees, and that k er B is as indicat ed.  By combining Corol lary 1.9, Prop ositi on 2.2 a nd Lemma 2.3, w e obtain t he follo wing result. Theorem 2.5. (a) Ther e is a r ational ly split homotopy fib er se quenc e K ( π ku ) → K ( k u ) π − → K ( Z ) and the tr ac e map tr : K ( k u ) → T H H ( k u ) i dentifies K ∗ ( π ku ) ⊗ Q ∼ = Q [ u ] { du } with its image in T H H ∗ ( k u ) ⊗ Q ∼ = Q [ u ] ⊗ E ( du ) . Her e | u | = 2 and | du | = 3 , so K ( π ku ) has P oinc ar´ e s e ries t 3 / (1 − t 2 ) . 10 CHRISTIA N AUSONI AND JOHN R OGNES (b) Si milarly, ther e ar e r ati onal ly split homotopy fib er se quenc es K ( π ko ) → K ( k o ) π − → K ( Z ) K ( π ℓ ) → K ( ℓ ) π − → K ( Z ( p ) ) and the tr ac e maps identify K ∗ ( π ko ) ⊗ Q ∼ = Q [ α ] { dα } K ∗ ( π ℓ ) ⊗ Q ∼ = Q [ v 1 ] { dv 1 } with their images in T H H ∗ ( k o ) ⊗ Q ∼ = Q [ α ] ⊗ E ( dα ) and T H H ∗ ( ℓ ) ⊗ Q ∼ = Q [ v 1 ] ⊗ E ( dv 1 ) , r esp e ctively. Henc e K ( π ko ) has Poinc ar ´ e seri e s t 5 / (1 − t 4 ) , wher e as K ( π ℓ ) has Poinc ar´ e se r i es t 2 p − 1 / (1 − t 2 p − 2 ) . R emark 2.6. The P oincar ´ e series of K ( Z ) i s 1 + t 5 / (1 − t 4 ) b y Borel’s cal culat ion [Bo74]. Hence the (commo n) P oincar ´ e series of K ( k u ) and A ( S 3 ) is 1 + t 3 / (1 − t 2 ) + t 5 / (1 − t 4 ) = 1 + ( t 3 + 2 t 5 ) / (1 − t 4 ) , whereas the Poincar ´ e series of K ( k o ) and A ( S 5 ) i s 1 + 2 t 5 / (1 − t 4 ). More generally , w e recov er the Poincar ´ e series 1 + t 5 / (1 − t 4 ) + t 2 n +1 / (1 − t 2 n ) of A ( S 2 n +1 ) for n ≥ 1, from [HS82, Cor. 1.2]. The group K 1 ( Z ( p ) ) is not finitely generated, so w e do not discuss the Poincar ´ e series of K ( ℓ ) . 2.7. P erio dic K -theory sp ectra. Let K U b e the p erio dic complex K -theory sp ectrum, K O the p erio dic real K -theory sp ectrum, and L = E (1) the Adams summand of K U ( p ) , for p an o dd prime. W e ha v e maps of comm utative S -algebras H Z π ← − k u ρ − → K U with asso ciated maps of “brav e new” affine sc hemes [TV, § 2] (2.8) Sp ec( Z ) π − → Sp ec( k u ) ρ ← − Sp ec( K U ) . Let i : S → S [Ω S 3 ] and c : S [Ω S 3 ] → S b e induced b y the inclusion ma p ∗ → S 3 and the collapse map S 3 → ∗ , resp ectively . W e hav e a map of horizon tal cofib er sequences (2.9) Σ 2 S [Ω S 3 ] x / / Σ 2 κ   S [Ω S 3 ] c / / κ   S λ   Σ 2 k u u / / k u π / / H Z where the top row exhibits S as a t w o-cell S [Ω S 3 ]-mo dule, and the b ottom row exhibits H Z as a tw o-cell k u -mo dule. (In eac h case, the t w o cells are in dimension zero and three.) There are a l gebraic K -t heory t ra nsfe r maps c ∗ : A ( ∗ ) → A ( S 3 ) and π ∗ : K ( Z ) → K ( ku ) (with a low er star, in accordance wit h the v ariance conv en tions from algebraic g eometry and (2.8)), that are induced b y the functors that view finite cell S -mo dules as finite cell S [Ω S 3 ]-mo dules, a nd finite cell H Z -mo dules as RA TIONAL ALGEBRAIC K-THEOR Y OF TOPOLO GICA L K-THEOR Y 11 finite cell k u -mo dules, respecti vely . In terms of retractive spaces, c ∗ is induced b y the ex a ct functor R f ( ∗ ) → R f ( S 3 ) that t a kes a p oin ted space X ⇄ ∗ to the retractive space X × S 3 ⇄ S 3 . The transfer maps are compatible, b y (2.9), so w e ha v e a commutativ e diagram with v ertical rati onal equi v alences (2.10) A ( ∗ ) c ∗ / / λ   A ( S 3 ) κ   K ( Z ) π ∗ / / K ( k u ) ρ / / K ( K U ) . The b ott om row is a homotopy fib er sequence b y the lo calizat i on theorem of [B M]. Lemma 2.11. The tr ansfer map c ∗ : A ( ∗ ) → A ( S 3 ) is nul l-homotopic, as a map of A ( ∗ ) -mo dule sp e ctr a. The tr ansfer map π ∗ : K ( Z ) → K ( k u ) i s r ational ly nul l- homotopic, again as a map of A ( ∗ ) -mo dule sp e ctr a. Pr o of. The pro jection form ula asserts that c ∗ is a n A ( S 3 )-mo dule map, where c : A ( S 3 ) → A ( ∗ ) makes A ( ∗ ) an A ( S 3 )-mo dule. Restricting t he mo dule struc- tures along i : A ( ∗ ) → A ( S 3 ), w e see that c ∗ is a map of A ( ∗ )-mo dule sp ectra, and the source is a free A ( ∗ )-mo dule of rank one. H ence i t suffices to sho w that c ∗ take s a generator of π 0 A ( ∗ ), represen ted sa y b y S 0 ⇄ ∗ , to zero i n π 0 A ( S 3 ) ∼ = Z . But c ∗ maps that generator to the cl ass of S 0 × S 3 ⇄ S 3 , whic h corresp onds to its Euler c haracteristic χ ( S 3 ) = 0. The conclusion for π ∗ follo ws from that for c ∗ , via the rational equiv alences λ and κ .  Note the utility of the comparison with A -theory at this point, since w e do not ha v e an S -algebra map K ( Z ) → K ( ku ) that is analogous t o i : A ( ∗ ) → A ( S 3 ). Theorem 2.12. Ther e ar e r ational ly split homotopy fib er se quenc es K ( k u ) ρ − → K ( K U ) ∂ − → B K ( Z ) K ( ℓ ) ρ − → K ( L ) ∂ − → B K ( Z ( p ) ) of infinite lo op sp ac es. Henc e the Poinc ar ´ e series of K ( K U ) is (1 + t ) + ( t 3 + 2 t 5 + t 6 ) / (1 − t 4 ) . Pr o of. The claims for K U follow b y combining Theorem 2.5(a) and Lemma 2.11. The pro of of the cla im for L i s completel y similar, using t hat H Z ( p ) is a tw o-cell ℓ -mo dule, with cells in dimension zero and (2 p − 1). B y [BM] there is a homoto p y fib er sequence K ( Z ( p ) ) → K ( ℓ ) → K ( L ).  R emark 2.13. W e do not know ho w to rel ate K ( k o ) with K ( K O ), so we do not ha v e a rational calculation of K ( K O ). Ho w ev er, K O → K U is a Z / 2-Galo is ex- tension of comm utative S -algebras, in the sense of [Ro, § 4.1], so it is plausible that K ( K O ) → K ( K U ) h Z / 2 is close to an equiv alence. Here Z / 2 acts on K U b y complex conjugation, and π ∗ ( K ( K U ) h Z / 2 ) ⊗ Q ∼ = [ K ∗ ( K U ) ⊗ Q ] Z / 2 . 12 CHRISTIA N AUSONI AND JOHN R OGNES The conjugation action on k u fixes K ( Z ), and acts on K ∗ ( π ku ) ⊗ Q ∼ = Q [ u ] { du } b y sign on u a nd du , hence fixes Q [ u 2 ] { udu } ∼ = Q [ α ] { dα } ∼ = K ∗ ( π ko ) ⊗ Q . So K ( k o ) → K ( k u ) h Z / 2 is a rational equi v alence. T he conjugation action al so fixes B K ( Z ) after rationalizat i on, so t he Poincar ´ e series o f K ( K U ) h Z / 2 is (1 + t ) + (2 t 5 + t 6 ) / (1 − t 4 ). R emark 2. 14. W e exp ect t hat c ∗ and π ∗ are essen tial (not null-homotopic) as maps of A ( S 3 )-mo dule sp ectra and K ( k u )-mo dule sp ectra, resp ectiv ely . In other w ords, w e exp ect that K ( k u ) → K ( K U ) → Σ K ( Z ) is a non-split ext ension of K ( k u ) - mo dule sp ectra. This exp ectation is to some exten t justified by the fact t hat the cofib er T H H ( k u | K U ) of t he T H H -transfer map π ∗ : T H H ( Z ) → T H H ( k u ) sits in a non-split extension T H H ( k u ) → T H H ( k u | K U ) → Σ T H H ( Z ) of T H H ( k u ) - mo dule sp ectra. See [Au05, 10.4], or [HM03, Lemma 2.3. 3] for a similar result in an algebraic case. § 3. Examples from s mooth bordism 3.1. Orien ted b ordism sp ectra. Let M U b e the complex b ordism sp ectrum, M S O the real orien ted b ordism sp ectrum, and M S p the sy mplectic b ordism spec- trum. These are al l connectiv e commutativ e S -algebras, given by the Thom sp ectra asso ciated to infinite lo op maps from B U , B S O and B S p to B S F = B S L 1 ( S ), re- sp ectiv ely . W e recall that H ∗ ( B U ) ∼ = Z [ b k | k ≥ 1] with | b k | = 2 k , whil e H ∗ ( B S O ; Z [1 / 2]) ∼ = H ∗ ( B S p ; Z [1 / 2]) ∼ = Z [1 / 2][ q k | k ≥ 1 ] with | q k | = 4 k . The Thom equiv al ence θ : M U ∧ M U → M U ∧ S [ B U ] induces an equiv alence H Z ∧ M U ≃ H Z ∧ S [ B U ] = H Z [ B U ]. Combine d with the Hurewicz map π : S → H Z w e obtain a c hain of maps of comm utativ e S -algebras M U − → H Z ∧ M U ≃ H Z [ B U ] ← − S [ B U ] , that are π 0 -isomorphisms and rati o nal equiv alences . There are simil ar chains M S O − → H Z ∧ M S O ≃ H Z [ B S O ] ← − S [ B S O ] and M S p − → H Z ∧ M S p ≃ H Z [ B S p ] ← − S [ B S p ] , and all induce rational equiv alences (3.2) K ( M U ) − → K ( Z [ B U ]) ← − A ( B B U ) K ( M S O ) − → K ( Z [ B S O ]) ← − A ( B B S O ) K ( M S p ) − → K ( Z [ B S p ]) ← − A ( B B S p ) of comm utative S -algebras. Here we view B U ≃ Ω B B U as t he Kan lo op group of B B U , Z [ B U ] is the asso ciated simplicial ring, and simi larly for B S O and B S p . Lemma 3.3. (a) The simplicial Q -algebr a R = Q [ B U ] with π ∗ R = H ∗ ( B U ; Q ) = Q [ b k | k ≥ 1] has Ho chschild homolo gy H H ∗ ( R ) ∼ = Q [ b k | k ≥ 1] ⊗ E ( db k | k ≥ 1) , with Poinc ar´ e series h ( t ) = Y k ≥ 1 1 + t 2 k +1 1 − t 2 k , RA TIONAL ALGEBRAIC K-THEOR Y OF TOPOLO GICA L K-THEOR Y 13 and Connes’ op er ator acts by B ( b k ) = db k . (b) The de Rham homolo gy H dR ∗ ( R ) ∼ = Q is c onc entr ate d in de gr e e zer o, so π R : R → Q is a de R ham e quiv a lenc e. (c) The Poinc ar ´ e series of ker( B ) ⊂ H H ∗ ( R ) is k ( t ) = 1 + th ( t ) 1 + t . (d) The simplicial Q -algebr a R so = Q [ B S O ] ≃ Q [ B S p ] , with π ∗ R so = Q [ q k | k ≥ 1] , has Ho chschi ld homolo gy H H ∗ ( R so ) ∼ = Q [ q k | k ≥ 1] ⊗ E ( dq k | k ≥ 1) . Its Poinc ar ´ e series is h so ( t ) = Q k ≥ 1 (1 + t 4 k +1 ) / (1 − t 4 k ) . The map R so → Q is a de R ham e quivalenc e, and k er( B ) ⊂ H H ∗ ( R so ) ha s Poinc ar ´ e seri es k so ( t ) = (1 + th so ( t )) / (1 + t ) . Pr o of. (a) In this case the sp ectral sequence ( 2.4) has E 2 ∗∗ = H H ∗ ( Q [ b k | k ≥ 1]) ∼ = Q [ b k | k ≥ 1] ⊗ E ( db k | k ≥ 1). The algebra generators are in filtrations 0 and 1, so E 2 = E ∞ . This term is free as a graded commu tative Q -algebra, so H H ∗ ( R ) is isomorphic to the E ∞ -term. (b) The homology of Q [ b k ] ⊗ E ( db k ) wit h resp ect t o B is just Q , for each k ≥ 1, so b y the K ¨ unneth theorem the de Rham homology of H H ∗ ( R ) is also just Q . (c) W rite H n for H H n ( R ) and K n for ker( B : H n → H n +1 ). Let h n = dim Q H n , so h ( t ) = P n ≥ 0 h n t n , and k n = dim Q K n . In vi ew of t he exa ct se quence 0 → Q → H 0 d − → H 1 d − → . . . d − → H n − 1 → K n → 0 w e find that 1 − ( − 1) n k n = h 0 − h 1 + · · · + ( − 1) n − 1 h n − 1 , so X n ≥ 0 t n k n − X n ≥ 0 ( − t ) n = th ( t ) − t 2 h ( t ) + · · · + ( − t ) m +1 h ( t ) + . . . . It foll o ws that the P oincar´ e series k ( t ) = P n ≥ 0 k n t n for ker( B ) satisfies k ( t ) − 1 / (1 + t ) = th ( t ) / (1 + t ). (d) The only change from the complex to the orien ted real and symplectic cases is in the grading of the algebra generators, whic h (as long as they remain in ev en degrees) pla ys no rol e for t he pro ofs.  Theorem 3.4. (a) Ther e is a r ational ly split homotopy fib er se quenc e K ( π M U ) → K ( M U ) π − → K ( Z ) and the tr ac e map tr : K ( M U ) → T H H ( M U ) i dentifies K ∗ ( π M U ) ⊗ Q with the p o s itive-de gr e e p art of k er( B ) in T H H ∗ ( M U ) ⊗ Q ∼ = Q [ b k | k ≥ 1] ⊗ E ( db k | k ≥ 1) , wher e | b k | = 2 k and B ( b k ) = db k . Henc e K ( π M U ) has P oinc ar´ e s e ries k ( t ) − 1 = th ( t ) − t 1 + t . 14 CHRISTIA N AUSONI AND JOHN R OGNES (b) Ther e ar e r ati o na l ly split homotopy fib er s e quenc es K ( π M S O ) → K ( M S O ) π − → K ( Z ) K ( π M S p ) → K ( M S p ) π − → K ( Z ) and the tr ac e maps identi fy b oth K ∗ ( π M S O ) ⊗ Q and K ∗ ( π M S p ) ⊗ Q with the p o s itive- de gr e e p art of k er( B ) in T H H ∗ ( M S O ) ⊗ Q ∼ = T H H ∗ ( M S p ) ⊗ Q ∼ = Q [ q k | k ≥ 1] ⊗ E ( dq k | k ≥ 1) , wher e | q k | = 4 k and B ( q k ) = dq k . Henc e K ( π M S O ) and K ( π M S p ) b oth have Poinc ar ´ e series k so ( t ) − 1 = ( th so ( t ) − t ) / ( 1 + t ) . R emark 3. 5 . Adding t he P oincar ´ e series of K ( Z ), as in Remark 2 . 6, we find that the Poincar ´ e series of K ( M U ) and A ( B B U ) i s t 5 1 − t 4 + 1 + th ( t ) 1 + t , whereas the P oincar ´ e series of K ( M S O ), A ( B B S O ) , K ( M S p ) and A ( B B S p ) is t 5 / (1 − t 4 ) + (1 + th so ( t )) / (1 + t ). § 4. Units, determinants and traces 4.1. Uni ts. F or eac h connective S -algebra A there is a natural map of spaces w : B GL 1 ( A ) → K ( A ) that factors as the infinite st a bilization map B GL 1 ( A ) → B GL ∞ ( A ), comp osed with the inclusion B GL ∞ ( A ) → B GL ∞ ( A ) + in to Quillen’s plus cons truction, and follo w ed by t he inclusion o f B GL ∞ ( A ) + ∼ = { 1 } × B GL ∞ ( A ) + in to K 0 ( π 0 A ) × B GL ∞ ( A ) + = K ( A ). R emark 4. 2. This w i s an E ∞ map wit h resp ect to the m ultiplicative E ∞ structure on K ( A ) that is induced b y the smash pro duct o ver A . How ever, we shall only w ork with the additive grouplike E ∞ structure o n K ( A ), which comes from vi ewi ng K ( A ) as the underlyi ng infinite lo op sp ace of the K -theory sp ectrum. So when w e refe r to infinite lo op structures below, we ar e thinking o f the additi v e ones. W e write B S L 1 ( A ) = B GL 1 ( π A ) for the homotop y fib er of the map B GL 1 ( A ) → B GL 1 ( π 0 A ) induced b y π A : A → H π 0 A . In the resulting dia gram (4.3) B S L 1 ( A ) w − → K ( A ) π − → K ( π 0 A ) the comp osite map has a preferred null-homotop y ( to the base p oi nt of the 1- comp onen t of K ( π 0 A )). The diagram i s a rational homotop y fib er sequence if and only if w : B S L 1 ( A ) → K ( π A ) is a rational equiv alence. No te that the natural inclusion { 1 } × B GL ∞ ( A ) + → K ( A ) induces a homot o p y equiv alence B GL ∞ ( π A ) + ≃ K ( π A ) , since K 0 ( A ) ∼ = K 0 ( π 0 A ). RA TIONAL ALGEBRAIC K-THEOR Y OF TOPOLO GICA L K-THEOR Y 15 4.4. De terminan ts. Suppose furthermore t hat A i s comm utative as an S -al gebra. One at tempt at pro ving that w is injective could b e to construct a map det : K ( A ) → B GL 1 ( A ) with the prop erty that de t ◦ w ≃ i d. How ev er, no suc h determinan t map exists in general, as the following adaption of [W a82, 3.7] shows. Example 4.5. When A = S , the map λ ◦ w : B F = B GL 1 ( S ) → A ( ∗ ) → K ( Z ) factors through B GL 1 ( Z ) ≃ K ( Z / 2 , 1), and π 2 ( λ ) : π 2 A ( ∗ ) → K 2 ( Z ) ∼ = Z / 2 is an isomorphism, so π 2 ( w ) : π 2 B F → π 2 A ( ∗ ) is the zero map. But π 2 B F ∼ = π 1 ( S ) ∼ = Z / 2 i s not ze ro, so π 2 ( w ) is not injective. In particular, w i s not sp lit injectiv e up to homotop y . Ho w ev er, it is p ossible to construct a rat ionalized determinant map. Recall from Subsection 1.1 t hat A Q is eq uiv alen t to H R for some natural ly determined comm utative simplici al Q -algebra R . Lemma 4.6. L et R b e a c ommutative simplici al ring. Ther e is a natur al infinite lo op map det : B GL ∞ ( R ) + → B GL 1 ( R ) that agr e es with the usual determinant map for dis cr ete c ommutative ri ngs, such that the c omp osi te with w : B GL 1 ( R ) → B GL ∞ ( R ) + e qual s the identity. Pr o of. The usual matrix determinan t det : M n ( R ) → R induces a simplicial group homomorphism GL n ( R ) → GL 1 ( R ) and a p oin ted map B GL n ( R ) → B GL 1 ( R ) for eac h n ≥ 0. These stabilize to a map B GL ∞ ( R ) → B GL 1 ( R ), which ext ends to an infinite lo op map det : B GL ∞ ( R ) + → B GL 1 ( R ) , unique up to homotop y , b y the m ultiplicative infinite loop structure on the target and the univ ersal prop ert y of Quillen’s plus construction. T o make the construction natural, w e fix a c hoice o f extension in the initial case R = Z , and define det R for general R as the dash ed pu shout map in the following diagram: B GL ∞ ( Z ) / /   B GL ∞ ( Z ) +   det Z ' ' N N N N N N N N N N N B GL ∞ ( R ) / / + + W W W W W W W W W W W W W W W W W W W W W W W B GL ∞ ( R ) + det R ' ' O O O O O O B GL 1 ( Z )   B GL 1 ( R ) (Recall that Quillen’s plus construction i s made fun ctorial b y demanding that the left hand square is a pushout.)  Prop osition 4.7. L et A b e a c onne cti ve c ommutative S -algebr a. Ther e is a natur al infinite lo op map det Q : B GL ∞ ( A ) + → B GL 1 ( A ) Q that agr e es with the r ationalize d determinant map for a c ommut ative ring R when A = H R , such that the c omp osite B GL 1 ( A ) w − → B GL ∞ ( A ) + det Q − − − → B GL 1 ( A ) Q 16 CHRISTIA N AUSONI AND JOHN R OGNES is homotopic to the r ationaliz a ti on map. Pr o of. W e define det Q as the dashed pullback map in the following diagram B GL ∞ ( A ) + / /   det Q ( ( Q Q Q Q Q Q B GL ∞ ( A Q ) + det ′ R ( ( P P P P P P P P P P P P B GL ∞ ( π 0 A ) + (det π 0 A ) Q ( ( Q Q Q Q Q Q Q Q Q Q Q Q B GL 1 ( A ) Q / /   B GL 1 ( A Q )   B GL 1 ( π 0 A ) Q / / B GL 1 ( π 0 A Q ) where the v ertical maps are induc ed b y the P ostnik o v se ction π : A → H π 0 A , and the horizontal maps are i nduce d by the rationalizat ion q : A → A Q . The right hand square is a homotopy pullbac k, since S L 1 ( A ) Q ≃ S L 1 ( A Q ). T o define t he map det ′ R , we t ak e R to b e a comm utative simpl i cial Q -algebra suc h t hat A Q ≃ H R a s comm utative H Q -algebras. A nat ural c hoice can b e made for R , as discussed in Subsection 1. 1, suc h that the iden tification π 0 A Q ∼ = π 0 R is the iden tity . Then det ′ R is the comp osite map B GL ∞ ( A Q ) + ≃ B GL ∞ ( R ) + det R − − − → B GL 1 ( R ) ≃ B GL 1 ( A Q ) , with det R from Lemma 4 .6. It strictl y cov ers the map det π 0 A Q , so the outer hexagon comm utes strictly . This defines the desired map det Q . T o compare det Q ◦ w and q : B GL 1 ( A ) → B GL 1 ( A ) Q , note that b oth maps ha v e the same comp osite to B GL 1 ( π 0 A ) Q , they hav e homoto pic comp osites t o B GL 1 ( A Q ), and al l comp osit es (and homotopies) to B GL 1 ( π 0 A Q ) are equal. Hence the maps to the homot op y pullbac k are homotopic, to o.  Theorem 4.8. (a) The r elative unit map B B U ⊗ = B GL 1 ( π ku ) w − → B GL ∞ ( π ku ) + ≃ K ( π ku ) is a r ational e quivalenc e, with r a ti onal homotopy inverse given by the r elative r a- tional determinant map det Q : B GL ∞ ( π ku ) + → B GL 1 ( π ku ) Q = ( B B U ⊗ ) Q . (b) The r elative unit maps B B O ⊗ = B GL 1 ( π ko ) → K ( π ko ) B W ⊗ = B GL 1 ( π ℓ ) → K ( π ℓ ) ar e r ational e quivalenc es (with r ational homotopy inverse de t Q in e ach c ase) . Pr o of. (a) By Prop osit ion 4.7 , the compo si te B B U ⊗ w − → K ( π ku ) det Q − − − → ( B B U ⊗ ) Q RA TIONAL ALGEBRAIC K-THEOR Y OF TOPOLO GICA L K-THEOR Y 17 is a rational eq ui v alence, so w is r a t ionally injectiv e. Here π ∗ B B U ⊗ ∼ = π ∗− 1 B U ⊗ has Poincar ´ e series t 3 / (1 − t 2 ), just like K ( π ku ) by Theorem 2.5 ( a). Thus w is a rational equiv alence. (b) The same pro of w orks for k o and ℓ , using that B B O ⊗ and B W ⊗ ha v e P oincar ´ e series t 5 / (1 − t 4 ) and t 2 p − 1 / (1 − t 2 p − 2 ), resp ectiv ely .  R emark 4.9. The analogous map w : B S L 1 ( M U ) → K ( π M U ) is ra tionally i njective, but not a rati onal equiv alence. F or the P oincar ´ e series of the source is t ( p ( t ) − 1) = t 3 + 2 t 5 + 3 t 7 + 5 t 9 + . . . , where p ( t ) = Q k ≥ 1 1 / (1 − t 2 k ), and the Poincar ´ e series of the tar g et is ( th ( t ) − t ) / ( 1 + t ) = t 3 + 2 t 5 + 3 t 7 + t 8 + 5 t 9 + . . . , b y The orem 3.4(a). These first differ in degree 8 , since π 8 B S L 1 ( M U ) ∼ = π 7 M U is trivial , but K 8 ( M U ) and K 8 ( π M U ) hav e r a nk one. A g enerator of the latter group maps to db 1 · d b 2 in k er( B ) ⊂ T H H ∗ ( M U ) ⊗ Q . In the same w a y , w : B S L 1 ( M S O ) → K ( π M S O ) and its symplectic v aria n t are rationally injectiv e, b ut not rational eq ui v alences. 4.10. T ra ces. Our origi nal strategy for pro ving that w : B GL 1 ( A ) → K ( A ) is rationally injectiv e for A = k u was to use the tra ce map tr : K ( A ) → T H H ( A ), in place of the rational determinan t map. B y [Sc hl04, § 4], there is a natural commu - tative diagram B GL 1 ( A ) w / /   K ( A )   tr % % L L L L L L L L L L B cy GL 1 ( A ) / / K cy ( A ) / / T H H ( A ) GL 1 ( A ) / / O O Ω ∞ A O O where B cy and K cy denote t he cyclic bar construction and cyclic K -theory , re- sp ectiv ely . The middle ro w is the geometric realizatio n of tw o cyclic maps, hence consists of circle equiv arian t spaces and maps. When A = k u , the resulti ng B -op erator on H ∗ ( B cy B GL 1 ( A ); Q ) takes primiti ve classes in the image from H ∗ ( GL 1 ( A ); Q ) ∼ = H ∗ ( B U ⊗ ; Q ) t o primit iv e classes gen- erating the image from H ∗ ( B GL 1 ( A ); Q ) ∼ = H ∗ ( B B U ⊗ ; Q ), so b y a diagram c hase w e can determine t he images of t he latter primi tiv e classes in H ∗ ( T H H ( A ); Q ). B y an a pp eal to the Milnor–Mo ore theorem [MM65, App .], this suffices to prov e that tr ◦ w is rational ly injectiv e in this case. In compari son with the rati onal determinan t approach take n ab o v e, this trace metho d in v olves more complicated calculat i ons. F or comm utative S -al g ebras, it is t herefore less at tractiv e. How ev er, for non-comm utative S -al gebras, the trace metho d ma y still b e useful, since no ( rational) determinan t ma p is likely to exist. W e hav e therefore sk etc hed the idea here, with a view to future applications. 18 CHRISTIA N AUSONI AND JOHN R OGNES § 5. Two -vector bundles and ellip tic o bjects The following discussion elab orates on the second author’s work with Baas and Dundas in [BDR04]. It is in tended to explain some of our in terest in Theorem 0.1. 5.1. Tw o-v ector bun dles. A 2-ve ctor bundle E of rank n ov er a base space X is represen ted b y a map X → | B GL n ( V ) | , w here V is the symmetric bimonoidal category of finite dimensional complex v ector spaces. A virt ual 2-v ector bund le E o v er X i s represen ted b y a map X → K ( V ), where K ( V ) the al gebraic K -theory of the 2-category of finitely generated free V -mo dules; see [BDR04, Thm. 4.10 ]. B y [BDRR], spectrificati on induces a weak equiv al ence Spt : K ( V ) → K ( k u ), so the 2- v ector bundles o v er X are geometric 0 -cycles for the cohomolo gy theory K ( k u ) ∗ ( X ). 5.2. Anomaly bu ndles. The preferred rational splitt ing of π : K ( ku ) → K ( Z ) defines an infinite lo op map det Q : K ( ku ) → ( B B U ⊗ ) Q , whic h ext ends the rational ization map o v er w : B B U ⊗ → K ( ku ) and agrees with the rel a tiv e ratio nal determinan t on K ( π ku ). ( W e do not know if there exists an in tegral determinan t map B GL ∞ ( k u ) + → B B U ⊗ in this case.) W e define t he rational determinan t bundle |E | = det( E ) of a virtual 2-v ector bundle represen t ed b y a map E : X → K ( V ) ≃ K ( k u ), as the comp osite map |E | : X E − → K ( k u ) det Q − − − → ( B B U ⊗ ) Q . W e define the rational anomaly bundle H → L X of E as the comp osite map H : L X L|E | − − → L ( B B U ⊗ ) Q r Q − → ( B U ⊗ ) Q , where r : L B B U ⊗ → B U ⊗ is the ret ra ction defined as t he infinite lo op cofib er of the constan t lo ops map B B U ⊗ → L B B U ⊗ . U p to rati onalization, H is a virtual v ector bundle of virtual dimension +1, i.e., a virtual line bundle. F urthermore, t he anomaly bundle rela t es the comp osition ⋆ of free l o ops, when defined, t o the tensor pro duct of virtual vec tor sp aces: t he square (5.3) L X × X L X ( H , H ) / / ⋆   ( B U ⊗ ) Q × ( B U ⊗ ) Q ⊗   L X H / / ( B U ⊗ ) Q comm utes up to coheren t i somorphism. 5.4. Gerb es. A 2-vector bundle o f rank 1 ov er X is t he same as a C ∗ -gerb e G , whic h is represen ted b y a ma p G : X → B B U (1). When viewed as a vi rtual 2- v ector bundle, via B B U (1) → B B U ⊗ → K ( k u ), the associat ed anomaly bundle is the complex line bundle o v er L X that is represen ted b y the comp osite L X LG − − → L B B U (1) r − → B U (1) . This i s precisely the a noma l y line bundle for G , as describ ed in [Br93, § 6.2]. Note that the rational anomal y bundles of virtual 2-vector bundles represen t general elemen ts in 1 + e K 0 ( L X ) ⊗ Q ⊂ K 0 ( L X ) ⊗ Q , whereas the anomaly li ne bundles of gerbes only represen t elemen t s in H 2 ( L X ). RA TIONAL ALGEBRAIC K-THEOR Y OF TOPOLO GICA L K-THEOR Y 19 5.5. S t ate spaces and action functionals. In ph ysical language, w e think o f a free lo op γ : S 1 → X as a clo sed string in a space-time X . F or a 2-vector bund le E → X , w e think of the fib er H γ (a virt ual v ector space) at γ of t he anomaly bundle H → L X as the state space of that string. Then the state space of a comp osite of tw o strings (or a disjoint union of t w o strings) is the tensor pro duct of the individual state spaces, as is usual i n quan tum mec hanics. Simila rly , the state space of an empt y set of strings is C . In the special case of anomaly line bundles for gerbes, the resulting state sp aces are only complex li nes, but in our generalit y they are virtual v ector spaces. These a r e m uc h closer t o the Hilb ert spaces usually considered in more analyt ical a pproac hes to this sub ject. There is evidence t hat a tw o-part differential-geometric structure ( ∇ 1 , ∇ 2 ) on E o v er X (somewhat l ik e a connection for a vector bun dle, but pro viding paral lel transp ort b oth for ob jects and for morphisms in the 2-v ector bundle) provides H → L X with a connection, and more generally an action functional S (Σ) : H ¯ γ 1 ⊗ · · · ⊗ H ¯ γ p → H γ 1 ⊗ · · · ⊗ H γ q , where Σ : F → X is a compact Riemann surface o ver X , with p incoming and q outgoing b oundary circles. The t ime dev elopmen t o f the ph ysical system i s t hen give n b y t he Euler–Lagra nge eq uat ions of the action functional. In a little more detail, the idea is that the primary form of parallel transp ort in ( E , ∇ 1 ) around γ provides an endo-fu nctor ˜ γ of the fiber category E x ∼ = V n o v er a c hosen p oin t x o f γ . More precisely , parall el transp ort only pro vides a zig-zag of functors connecting E x to itself, but the determi nant in ( B U ⊗ ) Q is st i ll we ll-defined. This “ holonom y” is then the fib er H γ = det( ˜ γ ) at γ of the anomaly bundle H . F or a moving string, sa y on the Riemann surface F , the secondary form of parall el transp ort ∇ 2 sp ecifies ho w the holonom y c hanges wit h t he string, and this de fines the connection ∇ on H → L X . In the gerb e case, this t heory has b een work ed o ut in [Br93, § 5.3], where ∇ 1 is called “connectiv e structure” and ∇ 2 is called “curving”. F or a closed surface F , S (Σ) : C → C is multiplication by a complex n um b er, whic h w ould only dep end on the rat ional t yp e o f E . Optimistically , t his asso ciation can pro duce a conformal inv arian t of F ov er X , which i n the case of gen us 1 surfaces w ould lead to an ellipti c mo dular form. Less naively , additional structure deriv ed from a string structure on X should account for the w eigh t of the mo dular form. With suc h structure, a 2-v ector bundle E with connective structure ∇ would q ualify as a Segal elliptic ob ject o v er X . 5.6. Op en strings. In t he presence of D -branes in the space-time X , w e can extend the anomaly bundle to also co ve r op en strings with end p oin ts restricted to lie on these D -branes; see [Mo04, § 3.4]. In this terminol ogy , t he (rat ional) determinan t bundle | E | → X pla ys the ro le of the B -field. By a (rati onal) D -bran e ( W , E ) in X w e will mean a subspace W ⊂ X together with a triv ialization E of the restricti on of the (rational ) determinan t bundle |E | to W . In terms of represen ting maps, E is a null-homotop y of the comp osite map W ⊂ X E − → K ( k u ) det Q − − − → ( B B U ⊗ ) Q . In similar terminolo gy , w e ma y refer to the determinan t bundle |E | → X as t he (rational) B -field . 20 CHRISTIA N AUSONI AND JOHN R OGNES When the B -field |E | is rationall y trivi al, then a second choice of trivi alization E amoun ts to a ch oice of n ull-homotop y of t he trivial ma p W → ( B B U ⊗ ) Q , or equiv al en tly to a map E : W → ( B U ⊗ ) Q . In o ther words, E is a virt ual vector bundle o v er W of virtual dimension +1, up t o rati onalization. In this case, the K - theory class of E → W in 1 + e K 0 ( W ) ⊗ Q is the “charge” of the D -brane ( W , E ). This conforms with the (early ) v i ew on D -branes as coming equipp ed wi th a charge [ E ] in top ologi cal K -theory . F or a general B -field |E | , the p ossible triv i alizations E of i ts restrictio n to W instead form a torsor under the group 1 + e K 0 ( W ) ⊗ Q . F or t w o suc h trivial i zations E and E ′ differ by a l o op of maps W → ( B B U ⊗ ) Q , or equiv alen tly a map E ′ − E : W → ( B U ⊗ ) Q . So [ E ′ − E ] is a top ologi cal K -theory cl a ss measuring the c harge difference b et w een the tw o D -branes ( W , E ) and ( W , E ′ ). Giv en t w o D -branes ( W 0 , E 0 ) and ( W 1 , E 1 ) in ( X , E ), we ha ve a comm utative diagram W 0 / / E 0   X |E |   W 1 o o E 1   P ( B B U ⊗ ) Q π / / ( B B U ⊗ ) Q P ( B B U ⊗ ) Q π o o where π : P Y → Y denotes the path space fibration co vering a based space Y . An op en string in X , constrained t o W 0 and W 1 at its ends, is a map γ : I → X wit h γ (0) ∈ W 0 and γ (1) ∈ W 1 . In other words, it is an elemen t in the homotopy pullbac k of the top ro w in the diagram abov e. Let Ω( X , W 0 , W 1 ) denote the space o f suc h op en strings. The homo t op y pullbac k of the low er ro w is Ω( B B U ⊗ ) Q ≃ ( B U ⊗ ) Q . Hence the 2-vector bundle E and the tw o D -branes sp ecify a map of homot o p y pullbac ks H : Ω( X , W 0 , W 1 ) → ( B U ⊗ ) Q that we call t he (rational , vi rtual) anomaly bundle of this space of op en strings. Again, we think of eac h fiber H γ at γ : ( I , 0 , 1) → ( X , W 0 , W 1 ) as the state space of that op en string. In t he presence of a suitable connection ( ∇ 1 , ∇ 2 ) on E → X , parallel transport in ( E , ∇ 1 ) alo ng γ induces a (zi g-zag) functor ˜ γ from E x to E y , with determinan t det( ˜ γ ) from the fib er of |E | at x = γ (0) to t he fiber at y = γ (1). The tri vializations of these tw o fib ers provided b y the D -brane data E 0 and E 1 , resp ectively , then agree up to a correction term, whic h is the fib er H γ in the anomaly bu ndle: det( ˜ γ )( E 0 ,x ) ∼ = H γ ⊗ E 1 ,y Again, t he secondary part of t he connection may induce a connection on H o v er Ω( X , W 0 , W 1 ), and mor e generally an action functional S (Σ), where no w Σ : F → X and the incoming and the outgoing parts of F are unions of circles and clo sed in terv al s. F or ex ample, a n op en st ri ng might spli t off a clo sed string. One adv an tage of t he ab o ve p erspective is that the state spaces of op en and closed strings arise in a compatible fashion, as the holonomy of parallel transp ort in t he 2-v ector bundle E , and this mak es t he construction of S ( Σ) feasible. The gerb e case is discussed i n [Br93, § 6.6]. RA TIONAL ALGEBRAIC K-THEOR Y OF TOPOLO GICA L K-THEOR Y 21 Reference s [AR02] Ch. Ausoni and J. 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