RATIONAL ALGEBRAIC K-THEORY
OF TOPOLOGICAL K-THEORY
Christian Ausoni and John Rognes
August 14th 2007
Abstract. We show that after rationalization there is a homotopy fiber se*
*quence
BBU ! K(ku) ! K(Z). We interpret this as a correspondence between the
virtual 2-vector bundles over a space X and their associated anomaly bund*
*les over
the free loop space LX. We also rationally compute K(KU) by using the loc*
*alization
sequence, and K(MU) by a method that applies to all connective S-algebras.
Introduction
We are interested in the algebraic K-theory K(ku) of the connective complex *
*K-
theory spectrum ku. By the calculations of [AR02, Thm. 0.4] and [Au], the "mod p
and v1" homotopy of K(ku) is purely v2-periodic, a distinctive homotopy theoret*
*ic
property it shares with the spectra representing elliptic cohomology [LRS95] and
topological modular forms [Ho02, x4]. The theory of 2-vector bundles from [BDR0*
*4]
and [BDRR] therefore exhibits K(ku) as a geometrically defined form of elliptic
cohomology. In Section 5 we outline how a 2-vector bundle with connecting data,
similar to a connection in a vector bundle, is thought to specify a 1 + 1-dimen*
*sional
conformal field theory. Since these 2-vector bundles are also effective cycles *
*for the
form of elliptic cohomology theory represented by K(ku), we have some justifica*
*tion
for referring to them as elliptic objects, as proposed by Segal [Se89].
As illustrated by the authors' calculations referred to above, the arithmeti*
*c and
homotopy-theoretic information captured by algebraic K-theory becomes more ac-
cessible after the introduction of suitable finite coefficients. However, for *
*the ex-
traction of C-valued numerical invariants from a conformal field theory, only t*
*he
rational homotopy type of K(ku) will matter. We are grateful to Ib Madsen and
Dennis Sullivan for insisting that for such geometric applications, we should f*
*irst
want to compute K(ku) rationally. To this end we can offer the following theore*
*m.
There is a unit inclusion map
w :BBU ! BGL1(ku) ! BGL1 (ku)+ ! K(ku) .
Let ss :K(ku) ! K(Z) be induced by the zero-th Postnikov section ku ! HZ. The
composite ss O w is the constant map to the base-point of the 1-component of K(*
*Z).
Typeset by AM S-T*
*EX
1
2 CHRISTIAN AUSONI AND JOHN ROGNES
Theorem 0.1. (a) After rationalization,
BBU -w! K(ku) -ss!K(Z)
is a split homotopy fiber sequence.
(b) The Poincar'e series of K(ku) is
3 t5 t3 + 2t5
1 + __t___1+-_t2___1=-1t4+ ________1.- t4
(c) There is a rational determinant map
detQ :BGL1 (ku)+ ! BGL1(ku)Q
that, in its relative form for ku ! HZ, rationally splits w.
By thePPoincar'e series of a space X of finite type, we mean the formal power
series n 0 rntn in Z[[t]], where rn is the rank of ssn(X). Theorem 0.1 is pro*
*ved by
assembling Theorem 2.5(a) and Theorem 4.8(a). The other parts of those theorems
prove similar results for K(ko) and K(`), where ` = BP <1>.
The splitting of K(ku)Q shows that for a (virtual) 2-vector bundle over X, r*
*ep-
resented by a map E :X ! K(ku), the rational information splits into two pieces.
The less interesting piece is the decategorified information carried by the dim*
*ension
bundle dim (E) = ss O E :X ! K(Z). The more interesting piece is the determinant
bundle
|E| = detQ O E :X ! (BBU )Q .
To specify |E| is equivalent to specifying a rational virtual vector bundle H :*
*LX !
(BU )Q over the free loop space LX = Map (S1, X), called the "anomaly bun-
dle", subject to a coherence condition relating the composition of free loops, *
*when
defined, to the tensor product of virtual vector spaces. See diagram (5.3).
The conclusion is that for rational purposes the information in a 2-vector b*
*un-
dle E over X is the same as that in its anomaly bundle H over LX (subject to the
indicated coherence condition, which we think of as implicit, and together with*
* the
dimension bundle dim (E) over X, which we tend to ignore). In physical language,
the fiber of the anomaly bundle at a free loop fl :S1 ! X plays the role of the
state space of fl viewed as a closed string in X. The advantage of 2-vector bun*
*dles
over their homotopy-theoretic alternatives, such as representing maps to classi*
*fying
spaces or bundles of ku-modules, is that they are geometrically modeled in term*
*s of
vector bundles, rather than virtual vector bundles. This seems to become an ess*
*en-
tial virtue when one wants to treat differential-geometric structures like conn*
*ections
on these bundles.
We also compute the rational algebraic K-theory K(KU) of the periodic complex
K-theory spectrum KU. To this end we evaluate the (rationalized) transfer map
ss* in the localization sequence
K(Z) -ss*!K(ku) -ae!K(KU)
predicted by the second author and established by Blumberg and Mandell [BM].
RATIONAL ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 3
Theorem 0.2. (a) There is a rationally split homotopy fiber sequence of infini*
*te
loop spaces
K(ku) -ae!K(KU) -@!BK(Z)
where ae is induced by the connective cover map ku ! KU, and BK(Z) denotes the
first connected delooping of K(Z).
(b) The Poincar'e series of K(KU) is
3 + 2t5 + t6
(1 + t) + t___________1.- t4
See Theorem 2.12 for our proof.
As stated, Theorems 0.1 and 0.2 only concern the algebraic K-theory of topo-
logical K-theory, but we develop our proofs in the greater generality of arbitr*
*ary
connective S-algebras. In Section 1 we observe how the calculation by Goodwillie
[Go86] of the relative rational algebraic K-theory for a 1-connected map R ! ss*
*0R
of simplicial rings (which generalizes earlier calculations by Hsiang and Staff*
*eldt
[HS82] for simplicial group rings), also applies to determine the relative rati*
*onal
algebraic K-theory for a 1-connected map A ! Hss0A of connective S-algebras.
The answer is given in terms of negative cyclic homology; see Theorem 1.5 and
Corollary 1.6.
When ss0A is close to Z, and A ! Hss0A is a "rational de Rham equivalence",
we get a very simple expression for the relative rational algebraic K-theory as*
* the
image of Connes' B-operator on Hochschild homology; see Proposition 1.8 and
Corollary 1.9. These hypotheses apply to a number of interesting examples of co*
*n-
nective S-algebras, including the K-theory spectra ku, ko and `, and the bordism
spectra MU, MSO and MSp. We work these examples out in Theorem 2.5 and
Theorem 3.4, respectively.
In Section 4 we consider the unit inclusion map w :BGL1(A) ! K(A). For
commutative A, the rationalization AQ is equivalent as a commutative HQ-algebra
to the Eilenberg-Mac Lane spectrum HR of a commutative simplicial Q-algebra,
so we can use the determinant GLn(R) ! GL1(R) to define a rational determinant
map
detQ: BGL1 (A)+ ! BGL1(A)Q .
We show in Proposition 4.7 that the composite detQ Ow is the rationalization ma*
*p,
and apply this in Theorem 4.8 to show that w induces a rational equivalence from
BBU to the homotopy fiber of ss :K(ku) ! K(Z), and similarly for ko and `. This
last step is a counting argument; it does not apply for MU or the other bordism
spectra.
Acknowledgments. In an earlier version of this paper, we emphasized a trace
map to T HH(ku) over the determinant map to (BBU )Q , in order to detect the
image of w in K(ku). We are grateful to Bjorn Dundas for reminding us of the
existence of determinants for commutative simplicial rings, which is half of th*
*e basis
for the existence of the map detQ defined in Lemma 4.6. We are also grateful to
Mike Mandell and Brooke Shipley for help with some of the references concerning
commutative simplicial Q-algebras given in Subsection 1.1.
4 CHRISTIAN AUSONI AND JOHN ROGNES
x1. Rational algebraic K-theory of connective S-algebras
1.1. S-algebras. We work in one of the modern symmetric monoidal categories
of spectra [EKMM97], [Ly99], [HSS00], [MMSS01], which we shall refer to as S-
modules. The monoids (resp. commutative monoids) in this category are called
S-algebras (resp. commutative S-algebras), and are equivalent to the A1 ring s*
*pec-
tra (resp. E1 ring spectra) considered since the 1970's. The Eilenberg-Mac Lane
functor H :R 7! HR maps the category of simplicial rings (resp. commutative
simplicial rings) to the category of S-algebras (resp. commutative S-algebras).
Schwede proved in [Schw99, 4.5] that H is part of a Quillen equivalence from
the category of simplicial rings to the category of connective HZ-algebras. The*
*re
is a similar equivalence between the category of commutative simplicial Q-algeb*
*ras
and the category of connective commutative HQ-algebras.
One form of the latter equivalence appears in [KM95, II.1.3]. In a little m*
*ore
detail, the category of connective commutative HQ-algebras is "connective Quill*
*en
equivalent" [MMSS01, p. 445] to the category of connective E1 HQ-ring spec-
tra [EKMM97, II.4], and connective E1 HQ-ring spectra are the E1 objects in
connective HQ-modules, which are Quillen equivalent to E1 simplicial Q-algebras
[Schw99, 4.4]. The monads defining E1 algebras and commutative algebras in sim-
plicial Q-modules are weakly equivalent, since for every j 0 the group homolo*
*gy
of j with coefficients in any Q-module is concentrated in degree zero. Hence E1
simplicial Q-algebras are Quillen equivalent to commutative simplicial Q-algebr*
*as
[Ma03, 6.7].
The homotopy categories of commutative simplicial rings and connective com-
mutative HZ-algebras are not equivalent.
1.2. Linearization. Let A be a connective S-algebra. We write ss = ssA :A !
Hss0A for its zero-th Postnikov section, and define the linearization map ~ =
~A :A ! HZ ^ A to be ss ^ id: A ~= S ^ A ! HZ ^ A. It is a ss0-isomorphism
and a rational equivalence of connective S-algebras. For each (simplicial or to*
*po-
logical) monoid G let S[G] = 1 G+ be its unreduced suspension spectrum. For
A = S[G], the linearization map ~: S[G] ! HZ ^ S[G] ~= HZ[G] agrees with the
map considered by Waldhausen [Wa78, p. 43].
In general, HZ ^ A is a connective HZ-algebra, so by the first Quillen equiv*
*a-
lence above there is a naturally associated simplicial ring R with HZ ^ A ' HR.
For connective commutative A, the rationalization AQ = HQ ^ A is a connective
commutative HQ-algebra, so by the second Quillen equivalence above there is a
naturally associated commutative simplicial Q-algebra R with AQ ' HR.
1.3. Algebraic K-theory. For a general S-algebra A, the algebraic K-theory
space K(A) can be defined as |hSoCA |, where CA is the category of finite cell*
* A-
module spectra and their retracts, So denotes Waldhausen's So-construction [Wa8*
*5,
x1.3], and |h(-)| indicates the nerve of the subcategory of weak equivalences. *
*By
iterating the So-construction, we may also view K(A) as a spectrum. For con-
nective S-algebras, K(A) can alternatively be defined in terms of Quillen's plu*
*s-
construction as K0(ss0A)xBGL1 (A)+ , and then the two definitions are equivalen*
*t,
see [EKMM97, VI.7.1]. We write K(R) for K(HR), and similarly for other functors.
Any map A ! A0 of connective S-algebras that is a ss0-isomorphism and a
rational equivalence induces a rational equivalence K(A) ! K(A0), by [Wa78, 2.2*
*].
The proof goes by observing that BGLn(A) ! BGLn(A0) is a rational equivalence
RATIONAL ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 5
for each n. In particular, for R with HZ ^ A ' HR there is a natural rational
equivalence ~: K(A) ! K(HZ ^ A) ' K(R). For X ' BG, Waldhausen writes
A(X) for K(S[G]), and ~: A(X) ! K(Z[G]) is a rational equivalence. In this case,
A(X) can also be defined as the algebraic K-theory of a category Rf(X) of suita*
*bly
finite retractive spaces over X, see [Wa85, x2.1].
1.4. Cyclic homology. There is a natural trace map tr :K(A) ! T HH(A) to
the topological Hochschild homology of A, see [BHM93, x3]. The target is a cycl*
*ic
object in the sense of Connes, hence carries a natural S1-action. There exists*
* a
model for the trace map that factors through the fixed points of this circle1ac*
*tion
[Du04], hence it also factors through the homotopy fixed points T HH(A)hS . We
get a natural commutative triangle
K(A) __ff__//T HH(A)hS1
LLL
LLL |F
trLLLL&& fflffl||
T HH(A)
where the Frobenius map F forgets about S1-homotopy invariance. For any sim-
plicial ring R there are natural isomorphisms T HH*(HRQ ) ~=HH*(R Q) (Hoch-
schild homology) and T HH*(HRQ )hS1 ~=HC-*(R Q) (negative cyclic homology).
See e.g. [EKMM97, IX.1.7] and [CJ90, 1.3(3)]. With these identifications, the t*
*ri-
angle above realizes the commutative diagram of [Go86, II.3.1]. In [Go86, II.3.*
*4],
Goodwillie proved:
Theorem 1.5. Let f :R ! R0 be a map of simplicial rings, with ss0R ! ss0R0 a
surjection with nilpotent kernel. Then
K(R)Q __ff__//HC- (R Q)
f || |f|
fflffl|ff fflffl|
K(R0)Q ______//HC- (R0 Q)
is homotopy Cartesian, i.e., the map of vertical homotopy fibers
ff :K(f)Q ! HC- (f Q)
is an equivalence.
Here we write K(f) for the homotopy fiber of K(R) ! K(R0), so that there is
a long exact sequence
. .!.K*+1 (R0) ! K*(f) ! K*(R) ! K*(R0) ! . .,.
and similarly for other functors from (S-)algebras to spaces. (Goodwillie writ*
*es
K(f) for a delooping of our K(f), but we need to emphasize fibers over cofibers*
*.)
We write K(R)Q for the rationalization of K(R), and similarly for other spaces *
*and
S-algebras.
6 CHRISTIAN AUSONI AND JOHN ROGNES
Corollary 1.6. Let g :A ! A0 be a map of connective S-algebras, with ss0A !
ss0A0 a surjection with nilpotent kernel, Then
K(A)Q __ff__//T HH(AQ )hS1
g|| |g|
fflffl|ff fflffl|1
K(A0)Q ______//T HH(A0Q)hS
is homotopy Cartesian, i.e., the map of vertical homotopy fibers
1
ff :K(g)Q ! T HH(gQ )hS
is an equivalence.
Proof. Given g :A ! A0we find f :R ! R0with HZ^A ' HR and HZ^A0' HR0
making the diagram
A ___~__//HR
g|| |Hf|
fflffl|~ fflffl|
A0 ______//HR0
homotopy commute. Then ~: K(A) ! K(R) is a rational equivalence and AQ '
H(R Q), so the square in the corollary is equivalent to the square in Goodwil*
*lie's
theorem.
1.7. De Rham homology. The (spectrum level) circle action on T HH(A) in-
duces a suspension operator d: T HH*(A) ! T HH*+1 (A), analogous to Connes'
operator B :HH*(R) ! HH*+1 (R). When AQ ' H(R Q), these operators are
compatible under the isomorphism T HH*(AQ ) ~=HH*(R Q). In general dd = dj
is not zero [He97, 1.4.4], where j is the stable Hopf map, but in the algebraic*
* case
BB = 0, so one can define the de Rham homology
HdR*(R) = ker(B)= im(B)
of a simplicial ring R as the homology of HH*(R) with respect to the B-operator.
For a map g :A ! A0of S-algebras, the homotopy fiber T HH(g) of T HH(A) !
T HH(A0) inherits a circle action and associated suspension operator. Similarly*
*, for
a map f :R ! R0 of simplicial rings there is a relative B-operator acting on the
term HH*(f) in the long exact sequence
. .!.HH*+1 (R0) ! HH*(f) ! HH*(R) ! HH*(R0) ! . .,.
and we define HdR*(f) to be the homology of HH*(f) with respect to this B-
operator. We say that f :R ! R0 is a de Rham equivalence if HdR*(f) = 0, and
that f is a rational de Rham equivalence if HdR*(f Q) = 0. If we assume that
HH*(R) ! HH*(R0) is surjective in each degree, then there is a long exact seque*
*nce
. .!.HdR*+1(R0) ! HdR*(f) ! HdR*(R) ! HdR*(R0) ! . .,.
in which case f is a de Rham equivalence if and only if HdR*(R) ! HdR*(R0) is an
isomorphism in every degree.
RATIONAL ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 7
Proposition 1.8. If f :R ! R0 is a de Rham equivalence, then there is an exact
sequence
0 ! HC-*(f) -F!HH*(f) -B!HH*+1 (f)
that identifies HC-*(f) with ker(B) HH*(f).
Proof. By analogy with the homotopy fixed point spectral sequence for T HH(g)hS*
*1,
there is a second quadrant homological spectral sequence
E2**= Q[t] HH*(f) =) HC-*(f)
with t 2 E2-2,0and d2(ti . x) = ti+1 . B(x) for all x 2 HH*(f), i 0. So E3***
*is
the sum of ker(B) HH*(f) in the zero-th column and a copy of HdR*(f) in each
even, negative column. By assumption the latter groups are all zero, so the spe*
*ctral
sequence collapses to the zero-th column at the E3-term. The Frobenius F is the
edge homomorphism for this spectral sequence, and the assertion follows.
Corollary 1.9. Let A be a connective S-algebra such that ss0A is any localizat*
*ion
of the integers, and let R be a simplicial Q-algebra with AQ ' HR.
(a) The homotopy fiber sequence
K(ssA ) ! K(A) -ssA-!K(ss0A)
is rationally split, where ssA :A ! Hss0A is the zero-th Postnikov section.
(b) There are equivalences
1 -
K(ssA )Q -ff!'T HH(ssAQ )hS ' HC (ssR ) ,
where ssR :R ! ss0R = Q is the zero-th Postnikov section.
Suppose furthermore that HdR*(R) ~=Q is trivial in positive degrees.
(c) The map ssR is a de Rham equivalence, and the Frobenius map identifies
HC-*(ssR ) with the positive-degree part of
ker(B) HH*(R) ~=T HH*(A) Q .
That part is also equal to im (B) T HH*(A) Q.
(d) The trace map tr :K(A) ! T HH(A) induces the composite identification of
K*(ssA ) Q with the positive-degree part of ker(B) T HH*(A) Q.
Proof. (a) Write ss0A = Z(P) for some (possibly empty) set of primes P . The un*
*it
map i: S ! A factors through S(P), and the composite map S(P) ! A ! Hss0A is
a ss0-isomorphism and a rational equivalence. Hence the composite
K(S(P)) ! K(A) -ssA-!K(ss0A)
is a rational equivalence.
(b) The map ssA :A ! Hss0A induces the identity on ss0, so ff is an equivale*
*nce
by Corollary 1.6. We recalled the second identification in Subsection 1.4. It i*
*s clear
that ss0R = ss0AQ = ss0A Q = Q.
(c) Since HH*(Q) = Q is trivial in positive degrees, the map HH*(R) !
HH*(Q) is surjective in each degree, so ssR is a de Rham equivalence if (and o*
*nly
8 CHRISTIAN AUSONI AND JOHN ROGNES
if) HdR*(R) ~= HdR*(Q) = Q is trivial in all positive degrees. The homotopy fi*
*ber
sequence
HH(ssR ) ! HH(R) ! HH(Q)
identifies HH*(ssR ) with the positive-degree part of HH*(R), so ker(B) HH*(s*
*sR )
is the positive-degree part of ker(B) HH*(R). The identification im(B) = ker(*
*B)
in positive degrees is of course equivalent to the vanishing of HdR*(R) in posi*
*tive
degrees.
(d) The trace map factors as tr = F O ff.
x2. Examples from topological K-theory
2.1. Connective K-theory spectra. Let ku be the connective complex K-
theory spectrum, ko the connective real K-theory spectrum, and ` = BP <1> the
Adams summand of ku(p), for p an odd prime. These are all commutative S-
algebras. We write 1 ku = BU x Z, 1 ko = BO x Z and 1 ` = W x Z(p)for
the underlying infinite loop spaces (see [Ma77, V.3-4]). The homotopy units form
infinite loop spaces, namely GL1(ku) = BU x { 1}, GL1(ko) = BO x { 1} and
GL1(`) = W xZx(p). The homotopy algebras are ss*ku = Z[u] with |u| = 2, ss*ko =
Z[j, ff, fi]=(2j, j3, jff, ff2- 4fi) with |j| = 1, |ff| = 4, |fi| = 8, and ss*`*
* = Z(p)[v1] with
|v1| = 2p - 2. The complexification map ko ! ku takes j to 0, ff to 2u2 and fi *
*to
u4. The inclusion ` ! ku(p)takes v1 to up-1 .
Proposition 2.2. (a) There are ss0-isomorphisms and rational equivalences
~ :S[ S3] ! S[K(Z, 2)] ! ku
of S-algebras, so kuQ ' HQ[ S3] as homotopy commutative HQ-algebras, where
Q[ S3] is a simplicial Q-algebra, and kuQ ' HQ[K(Z, 2)] as commutative HQ-
algebras, where Q[K(Z, 2)] is a commutative simplicial Q-algebra.
(b) There are ss0-isomorphisms and rational equivalences ~ff:S[ S5] ! ko and
~v1:S(p)[ S2p-1 ] ! `, so koQ ' HQ[ S5] and `Q ' HQ[ S2p-1 ] as HQ-algebras,
where Q[ S5] and Q[ S2p-1 ] are simplicial Q-algebras.
(c) In particular, there is a rational equivalence ~ :A(S3) ! K(ku) of S-
algebras, a rational equivalence A(K(Z, 3)) ! K(ku) of commutative S-algebras,
and a rational equivalence ~ff:A(S5) ! K(ko) of S-modules.
Proof. (a) Let BS3 ! K(Z, 4) represent a generator of H4(BS3). It induces a
double loop map S3 ! K(Z, 2), such that the composite S2 ! S3 ! K(Z, 2)
represents a generator of ss2K(Z, 2). The inclusions K(Z, 2) ' BU(1) ! BU !
GL1(ku) are infinite loop maps, and the generator of ss2K(Z, 2) maps to a gener-
ator of ss2GL1(ku). By adjunction we have an E2 ring spectrum map S[ S3] !
S[K(Z, 2)] and an E1 ring spectrum map S[K(Z, 2)] ! ku, with composite the E2
ring spectrum map ~: S[ S3] ! ku.
These are rational equivalences, because ss*S[ S3] Q ~= H*( S3; Q) ~= Q[x],
H*(K(Z, 2); Q) ~=Q[b] and ss*ku Q = Q[u], with ~ mapping x via b to u. We may
take the Kan loop group of S3 (a simplicial group, see e.g. [Wa96]) as our model
for S3, and rigidify ~ to a map of S-algebras. Following [FV], there remains an
E1 = A1 operad action on these S-algebras and ~, which in particular implies t*
*hat
~: A(S3) ! K(ku) is homotopy commutative.
RATIONAL ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 9
(b) For the real case, let S4 ! BO GL1(ko) represent a generator of
ss4GL1(ko). By the loop structure on the target, it extends to a loop map S5 !
GL1(ko), with left adjoint an A1 ring spectrum map ~ff:S[ S5] ! ko. It is a ra*
*tio-
nal equivalence, because ss*S[ S5] Q ~=H*( S5; Q) ~=Q[y] and ss*ko Q = Q[ff*
*],
with ~ffmapping y to ff. We interpret S5 as the Kan loop group, and form the
simplicial Q-algebra Q[ S5] as its rational group ring. The Adams summand case
is entirely similar, starting with a map S2p-2 ! W GL1(`).
(c) By [FV] and naturality there is an A1 operad action on the induced map *
*of
spectra A(S3) = K(S[ S3]) ! K(ku) (rather than of spaces), which we can rigidify
to a map of S-algebras. The S-algebra multiplication A(S3) ^ A(S3) ! A(S3) is
induced by the group multiplication S3 x S3 ! S3.
Lemma 2.3. (a) For any integer n 1 the simplicial Q-algebra R = Q[ S2n+1 ]
has Hochschild homology
HH*(R) ~=Q[x] E(dx)
with |x| = 2n, where Connes' B-operator satisfies B(x) = dx. Here E(-) denotes
the exterior algebra. (b) The de Rham homology HdR*(R) ~= Q is concentrated in
degree zero, so ssR :R ! Q is a de Rham equivalence. (c) The positive-degree pa*
*rt
of ker(B) HH*(R) is
Q[x]{dx} = Q{dx, x dx, x2 dx, . .}..
Proof. (a) The Hochschild filtration on the bisimplicial Q-algebra HH(R) yields*
* a
spectral sequence
(2.4) E2**= HH*(ss*(R)) =) HH*(R) ,
and ss*(R) = Q[x] with |x| = 2n. The Hochschild homology of this graded com-
mutative ring is Q[x] E(dx), where dx 2 E21,2nis the image of x under Connes'
B-operator. The spectral sequence collapses at that stage, for bidegree reasons.
(b,c) The B-operator is a derivation, hence takes xm to mxm-1 dx for all m*
* 0.
It follows easily that the de Rham homology is trivial in positive degrees, and*
* that
ker B is as indicated.
By combining Corollary 1.9, Proposition 2.2 and Lemma 2.3, we obtain the
following result.
Theorem 2.5. (a) There is a rationally split homotopy fiber sequence
K(ssku) ! K(ku) -ss!K(Z)
and the trace map tr :K(ku) ! T HH(ku) identifies
K*(ssku) Q ~=Q[u]{du}
with its image in T HH*(ku) Q ~=Q[u] E(du). Here |u| = 2 and |du| = 3, so
K(ssku) has Poincar'e series t3=(1 - t2).
10 CHRISTIAN AUSONI AND JOHN ROGNES
(b) Similarly, there are rationally split homotopy fiber sequences
K(ssko) ! K(ko) -ss!K(Z)
K(ss`) ! K(`) -ss!K(Z(p))
and the trace maps identify
K*(ssko) Q ~= Q[ff]{dff}
K*(ss`) Q ~= Q[v1]{dv1}
with their images in T HH*(ko) Q ~=Q[ff] E(dff) and T HH*(`) Q ~=Q[v1]
E(dv1), respectively. Hence K(ssko) has Poincar'e series t5=(1 - t4), whereas K*
*(ss`)
has Poincar'e series t2p-1=(1 - t2p-2).
Remark 2.6. The Poincar'e series of K(Z) is 1 + t5=(1 - t4) by Borel's calculat*
*ion
[Bo74]. Hence the (common) Poincar'e series of K(ku) and A(S3) is
1 + t3=(1 - t2) + t5=(1 - t4) = 1 + (t3 + 2t5)=(1 - t4) ,
whereas the Poincar'e series of K(ko) and A(S5) is 1 + 2t5=(1 - t4). More gener*
*ally,
we recover the Poincar'e series 1 + t5=(1 - t4) + t2n+1 =(1 - t2n) of A(S2n+1 )*
* for
n 1, from [HS82, Cor. 1.2]. The group K1(Z(p)) is not finitely generated, so *
*we
do not discuss the Poincar'e series of K(`).
2.7. Periodic K-theory spectra. Let KU be the periodic complex K-theory
spectrum, KO the periodic real K-theory spectrum, and L = E(1) the Adams
summand of KU(p), for p an odd prime. We have maps of commutative S-algebras
HZ- ssku -ae!KU
with associated maps of "brave new" affine schemes [TV, x2]
(2.8) Spec(Z) -ss!Spec(ku)- aeSpec(KU) .
Let i: S ! S[ S3] and c: S[ S3] ! S be induced by the inclusion map * ! S3
and the collapse map S3 ! *, respectively. We have a map of horizontal cofiber
sequences
(2.9) 2S[ S3] __x__//_S[ S3]__c___//S
||2~ |~| |~|
fflffl| u fflffl|ss fflffl|
2ku _________//_ku_______//HZ
where the top row exhibits S as a two-cell S[ S3]-module, and the bottom row
exhibits HZ as a two-cell ku-module. (In each case, the two cells are in dimens*
*ion
zero and three.) There are algebraic K-theory transfer maps c* :A(*) ! A(S3) and
ss* :K(Z) ! K(ku) (with a lower star, in accordance with the variance conventio*
*ns
from algebraic geometry and (2.8)), that are induced by the functors that view
finite cell S-modules as finite cell S[ S3]-modules, and finite cell HZ-modules*
* as
RATIONAL ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 11
finite cell ku-modules, respectively. In terms of retractive spaces, c* is ind*
*uced
by the exact functor Rf(*) ! Rf(S3) that takes a pointed space X AE * to the
retractive space X x S3 AE S3. The transfer maps are compatible, by (2.9), so we
have a commutative diagram with vertical rational equivalences
(2.10) A(*) ___c*_//A(S3)
|~| ~||
fflffl|ss* fflffl|ae
K(Z) ______//K(ku)______//K(KU) .
The bottom row is a homotopy fiber sequence by the localization theorem of [BM].
Lemma 2.11. The transfer map c* :A(*) ! A(S3) is null-homotopic, as a map
of A(*)-module spectra. The transfer map ss* :K(Z) ! K(ku) is rationally null-
homotopic, again as a map of A(*)-module spectra.
Proof. The projection formula asserts that c* is an A(S3)-module map, where
c: A(S3) ! A(*) makes A(*) an A(S3)-module. Restricting the module struc-
tures along i: A(*) ! A(S3), we see that c* is a map of A(*)-module spectra, and
the source is a free A(*)-module of rank one. Hence it suffices to show that c**
* takes
a generator of ss0A(*), represented say by S0 AE *, to zero in ss0A(S3) ~=Z. Bu*
*t c*
maps that generator to the class of S0 x S3 AE S3, which corresponds to its Eul*
*er
characteristic O(S3) = 0.
The conclusion for ss* follows from that for c*, via the rational equivalenc*
*es ~
and ~.
Note the utility of the comparison with A-theory at this point, since we do *
*not
have an S-algebra map K(Z) ! K(ku) that is analogous to i: A(*) ! A(S3).
Theorem 2.12. There are rationally split homotopy fiber sequences
K(ku) -ae!K(KU) -@!BK(Z)
K(`) -ae!K(L) -@!BK(Z(p))
of infinite loop spaces. Hence the Poincar'e series of K(KU) is
(1 + t) + (t3 + 2t5 + t6)=(1 - t4) .
Proof. The claims for KU follow by combining Theorem 2.5(a) and Lemma 2.11.
The proof of the claim for L is completely similar, using that HZ(p) is a two-c*
*ell
`-module, with cells in dimension zero and (2p - 1). By [BM] there is a homotopy
fiber sequence K(Z(p)) ! K(`) ! K(L).
Remark 2.13. We do not know how to relate K(ko) with K(KO), so we do not
have a rational calculation of K(KO). However, KO ! KU is a Z=2-Galois ex-
tension of commutative S-algebras, in the sense of [Ro, x4.1], so it is plausib*
*le
that K(KO) ! K(KU)hZ=2 is close to an equivalence. Here Z=2 acts on KU by
complex conjugation, and
ss*(K(KU)hZ=2) Q ~=[K*(KU) Q]Z=2 .
12 CHRISTIAN AUSONI AND JOHN ROGNES
The conjugation action on ku fixes K(Z), and acts on K*(ssku) Q ~=Q[u]{du} by
sign on u and du, hence fixes Q[u2]{udu} ~=Q[ff]{dff} ~=K*(ssko) Q. So K(ko) !
K(ku)hZ=2 is a rational equivalence. The conjugation action also fixes BK(Z) af*
*ter
rationalization, so the Poincar'e series of K(KU)hZ=2 is (1 + t) + (2t5 + t6)=(*
*1 - t4).
Remark 2.14. We expect that c* and ss* are essential (not null-homotopic) as ma*
*ps
of A(S3)-module spectra and K(ku)-module spectra, respectively. In other words,
we expect that K(ku) ! K(KU) ! K(Z) is a non-split extension of K(ku)-
module spectra. This expectation is to some extent justified by the fact that *
*the
cofiber T HH(ku|KU) of the T HH-transfer map ss* :T HH(Z) ! T HH(ku) sits
in a non-split extension T HH(ku) ! T HH(ku|KU) ! T HH(Z) of T HH(ku)-
module spectra. See [Au05, 10.4], or [HM03, Lemma 2.3.3] for a similar result *
*in
an algebraic case.
x3. Examples from smooth bordism
3.1. Oriented bordism spectra. Let MU be the complex bordism spectrum,
MSO the real oriented bordism spectrum, and MSp the symplectic bordism spec-
trum. These are all connective commutative S-algebras, given by the Thom spectra
associated to infinite loop maps from BU, BSO and BSp to BSF = BSL1(S), re-
spectively. We recall that
H*(BU) ~=Z[bk | k 1]
with |bk| = 2k, while H*(BSO; Z[1=2]) ~= H*(BSp; Z[1=2]) ~= Z[1=2][qk | k 1]
with |qk| = 4k.
The Thom equivalence ` :MU ^ MU ! MU ^ S[BU] induces an equivalence
HZ ^ MU ' HZ ^ S[BU] = HZ[BU]. Combined with the Hurewicz map ss :S !
HZ we obtain a chain of maps of commutative S-algebras
MU -! HZ ^ MU ' HZ[BU]- S[BU] ,
that are ss0-isomorphisms and rational equivalences. There are similar chains
MSO -! HZ ^ MSO ' HZ[BSO] - S[BSO] and MSp -! HZ ^ MSp '
HZ[BSp]- S[BSp], and all induce rational equivalences
K(MU) -! K(Z[BU])- A(BBU)
(3.2) K(MSO) -! K(Z[BSO])- A(BBSO)
K(MSp) -! K(Z[BSp])- A(BBSp)
of commutative S-algebras. Here we view BU ' BBU as the Kan loop group of
BBU, Z[BU] is the associated simplicial ring, and similarly for BSO and BSp.
Lemma 3.3. (a) The simplicial Q-algebra R = Q[BU] with ss*R = H*(BU; Q) =
Q[bk | k 1] has Hochschild homology
HH*(R) ~=Q[bk | k 1] E(dbk | k 1) ,
with Poincar'e series Y 2k+1
h(t) = 1_+_t____2k,
k 1 1 - t
RATIONAL ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 13
and Connes' operator acts by B(bk) = dbk.
(b) The de Rham homology HdR*(R) ~= Q is concentrated in degree zero, so
ssR :R ! Q is a de Rham equivalence.
(c) The Poincar'e series of ker(B) HH*(R) is
k(t) = 1_+_th(t)_1.+ t
(d) The simplicial Q-algebra Rso = Q[BSO] ' Q[BSp], with ss*Rso = Q[qk |
k 1], has Hochschild homology
HH*(Rso) ~=Q[qk | k 1] E(dqk | k 1) .
Q
Its Poincar'e series is hso(t) = k 1 (1 + t4k+1)=(1 - t4k). The map Rso ! Q
is a de Rham equivalence, and ker(B) HH*(Rso) has Poincar'e series kso(t) =
(1 + thso(t))=(1 + t).
Proof. (a) In this case the spectral sequence (2.4) has E2**= HH*(Q[bk | k 1]*
*) ~=
Q[bk | k 1] E(dbk | k 1). The algebra generators are in filtrations 0 and*
* 1,
so E2 = E1 . This term is free as a graded commutative Q-algebra, so HH*(R) is
isomorphic to the E1 -term.
(b) The homology of Q[bk] E(dbk) with respect to B is just Q, for each k *
* 1,
so by the K"unneth theorem the de Rham homology of HH*(R) is also just Q.
(c) WritePHn for HHn(R) and Kn for ker(B :Hn ! Hn+1 ). Let hn = dim QHn,
so h(t) = n 0 hntn, and kn = dim QKn. In view of the exact sequence
0 ! Q ! H0 -d!H1 -d!. .-.d!Hn-1 ! Kn ! 0
we find that 1 - (-1)nkn = h0 - h1 + . .+.(-1)n-1 hn-1 , so
X X
tnkn - (-t)n = th(t) - t2h(t) + . .+.(-t)m+1 h(t) + . ...
n 0 n 0
P
It follows that the Poincar'e series k(t) = n 0 kntn for ker(B) satisfies k(*
*t) -
1=(1 + t) = th(t)=(1 + t).
(d) The only change from the complex to the oriented real and symplectic cas*
*es
is in the grading of the algebra generators, which (as long as they remain in e*
*ven
degrees) plays no role for the proofs.
Theorem 3.4. (a) There is a rationally split homotopy fiber sequence
K(ssMU ) ! K(MU) -ss!K(Z)
and the trace map tr :K(MU) ! T HH(MU) identifies K*(ssMU ) Q with the
positive-degree part of ker(B) in
T HH*(MU) Q ~=Q[bk | k 1] E(dbk | k 1) ,
where |bk| = 2k and B(bk) = dbk. Hence K(ssMU ) has Poincar'e series
k(t) - 1 = th(t)_-_t_1.+ t
14 CHRISTIAN AUSONI AND JOHN ROGNES
(b) There are rationally split homotopy fiber sequences
K(ssMSO ) ! K(MSO) -ss!K(Z)
K(ssMSp ) ! K(MSp) -ss!K(Z)
and the trace maps identify both K*(ssMSO ) Q and K*(ssMSp ) Q with the positi*
*ve-
degree part of ker(B) in
T HH*(MSO) Q ~=T HH*(MSp) Q ~=Q[qk | k 1] E(dqk | k 1) ,
where |qk| = 4k and B(qk) = dqk. Hence K(ssMSO ) and K(ssMSp ) both have
Poincar'e series kso(t) - 1 = (thso(t) - t)=(1 + t).
Remark 3.5. Adding the Poincar'e series of K(Z), as in Remark 2.6, we find that
the Poincar'e series of K(MU) and A(BBU) is
__t5__+ 1_+_th(t)_,
1 - t4 1 + t
whereas the Poincar'e series of K(MSO), A(BBSO), K(MSp) and A(BBSp) is
t5=(1 - t4) + (1 + thso(t))=(1 + t).
x4. Units, determinants and traces
4.1. Units. For each connective S-algebra A there is a natural map of spaces
w :BGL1(A) ! K(A)
that factors as the infinite stabilization map BGL1(A) ! BGL1 (A), composed
with the inclusion BGL1 (A) ! BGL1 (A)+ into Quillen's plus construction, and
followed by the inclusion of BGL1 (A)+ ~= {1} x BGL1 (A)+ into K0(ss0A) x
BGL1 (A)+ = K(A).
Remark 4.2. This w is an E1 map with respect to the multiplicative E1 structu*
*re
on K(A) that is induced by the smash product over A. However, we shall only work
with the additive grouplike E1 structure on K(A), which comes from viewing K(A)
as the underlying infinite loop space of the K-theory spectrum. So when we refer
to infinite loop structures below, we are thinking of the additive ones.
We write BSL1(A) = BGL1(ssA ) for the homotopy fiber of the map BGL1(A) !
BGL1(ss0A) induced by ssA :A ! Hss0A. In the resulting diagram
(4.3) BSL1(A) -w!K(A) -ss!K(ss0A)
the composite map has a preferred null-homotopy (to the base point of the 1-
component of K(ss0A)). The diagram is a rational homotopy fiber sequence if and
only if w :BSL1(A) ! K(ssA ) is a rational equivalence. Note that the natural
inclusion {1} x BGL1 (A)+ ! K(A) induces a homotopy equivalence
BGL1 (ssA )+ ' K(ssA ) ,
since K0(A) ~=K0(ss0A).
RATIONAL ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 15
4.4. Determinants. Suppose furthermore that A is commutative as an S-algebra.
One attempt at proving that w is injective could be to construct a map det:K(A)*
* !
BGL1(A) with the property that detOw ' id. However, no such determinant map
exists in general, as the following adaption of [Wa82, 3.7] shows.
Example 4.5. When A = S, the map ~ O w :BF = BGL1(S) ! A(*) ! K(Z)
factors through BGL1(Z) ' K(Z=2, 1), and ss2(~): ss2A(*) ! K2(Z) ~= Z=2 is an
isomorphism, so ss2(w): ss2BF ! ss2A(*) is the zero map. But ss2BF ~= ss1(S) ~=
Z=2 is not zero, so ss2(w) is not injective. In particular, w is not split inje*
*ctive up
to homotopy.
However, it is possible to construct a rationalized determinant map. Recall
from Subsection 1.1 that AQ is equivalent to HR for some naturally determined
commutative simplicial Q-algebra R.
Lemma 4.6. Let R be a commutative simplicial ring. There is a natural infinite
loop map
det :BGL1 (R)+ ! BGL1(R)
that agrees with the usual determinant map for discrete commutative rings, such
that the composite with w :BGL1(R) ! BGL1 (R)+ equals the identity.
Proof. The usual matrix determinant det: Mn(R) ! R induces a simplicial group
homomorphism GLn(R) ! GL1(R) and a pointed map BGLn(R) ! BGL1(R) for
each n 0. These stabilize to a map BGL1 (R) ! BGL1(R), which extends to an
infinite loop map
det: BGL1 (R)+ ! BGL1(R) ,
unique up to homotopy, by the multiplicative infinite loop structure on the tar*
*get
and the universal property of Quillen's plus construction. To make the construc*
*tion
natural, we fix a choice of extension in the initial case R = Z, and define det*
*R for
general R as the dashed pushout map in the following diagram:
BGL1 (Z) _____//_BGL1 (Z)+
NN
| | NNNdetZN
| | NNNN
fflffl| fflffl| N''
BGL1 (R) _____//_WBGL1 (R)+ BGL1(Z)
WWWWW O O det
WWWWWW O OR |
WWWWWW O O |
WWW WW++'' fflffl|
BGL1(R)
(Recall that Quillen's plus construction is made functorial by demanding that t*
*he
left hand square is a pushout.)
Proposition 4.7. Let A be a connective commutative S-algebra. There is a natur*
*al
infinite loop map
detQ :BGL1 (A)+ ! BGL1(A)Q
that agrees with the rationalized determinant map for a commutative ring R when
A = HR, such that the composite
BGL1(A) -w!BGL1 (A)+ -detQ--!BGL1(A)Q
16 CHRISTIAN AUSONI AND JOHN ROGNES
is homotopic to the rationalization map.
Proof. We define detQ as the dashed pullback map in the following diagram
BGL1 (A)+ Q _______//BGL1 (AQ )+
PP
| Q QdetQ PPPdet0RPP
| Q Q PPPP
fflffl| Q(( P((
BGL1 (ss0A)+ BGL1(A)Q _______//_BGL1(AQ )
QQQ
QQQQ | |
(detss0A)QQQ((QQQfflffl|| |fflffl|
BGL1(ss0A)Q _____//_BGL1(ss0AQ )
where the vertical maps are induced by the Postnikov section ss :A ! Hss0A, and
the horizontal maps are induced by the rationalization q :A ! AQ . The right ha*
*nd
square is a homotopy pullback, since SL1(A)Q ' SL1(AQ ).
To define the map det0R, we take R to be a commutative simplicial Q-algebra
such that AQ ' HR as commutative HQ-algebras. A natural choice can be made
for R, as discussed in Subsection 1.1, such that the identification ss0AQ ~= ss*
*0R is
the identity. Then det0Ris the composite map
BGL1 (AQ )+ ' BGL1 (R)+ -detR--!BGL1(R) ' BGL1(AQ ) ,
with detR from Lemma 4.6. It strictly covers the map detss0AQ, so the outer hex*
*agon
commutes strictly. This defines the desired map detQ.
To compare detQ Ow and q :BGL1(A) ! BGL1(A)Q , note that both maps
have the same composite to BGL1(ss0A)Q , they have homotopic composites to
BGL1(AQ ), and all composites (and homotopies) to BGL1(ss0AQ ) are equal. Hence
the maps to the homotopy pullback are homotopic, too.
Theorem 4.8. (a) The relative unit map
BBU = BGL1(ssku) -w!BGL1 (ssku)+ ' K(ssku)
is a rational equivalence, with rational homotopy inverse given by the relative*
* ra-
tional determinant map
detQ: BGL1 (ssku)+ ! BGL1(ssku)Q = (BBU )Q .
(b) The relative unit maps
BBO = BGL1(ssko) ! K(ssko)
BW = BGL1(ss`) ! K(ss`)
are rational equivalences (with rational homotopy inverse detQ in each case).
Proof. (a) By Proposition 4.7, the composite
BBU -w! K(ssku) -detQ--!(BBU )Q
RATIONAL ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 17
is a rational equivalence, so w is rationally injective. Here ss*BBU ~=ss*-1*
* BU
has Poincar'e series t3=(1 - t2), just like K(ssku) by Theorem 2.5(a). Thus w *
*is a
rational equivalence.
(b) The same proof works for ko and `, using that BBO and BW have
Poincar'e series t5=(1 - t4) and t2p-1=(1 - t2p-2), respectively.
Remark 4.9. The analogous map w :BSL1(MU) ! K(ssMU ) is rationally injective,
but not a rational equivalence. For the Poincar'e series of the source is
t(p(t) - 1) = t3 + 2t5 + 3t7 + 5t9 + . .,.
Q
where p(t) = k 1 1=(1 - t2k), and the Poincar'e series of the target is
(th(t) - t)=(1 + t) = t3 + 2t5 + 3t7 + t8 + 5t9 + . .,.
by Theorem 3.4(a). These first differ in degree 8, since ss8BSL1(MU) ~=ss7MU is
trivial, but K8(MU) and K8(ssMU ) have rank one. A generator of the latter gro*
*up
maps to db1 . db2 in ker(B) T HH*(MU) Q.
In the same way, w :BSL1(MSO) ! K(ssMSO ) and its symplectic variant are
rationally injective, but not rational equivalences.
4.10. Traces. Our original strategy for proving that w :BGL1(A) ! K(A) is
rationally injective for A = ku was to use the trace map tr :K(A) ! T HH(A), in
place of the rational determinant map. By [Schl04, x4], there is a natural comm*
*u-
tative diagram
BGL1(A) __w____//_K(A)L
| | LLLLtrL
| | LLL
fflffl| fflffl| L%%
BcyGL1(A)OO _____//_Kcy(A)_____//_T HH(A)OO
| |
| |
| |
GL1(A) _______________________//_ 1 A
where Bcy and Kcy denote the cyclic bar construction and cyclic K-theory, re-
spectively. The middle row is the geometric realization of two cyclic maps, hen*
*ce
consists of circle equivariant spaces and maps.
When A = ku, the resulting B-operator on H*(BcyBGL1(A); Q) takes primitive
classes in the image from H*(GL1(A); Q) ~=H*(BU ; Q) to primitive classes gen-
erating the image from H*(BGL1(A); Q) ~=H*(BBU ; Q), so by a diagram chase
we can determine the images of the latter primitive classes in H*(T HH(A); Q). *
*By
an appeal to the Milnor-Moore theorem [MM65, App.], this suffices to prove that
tr O w is rationally injective in this case.
In comparison with the rational determinant approach taken above, this trace
method involves more complicated calculations. For commutative S-algebras, it
is therefore less attractive. However, for non-commutative S-algebras, the tra*
*ce
method may still be useful, since no (rational) determinant map is likely to ex*
*ist.
We have therefore sketched the idea here, with a view to future applications.
18 CHRISTIAN AUSONI AND JOHN ROGNES
x5. Two-vector bundles and elliptic objects
The following discussion elaborates on the second author's work with Baas and
Dundas in [BDR04]. It is intended to explain some of our interest in Theorem 0.*
*1.
5.1. Two-vector bundles. A 2-vector bundle E of rank n over a base space X
is represented by a map X ! |BGLn(V)|, where V is the symmetric bimonoidal
category of finite dimensional complex vector spaces. A virtual 2-vector bundle*
* E
over X is represented by a map X ! K(V), where K(V) the algebraic K-theory
of the 2-category of finitely generated free V-modules; see [BDR04, Thm. 4.10].*
* By
[BDRR], spectrification induces a weak equivalence Spt :K(V) ! K(ku), so the 2-
vector bundles over X are geometric 0-cycles for the cohomology theory K(ku)*(X*
*).
5.2. Anomaly bundles. The preferred rational splitting of ss :K(ku) ! K(Z)
defines an infinite loop map
detQ :K(ku) ! (BBU )Q ,
which extends the rationalization map over w :BBU ! K(ku) and agrees with
the relative rational determinant on K(ssku). (We do not know if there exists *
*an
integral determinant map BGL1 (ku)+ ! BBU in this case.) We define the
rational determinant bundle |E| = det(E) of a virtual 2-vector bundle represent*
*ed
by a map E :X ! K(V) ' K(ku), as the composite map
|E|: X -E!K(ku) -detQ--!(BBU )Q .
We define the rational anomaly bundle H ! LX of E as the composite map
H :LX -L|E|-!L(BBU )Q -rQ!(BU )Q ,
where r :LBBU ! BU is the retraction defined as the infinite loop cofiber of
the constant loops map BBU ! LBBU . Up to rationalization, H is a virtual
vector bundle of virtual dimension +1, i.e., a virtual line bundle. Furthermore*
*, the
anomaly bundle relates the composition ? of free loops, when defined, to the te*
*nsor
product of virtual vector spaces: the square
(H,H)
(5.3) LX xX LX _____//_(BU )Q x (BU )Q
? || ||
fflffl| H fflffl|
LX _______________//(BU )Q
commutes up to coherent isomorphism.
5.4. Gerbes. A 2-vector bundle of rank 1 over X is the same as a C*-gerbe G,
which is represented by a map G :X ! BBU(1). When viewed as a virtual 2-
vector bundle, via BBU(1) ! BBU ! K(ku), the associated anomaly bundle is
the complex line bundle over LX that is represented by the composite
LX -LG-!LBBU(1) -r!BU(1) .
This is precisely the anomaly line bundle for G, as described in [Br93, x6.2]. *
*Note
that the rational anomaly bundles of virtual 2-vector bundles represent general
elements in
1 + eK0(LX) Q K0(LX) Q ,
whereas the anomaly line bundles of gerbes only represent elements in H2(LX).
RATIONAL ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 19
5.5. State spaces and action functionals. In physical language, we think of a
free loop fl :S1 ! X as a closed string in a space-time X. For a 2-vector bund*
*le
E ! X, we think of the fiber Hfl(a virtual vector space) at fl of the anomaly
bundle H ! LX as the state space of that string. Then the state space of a
composite of two strings (or a disjoint union of two strings) is the tensor pro*
*duct of
the individual state spaces, as is usual in quantum mechanics. Similarly, the s*
*tate
space of an empty set of strings is C. In the special case of anomaly line bund*
*les
for gerbes, the resulting state spaces are only complex lines, but in our gener*
*ality
they are virtual vector spaces. These are much closer to the Hilbert spaces usu*
*ally
considered in more analytical approaches to this subject.
There is evidence that a two-part differential-geometric structure (r1, r2) *
*on
E over X (somewhat like a connection for a vector bundle, but providing parallel
transport both for objects and for morphisms in the 2-vector bundle) provides
H ! LX with a connection, and more generally an action functional
S( ): H~fl1 . . .H~flp! Hfl1 . . .Hflq,
where : F ! X is a compact Riemann surface over X, with p incoming and q
outgoing boundary circles. The time development of the physical system is then
given by the Euler-Lagrange equations of the action functional.
In a little more detail, the idea is that the primary form of parallel trans*
*port in
(E, r1) around fl provides an endo-functor "flof the fiber category Ex ~=Vn ove*
*r a
chosen point x of fl. More precisely, parallel transport only provides a zig-z*
*ag of
functors connecting Ex to itself, but the determinant in (BU )Q is still well-*
*defined.
This "holonomy" is then the fiber Hfl= det("fl) at fl of the anomaly bundle H. *
*For
a moving string, say on the Riemann surface F , the secondary form of parallel
transport r2 specifies how the holonomy changes with the string, and this defin*
*es
the connection r on H ! LX. In the gerbe case, this theory has been worked out *
*in
[Br93, x5.3], where r1 is called "connective structure" and r2 is called "curvi*
*ng".
For a closed surface F , S( ): C ! C is multiplication by a complex number,
which would only depend on the rational type of E. Optimistically, this associa*
*tion
can produce a conformal invariant of F over X, which in the case of genus 1 sur*
*faces
would lead to an elliptic modular form. Less naively, additional structure deri*
*ved
from a string structure on X should account for the weight of the modular form.
With such structure, a 2-vector bundle E with connective structure r would qual*
*ify
as a Segal elliptic object over X.
5.6. Open strings. In the presence of D-branes in the space-time X, we can
extend the anomaly bundle to also cover open strings with end points restricted
to lie on these D-branes; see [Mo04, x3.4]. In this terminology, the (rationa*
*l)
determinant bundle |E| ! X plays the role of the B-field.
By a (rational) D-brane (W, E) in X we will mean a subspace W X together
with a trivialization E of the restriction of the (rational) determinant bundle*
* |E| to
W. In terms of representing maps, E is a null-homotopy of the composite map
W X -E!K(ku) -detQ--!(BBU )Q .
In similar terminology, we may refer to the determinant bundle |E| ! X as the
(rational) B-field.
20 CHRISTIAN AUSONI AND JOHN ROGNES
When the B-field |E| is rationally trivial, then a second choice of triviali*
*zation
E amounts to a choice of null-homotopy of the trivial map W ! (BBU )Q , or
equivalently to a map E :W ! (BU )Q . In other words, E is a virtual vector
bundle over W of virtual dimension +1, up to rationalization. In this case, the*
* K-
theory class of E ! W in 1 + eK0(W) Q is the "charge" of the D-brane (W, E).
This conforms with the (early) view on D-branes as coming equipped with a charge
[E] in topological K-theory.
For a general B-field |E|, the possible trivializations E of its restriction*
* to W
instead form a torsor under the group 1+Ke0(W) Q. For two such trivializations E
and E0 differ by a loop of maps W ! (BBU )Q , or equivalently a map E0-E :W !
(BU )Q . So [E0-E] is a topological K-theory class measuring the charge differ*
*ence
between the two D-branes (W, E) and (W, E0).
Given two D-branes (W0, E0) and (W1, E1) in (X, E), we have a commutative
diagram
W0 ______________//Xoo____________W1
E0 || ||E|| |E1|
fflffl| ss fflffl| ss fflffl|
P (BBU )Q ______//(BBU )Q oo____P (BBU )Q
where ss :P Y ! Y denotes the path space fibration covering a based space Y . An
open string in X, constrained to W0 and W1 at its ends, is a map fl :I ! X with
fl(0) 2 W0 and fl(1) 2 W1. In other words, it is an element in the homotopy pul*
*lback
of the top row in the diagram above. Let (X, W0, W1) denote the space of such
open strings. The homotopy pullback of the lower row is (BBU )Q ' (BU )Q .
Hence the 2-vector bundle E and the two D-branes specify a map of homotopy
pullbacks
H : (X, W0, W1) ! (BU )Q
that we call the (rational, virtual) anomaly bundle of this space of open strin*
*gs.
Again, we think of each fiber Hflat fl :(I, 0, 1) ! (X, W0, W1) as the state sp*
*ace
of that open string.
In the presence of a suitable connection (r1, r2) on E ! X, parallel transpo*
*rt
in (E, r1) along fl induces a (zig-zag) functor "flfrom Ex to Ey, with determin*
*ant
det("fl) from the fiber of |E| at x = fl(0) to the fiber at y = fl(1). The triv*
*ializations
of these two fibers provided by the D-brane data E0 and E1, respectively, then
agree up to a correction term, which is the fiber Hflin the anomaly bundle:
det("fl)(E0,x) ~=Hfl E1,y
Again, the secondary part of the connection may induce a connection on H over
(X, W0, W1), and more generally an action functional S( ), where now : F ! X
and the incoming and the outgoing parts of F are unions of circles and closed
intervals. For example, an open string might split off a closed string. One adv*
*antage
of the above perspective is that the state spaces of open and closed strings ar*
*ise in
a compatible fashion, as the holonomy of parallel transport in the 2-vector bun*
*dle
E, and this makes the construction of S( ) feasible. The gerbe case is discusse*
*d in
[Br93, x6.6].
RATIONAL ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 21
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Institute of Mathematics, University of Bonn, DE-53115 Bonn, Germany
E-mail address: ausoni@math.uni-bonn.de
Department of Mathematics, University of Oslo, NO-0316 Oslo, Norway
E-mail address: rognes@math.uio.no