An approximation of the e-invariant in the stable homotopy category
aa r X i v : . [ m a t h . K T ] O c t An approximation of the e -invariant in the stablehomotopy category Yi-Sheng Wang ∗ September 24, 2018
Abstract
In their construction of the topological index for flat vector bundles,Atiyah, Patodi and Singer associate to each flat vector bundle a particu-lar C / Z - K -theory class. This assignment determines a map, up to weakhomotopy, from K a C , the 0-connective algebraic K -theory space of thecomplex numbers, to F t, C / Z , the homotopy fiber of the Chern character.In this paper we give evidence for the conjecture that this map can berepresented by an infinite loop map. The result of the paper implies arefined Bismut-Lott index theorem for a compact smooth bundle E → B with the fundamental group π ( E, ∗ ) finite, for every point ∗ ∈ E . Thispaper is a continuation of the author’s paper “Topological K -theory withcoefficients and the e -invariant”. Contents K a → K t . . . . . . . . 63 Infinitely many different liftings . . . . . . . . . . . . 114 The maps e , ch rel , e ♮h and t ♮h . . . . . . . . . . . . . . 125 Index theorems for flat vector bundles . . . . . . . . . 186 The Adams e -invariant. . . . . . . . . . . . . . . . 20Bibliography . . . . . . . . . . . . . . . . . . . . . . 21 Throughout the paper, we use the Quillen model category of pointed k -spaces Top ∗ as our convenient category of topological spaces. Given X, Y ∈ Top ∗ ,[ X, Y ] denotes the homotopy classes of maps from X to Y in Top ∗ .In their construction of the topological index for flat vector bundles, Atiyahet al. associate to each flat vector bundle a C / Z - K -theory class [APS76, p.89],and this assignment gives a homomorphism ¯ e APS : ˜ K ( M, C ) → [ M, F t, C / Z ],where ˜ K ( M, C ) is the abelian group of zero dimensional virtual flat vector bun-dles over M , a compact smooth manifold [Wan17b, Remark 4 . . e : K a C → ∗ [email protected] t, C / Z [JW95] and [Wan17b]. The construction can be summarized in the com-mutative diagram below:˜ K ( M, C ) [ M, F t, C / Z ][ M, K a C ] e ∗ ¯ e APS (1)It is also proved in [JW95, Theorem 3 .
1] (see [Wan17b, Theorem 4 . . e as follows: K a C F t, C / Z Ω K t, R H odd R e Im ch ⊗ R Bo where H odd R := Q i odd K ( R , i ) and K ( A, i ) denotes the Eilenberg-Maclane spaceof an abelian group A in degree i . The maps Im : F t, C / Z → Ω K t, R and ch ⊗ R :Ω K t, R ∼ −→ H odd R are induced by the homomorphism C / Z → R a + ib b and the Chern character, respectively, where K t, R is the infinite loop spacerepresenting 0-connective complex topological K -theory with coefficients in R .Now the Bismut-Lott index theorem [BL95] says, for every compact smoothfiber bundle E → B , the following diagram of abelian groups commutes:˜ K ( E, C )˜ K ( B, C ) [ E, H odd R ][ B, H odd R ] π ! tr ∗ BG ¯Bo¯Bo where π ! is given by taking the fiberwise homology of E → B with coefficients ina flat vector bundle, tr BG is the Beck-Gottlieb transfer and ¯Bo is the composition˜ K ( − , C ) → [ − , K a C ] Bo ∗ −−→ [ − , H odd R ] . We conjecture that there should be a finer index theorem in terms of ¯ e APS : Conjecture 1.1.
The following diagram is commutative ˜ K ( E, C )˜ K ( B, C ) [ E, F t, C / Z ][ B, F t, C / Z ] π ! tr ∗ BG ¯ e APS ¯ e APS ⊗ R , ∗ ◦ J ∗ .In view of the diagram (1) and the Dwyer-Weiss-Williams index theorem[DWW03], which entails commutativity of the diagram below:˜ K ( E, C )˜ K ( B, C ) [ E, K a ( C )][ B, K a ( C )] π ! tr ∗ BG (2)Conjecture 1.1 ensues from the following conjecture: Conjecture 1.2.
The map e is weakly homotopic to an infinite loop map. The results of this paper give evidence in support of Conjecture 1.2.
Main results
Theorem 1.3.
Let K a C → K t be the (canonical) comparison map from the -connective algebraic K -theory space of the complex numbers to the -connectivecomplex topological K -theory space. Then there exists an infinite loop map e ♮h ,unique up to phantom maps, such that the composition K a C e ♮h −→ F t, C / Z → K t and the (canonical) comparison map K a C → K t are homotopic as infinite loopmaps and the map e ♮h satisfies e ♮h, ∗ = e ∗ : π ∗ ( K a C ) → π ∗ ( F t, C / Z ) ,e ♮h, ∗ | Tor = e ∗ | Tor : Tor[
L, K a C ] → [ L, F t, C / Z ] , ch ⊗ R ◦ Im ◦ e ♮h = Bo ∈ Ho( P ) , for every finite CW -complex L . Theorem 1.3 shows that, in a certain sense, the map e ♮h is the unique approx-imation of the map e in the stable homotopy category, and it also implies thefollowing index theorem, which is slightly weaker than Conjecture 1.1: Theorem 1.4.
Let E → B be a smooth compact fiber bundle. If the fundamentalgroup π ( E, ∗ ) is finite, for any base point ∗ ∈ E , then the diagram ˜ K ( E, C )˜ K ( B, C ) [ E, F t, C / Z ][ B, F t, C / Z ] π ! tr ∗ BG ¯ e APS ¯ e APS (3)3 ommutes
The next theorem shows that the map e can be viewed as a generalizationof the Adams e -invariant. Theorem 1.5.
Let e Adams , ∗ : π s ∗ = π ∗ ( B Σ + ∞ ) → Q / Z ⊂ C / Z be the Adams e -invariant. Then we have e ∗ ◦ ι ∗ = e Adams , ∗ , where B Σ + ∞ is the plus construction of the classifying space of the infinitesymmetric group Σ ∞ and ι is the map induced from the canonical embedding Σ ∞ → GL( C ) , where GL( C ) is the infinite general linear group. Outline of the paper
To approximate the map e : K a C → F t, C / Z by infinite loop maps, we considerliftings—dashed arrow—of the following diagram: K a C F t, C / Z K t H ev C e C h ch where K a C → K t is the (canonical) comparison map, ch is the Chern character, H ev C := Q i even K ( C , i ), and the sequence F t, C / Z → K t ch −→ H ev C is a homotopy fiber sequence. In Section 1, we shall see the existence of liftings e C h in the stable homotopy category and how their induced homomorphisms allrestrict to e ∗ on the torsion subgroup of [ L, K a C ], for every finite CW -complex L . We show there are infinitely many different liftings in the stable homotopycategory in Section 2. In Section 3, we investigate the relation between liftings e C h and their induced maps t C h : K rel C → H odd C , where K rel C is the relativealgebraic K -theory space of the complex numbers, the homotopy fiber of thecomparison map K a C → K t , and H odd C := Q i odd K ( C , i ). Utilizing the relationbetween e C h and t C h , we construct the map e ♮h and prove the main theorem (1.3).Finally, we apply the results obtained in the previous sections to prove the indextheorem (1.4) and the comparison theorem (1.5) in Sections 4 and 5. Relation and comparison to Bunke’s regulators
In [Bun14, Section 2 .
5] and [Bun17, Definition 13 . K a C to F t, C / Z , which the comparison map K a C → K t factors over, are constructeddirectly in the stable homotopy category via the technique of the ∞ -categorical This relation between the map e and the Adams e -invariant has been claimed withoutproof in [JW95, p.930]. The name e is also due to Jones and Westbury. K -theory developed in [BNV16] and [BT15]. Notice that theirmethod actually give maps between ( − − K -theory. In fact, it isthe use of the geometries, along with the ∞ -categorical technique, gives thepreferred homotopies needed to define the maps from K a C to F t, C / Z . Thoughthe constructions of these two maps and the map e are similar in many respects,the precise relation between them is not entirely clear, and one might need aspace level comparison in order to unravel it (see [Bun17, Remark 12 . e in the stable homotopy category. Sincethere is no geometries of vector bundles involved in this approach, it is not easyto find a preferred homotopy. Therefore, instead of searching for a preferredhomotopy, we study all the maps from K a C to F t, C / Z over which the compar-ison map K a C → K t factors—there are infinitely many such maps (Theorem3.1). The idea is to find the homotopy properties that distinguish these mapsfrom each other (up to weak homotopy). It proceeds roughly as follows: Wefirst show all such maps restrict to the same map on the torsion part of K a C (Theorem 2.3 and Corollary 4.2) and hence what determines each of them is itsrestriction to the rational part of K a C . This observation, in view of the struc-ture theorem of (relative) algebraic K -theory (Corollaries 4.1 and 4.3), leads usto consider relative K -theory of the complex numbers and the approximation t ♮h of the Chern-Simons class associated to the Chern character in the stable homo-topy category. Note that the Chern-Simons classes give a map from K rel C to H odd C (see [Wan17b, Section 4 . K a C , can be considered as a (weak) lifting of the map e and the map t ♮h is anapproximation of the Chern-Simons class in the sense that it induces the samehomomorphism that Chern-Simons class induces between the homotopy groupsof K rel C and H odd C . Then, using the connection between t C h : K rel C → H odd C and e C h : K a C → F t, C / Z (the diagram (14), Corollary 4.5 and Lemma 4.6), wefind the approximation e ♮h of the map e and prove its uniqueness (Theorem 4.8).In particular, if one can show either of the maps constructed in [Bun14,Section 2 .
5] and [Bun17, Definition 13 .
13] is weakly homotopic to the map e ,then the maps e ♮h and e are weakly homotopic as well. The latter can also bededuced if the map t ♮h indeed realizes the Chern-Simons class, for every compactsmooth manifold (Lemma 4.6). Notation and conventions
In this paper, we use the homotopy category of prespectra Ho( P ) (see [Wan17b,Appendix]) as our model for the stable homotopy category, and as in [Wan17b],bold letters are reserved for prespectra and maps between them. Given twoinfinite loop spaces E = Ω ∞ E and F = Ω ∞ F , where E , F ∈ P , we let[ E , F ] Ho( P ) := [ E , F ] Ho( P ) , the abelian group of maps between cofibrant-fibrant replacements of E and F . When we say a diagram of infinite loopspaces is commutative, it commutes in Ho( P ), unless otherwise specified.Since, in most cases, our methods work in a more general setting that in-5ludes algebraic K -theory of the real numbers, real topological K -theory andtopological K -theory with coefficients in Q / Z or R / Z , we introduce the followingnotation for the sake of convenience: Notation 1.6. F ′ = R or C , and F = Q or F ′ . K a F ′ : The infinite loop space of the Ω - CW -prespectrum K a F ′ that represents -connective algebraic K -theory of the complex numbers (resp. the realnumbers). We often drop the field F ′ from the notation when both casesapply. K t : The infinite loop space of the Ω - CW -prespectrum K t that represents -connective complex (resp. real) topological K -theory.When a statement is true for both R and C , we do not specify the field.For instance, the map K a → K t could mean the comparison map fromthe -connective algebraic K -theory space of the complex numbers to the -connective complex topological K -theory space or the comparison mapfrom the -connective algebraic K -theory space of the real numbers to the -connective real topological K -theory space. If a statement applies tojust one case, we specify only the field used in algebraic K -theory. Forexample, the map K a C → K t stands for the comparison map from the -connective algebraic K -theory space of the complex numbers to the -connective complex topological K -theory space. X F : The infinite loop space of the prespectrum X ∧ M F , or equivalently the zerocomponent of its fibrant replacement, where X is a CW -prespectrum. Inthe case where X = K t , we have the homotopy equivalence K t, F ≃ H ev F given by the Chern character. F t, F / Z : The homotopy fiber of K t → K t, F , or equivalently the infinite loop spaceof the prespectrum Ω( K t ∧ M F / Z ) (see [Wan17b, Lemma A. . - ]). F a, Q / Z : The homotopy fiber of K a → K a, Q , or equivalently the infinite loop spaceof the prespectrum Ω( K a ∧ M Q / Z ) (see [Wan17b, Lemma A. . - ]). K rel F ′ : The homotopy fiber of K a F ′ → K t . It is the infinite loop space representingrelative K -theory of F ′ . The field F ′ is dropped from the notation when astatement holds for both R and C . K a → K t The comparison map K a → K t along with the generalized Chern character E ∧ M Z → E ∧ M F induced by the inclusion Z ֒ → F , where E ∈ P , gives us thefollowing commutative diagram of prespectra.6( K a ∧ M Z )Ω( K a ∧ M Q ) Fib ( f ) K a ∧ M Z K a ∧ M Q Ω( K t ∧ M Z )Ω( K t ∧ M Q ) Fib ( g ) K t ∧ M Z K t ∧ M Q Ω( K t ∧ M Z )Ω( K t ∧ M F ′ ) Fib ( h ) K t ∧ M Z K t ∧ M F ′ Ω ff Ω gg Ω hh Su ∼ id where Su stands for the π ∗ -isomorphism in [Wan17b, Corollary 2 . . P [Wan17b, Theorem A. . CW -prespectrum in the diagram above with an equivalent fibrant-cofibrant prespec-trum. Then, applying the infinite loop functor Ω ∞ , we obtain the followingdiagram of homotopy fiber sequences.Ω K a Ω K a, Q F a, Q / Z K a K a, Q Ω K t Ω K t, Q F t, Q / Z K t K t, Q Ω K t Ω K t, F ′ F t, F ′ / Z K t K t, F ′ Ω ff Ω gg Ω hh Su ∼ id (4)7ow, since K t ∧ M Q is rational, the following composition K a ∧ M Z → K t ∧ M Z → K t ∧ M Q is determined by their induced homomorphisms (see [Wan17b, Lemma 2 . .
7] or[Rud08, Theorem 5.8 and 7.11]): π ∗ ( K a ∧ M Z ) → π ∗ ( K t ∧ M Z ) → π ∗ ( K t ∧ M Q ) = π ∗ ( K t ) ⊗ Q . (5)Since, up to torsion groups, the algebraic K -groups of the real (resp. complex)numbers are divisible [Sus84, Theorem 4 . K a → K t → K t, Q is null-homotopic as an infinite loop map (or inHo( P )) . Lemma 2.1.
Liftings of the comparison map K a → K t with respect to thehomotopy fiber sequence F t, F / Z → K t → K t, F exist, denoted by e F h , and they fitinto the commutative diagram below. Ω K a Ω K a, Q F a, Q / Z K a K a, Q Ω K t Ω K t, Q F t, Q / Z K t K t, Q Ω K t Ω K t, F ′ F t, F ′ / Z K t K t, F ′ Ω ff Ω gg Ω hh Su ∼ jide F ′ h e Q h Proof.
As shown in the discussion preceding the lemma, the composition K a → K t → K t, F is null-homotopic, and choosing a null-homotopy gives us a lifting e F h . To see the diagram is commutative, we note F a, Q / Z has all its homotopygroups are torsion groups. That is because both coker( π n (Ω f )) and ker( π n ( f ))are torsion groups, for n ≥
1, and π n ( F a, Q / Z ) fits into the short exact sequence0 → coker( π n (Ω f )) → π n ( F a, Q / Z ) → ker( π n ( f )) → . Therefore the rationalization of F a, Q / Z is contractible and the abelian group[ F a, Q / Z , Ω K t, F ] Ho( P ) is trivial. This implies the commutativity of the followingtwo triangles: Another argument without using Suslin’s result can be found in [Wei84, Proposition 2]. a, Q / Z K a F t, Q / Z Su e Q h F a, Q / Z K a F t, F ′ / Z j ◦ Su e F ′ h Thus, the lemma is proved.
Theorem 2.2.
Given a pointed topological space X and liftings e Q h and e F ′ h asabove, then they induce the following isomorphisms: e Q h, ∗ | Tor : Tor[
X, K a ] ∼ −→ [ X, F t, Q / Z ] e F ′ h, ∗ | Tor : Tor[
X, K a ] ∼ −→ Tor[
X, F t, F ′ / Z ] ≃ [ X, F t, Q / Z ] . Proof.
Step j ∗ induces an isomorphismcoker((Ω g ) ∗ ) → Tor((coker(Ω h ) ∗ ) . To see this, we first recall some facts in homological algebra: Given an abeliangroup A , there is a short exact sequence0 → A T → A → A/A T → , where A T is the torsion subgroup of A . Now, by the right exactness of thetensor product, and the fact that A T ⊗ F / Z = 0, we see the homomorphism A ⊗ F / Z → A/A T ⊗ F / Z (6)is an isomorphism. Moreover, since A/A T is torsion free and hence flat, thereis another short exact sequence0 → A/A T ⊗ Q / Z → A/A T ⊗ F ′ / Z → A/A T ⊗ F ′ / Q → . Because
A/A T and F ′ / Q both are flat, the tensor product A/A T ⊗ F ′ / Q is flatand hence torsion free. Applying the left exactness of Tor, we further obtainthe following isomorphism A/A T ⊗ Q / Z → Tor(
A/A T ⊗ F ′ / Z ) . (7)Return to the theorem and let A be the abelian group [ X, Ω K t ]. Then we seethe isomorphisms (6) and (7) give the isomorphism claimedcoker((Ω g ) ∗ ) ∼ = A/A T ⊗ Q / Z ∼ −→ Tor(
A/A T ⊗ F ′ / Z ) ∼ = Tor(coker((Ω h ) ∗ ) . Step : We claim that the homomorphism[
X, F t, Q / Z ] → Tor[
X, F t, F / Z ]is an isomorphism. This can be seen from the following two diagrams of exactsequences: The first one is obtained from the diagram (4):90 coker((Ω g ) ∗ )coker((Ω h ) ∗ ) [ X, F t, Q / Z ][ X, F t, F ′ / Z ] ker( g ∗ )ker( h ∗ ) 00 , Applying the functor Tor, we obtain the second one:00 coker((Ω g ) ∗ )Tor(coker((Ω h ) ∗ ) [ X, F t, Q / Z ]Tor([ X, F t, F ′ / Z ]) ker( g ∗ )ker( h ∗ ) 00 l Note that the functor Tor does not always preserve short exact sequences, but,in this case, it does. The only thing to check is the surjectivity of l , yet it followsquickly from the surjectivity of the homomorphism[ X, F t, Q / Z ] → Tor([
X, F t, F / Z ]) → ker( h ∗ ) = ker( g ∗ ) . By the short five lemma, we see the homomorphism[
X, F t, Q / Z ] → Tor[
X, F t, F / Z ] (8)is indeed an isomorphism. Step : Consider the commutative diagram below:[
X, K a ][ X, F a, Q / Z ] Tor[ X, K a ] [ X, F t, Q / Z ] t Su ∗ ∼ e Q h, ∗ | Tor e Q h, ∗ and note the homomorphism [ X, F a, Q / Z ] → [ X, K a ] factors through the homo-morphism t . Since Su ∗ is an isomorphism by Suslin’s theorem (see [Wan17b,Theorem 2 . . t is injective. On the other hand, t is surjective by itsdefinition, and thus t is an isomorphism. Now, in view of the diagram above,we see the homomorphism e Q h, ∗ | Tor has to be an isomorphism as well.Combining with the isomorphism (8), the second assertion can be deducedfrom the following commutative diagram:[
X, K a ][ X, F a, Q / Z ] Tor[ X, K a ][ X, F t, Q / Z ] Tor([ X, F t, F ′ / Z ]) [ X, F F ′ / Z ] t j ∗ ∼ Su ∗ ∼ e F ′ h, ∗ | Tor e F ′ h, ∗
10n fact, we can see from the diagrams above the homomorphisms e F ′ h, ∗ | Tor and e Q h, ∗ | Tor are identical to the compositions j ∗ ◦ Su ∗ ◦ t − and Su ∗ ◦ t − , respectively.This means any lifting induces the same isomorphism on the torsion subgroupTor[ X, K a ]. On the other hand, when X = L , a finite CW -complex, the e -invariant satisfies the same commutative diagram [Wan17b, Lemma 4 . . L, F a, Q / Z ][ L, K a C ] [ L, F t, Q / Z ] [ L, F t, C / Z ] Su ∗ ∼ e ∗ Hence, we have shown the following theorem:
Theorem 2.3.
Every lifting e F h : K a → F t, F / Z of the comparison map K a → K t induces the same isomorphism e F h, ∗ | Tor : Tor[
X, K a ] ∼ −→ Tor[
X, F t, F / Z ] . When X = L , a finite CW -complex, and in the case of the complex numbers, e F h, ∗ restricts to the e -invariant on the torsion subgroup of [ L, K a C ] . In the last section we have seen the existence of liftings of the comparison map K a → K t . We shall see in this section, in the case of the complex numbers, there are in-finitely many different liftings of the comparison map K a C → K t in the categoryHo( P ).Recall that the number of different liftings is measured by the size of thesubgroup Im([ K a C , Ω K t, F ] Ho( P ) ) ⊂ [ K a C , F t, F / Z ] Ho( P ) . (9)This results from the following long exact sequence ... → [ K a C , Ω K t, F ] Ho( P ) → [ K a C , F t, F / Z ] Ho( P ) → [ K a C .K t ] Ho( P ) → .... Since Ω K t, F is rational, by [Wan17b, Lemma 2 . . K a C , Ω K t, F ] Ho( P ) ∼ −→ Hom ( π ∗ ( K a C ) ⊗ Q , π ∗ (Ω K t, F )) , (10)where Hom ( A ∗ , B ∗ ) is the abelian group of homogeneous homomorphisms ofdegree 0 between graded abelian groups A ∗ and B ∗ . Now it is known that theabelian group π ∗ (Ω K t, F ) = ( F ∗ = odd ∗ = even, π ∗ ( K a C ) ⊗ Q is not determined yet. Nevertheless, according to [Jah99, Sec.4-5], we have π ∗ ( K a C ) ⊗ Q is a non-trivial Q -vector space, when ∗ is odd. In fact, Jahrenconstructs a homomorphism from π ∗ ( K a C ) → R , for ∗ is odd and proves thatthis homomorphism reduces to the Borel classes after precomposing the homo-morphisms induced by the conjugate embeddings of a number field in C andtensoring R . Hence, when ∗ is odd, π ∗ ( K a C ) ⊗ Q cannot be trivial.In view of (9) and (10), we know if one can construct infinitely many differenthomomorphisms π ∗ ( K a C ) → π ∗ (Ω K t, F )such that, after composing with the homomorphism π ∗ (Ω K t, F ) → π ∗ ( F t, F / Z ) , they remain different, then we obtain infinitely many different liftings. Weprovide one construction here: Let ∗ be an odd number and pick up a non-trivialelement x ∈ π ∗ ( K a C ) ⊗ Q . Assign to it the number n ∈ F with n ∈ N \ { } .Then extend this assignment to a homomorphism π ∗ ( K a C ) ⊗ Q → F = π ∗ (Ω F t, F ) . It is not to difficult to find an extension of this assignment. For instance, one canchoose an inner product on π ∗ ( K a C ) ⊗ Q and let < x > ⊥ go to zero. Therefore,we have shown that the subgroupIm([ K a C , Ω K t, F ] Ho( P ) ) ⊂ [ K a C , F t, F / Z ] Ho( P ) contains at least countably infinitely many different elements. Theorem 3.1.
There are infinitely many different liftings K a C → F t, F / Z of thecomparison map K a C → K t such that the following diagram commutes: F a, Q / Z K a C F t, F / Z K t Proof.
This results from the discussion preceding the theorem and the fact thatthe rationalization of F a, Q / Z is contractible. Remark 3.2.
The same method does not work in the case of the real numbers.In fact, by the theorem of Jahren [Jah99], we can only conclude π ∗ ( K a R ) ⊗ Q isnon-trivial when ∗ = 4 k − . On the other hand, we have π ∗ (Ω K t, F ) = F when ∗ = 4 k − and otherwise, where k ∈ N . e , ch rel , e ♮h and t ♮h In this section, we construct the maps e ♮h and t ♮h and study their relation withthe maps e and ch rel in [Wan17b, Sections 4 . . K a and K rel . 12 orollary 4.1. There exists a homotopy equivalence of infinite loop spaces: K a ∼ −→ K a, Q × F t, Q / Z . Proof.
By Lemma 2.1, there exists a lifting e Q h : K a → F t, Q / Z of the comparisonmap K a → K t . Combining with the rationalization u Q : K a → K a, Q , we obtaina homotopy equivalence of infinite loop spaces K a ( u Q ,e Q h ) −−−−−→ K a, Q × F t, Q / Z . Corollary 4.2.
Given a lifting e F h : K a → F t, F / Z , the composition F t, Q / Z i −→ K a, Q × F t, Q / Z ≃ K a e F h −→ F t, F / Z is homotopic, as an infinite loop map, to the canonical map j : F t, Q / Z → F t, F / Z induced by the inclusion Q / Z ֒ → C / Z , where i is the inclusion into the secondcomponent and K a ≃ K a, Q × F t, Q / Z is the homotopy equivalence given by alifting e Q h .In other words, what determines a lifting e F h is its restriction to the divisiblepart K a, Q .Proof. This follows from the commutative diagram below F t, Q / Z K a, Q × F t, Q / Z F a, Q / Z K a F t, F / Z ≀≀ i Su ∼ ( u Q , e Q h ) ∼ e F h j (11)Recall that K rel is the homotopy fiber of the comparison map K a → K t . Corollary 4.3.
There exists a homotopy equivalence of infinite loop spaces: K rel ∼ −→ K a, Q × Ω K t, Q . roof. Choose a lifting e Q h and hence a homotopy equivalence K a ( u Q ,e Q h ) −−−−−→ K a, Q × F t, Q / Z , and consider the commutative diagram below:Ω K t K rel K a K t Ω K t K a, Q × Ω K t, Q K a, Q × F t, Q / Z K t ( u Q ◦ π, t Q h )( u Q , e Q h ) π p (12)where p is the composition K a, Q × F t, Q / Z π −→ F t, Q / Z → K t ,π is the projection onto the second component and t Q h is an infinite loop mapinduced by e Q h and a filler (homotopy) of the following triangle: K a F t, Q / Z K te Q h (13)Since ( u Q , e Q h ) : K a → K a, Q × F t, Q / Z is a homotopy equivalence, in view of thecommutative diagram (12), we see ( u Q ◦ π, t Q h ) is also a homotopy equivalence.The next lemma shows what determines t F h is its restriction to K a, Q . Lemma 4.4.
Let t F h : K rel → Ω K t, F be a lifting of the composition e Q h ◦ π : K rel → K a → F t, F / Z with respect to the fiber sequence Ω K t, F → F t, F / Z → K t . Then in
Ho( P ) the composition l : Ω K t, Q i −→ K a, Q × Ω K t, Q → Ω K t, F is homotopic to the canonical map Ω K t, Q j ′ −→ Ω K t, F induced from the inclusion Q ֒ → F . roof. This amounts to show the commutative diagram below:Ω K t, Q K a, Q × Ω K t, Q K rel Ω K t, F ≀≀ i ( u Q ◦ π, t Q h ) ∼ t F h j ′ Combining with the diagram (11), we see that l and j are homotopic aftercomposing with the map Ω K t, F → F t, F / Z . It means l and j differ by a mapΩ K t, Q → Ω K t, F that factors through Ω K t . However, given any map Ω K t, Q → Ω K t , the induced homomorphism π ∗ (Ω K t, Q ) → π ∗ (Ω K t ) is always trivial andhence the composition π ∗ (Ω K t, Q ) → π ∗ (Ω K t ) → π ∗ (Ω K t, F ) is also trivial. SinceΩ K t, F is rational, we see any map Ω K t, Q → Ω K t, F that factors through Ω K t isnull-homotopic [Wan17b, Lemma 2 . . l and j have to be homotopicin Ho( P ).A priori, the map t F h depends on the choice of fillers of the diagram (13), thefollowing shows, in effect, every filler induces the same t F h in Ho( P ). Corollary 4.5.
Given a lifting e F h , there is a unique map t F h : K rel → Ω K t, F making the following diagram commute: Ω K t K rel K a K t Ω K t Ω K t, F F t, F / Z K tt F h e F h iπ (14) Proof.
Suppose there is another map t F , ′ h which also fits into the commutativediagram (14), then the difference between t F h and t F , ′ h is measured by an elementin the image Im([ K rel , Ω K t ] Ho( P ) ) ⊂ [ K rel , F t, F / Z ] Ho( P ) . (15)15ince any map K rel → Ω K t induces the trivial homomorphism between homo-topy groups, the subgroup (15) should be trivial, and hence, the maps t F h and t F , ′ h must be homotopic in Ho( P ).Now if we fix the map t F h in the diagram (14) instead, we get a slightly weakerresult. Recall first that, given E a CW -prespectrum and F an Ω-prespectrum,by a phantom map f : E → F in P , we understand its restriction to any finite CW -subprespectrum is null-homotopic. Lemma 4.6.
The map t F h determines the lifting e F h up to phantom maps. Namely,if there is another lifting e F , ′ h such that the pair ( t F h , e F , ′ h ) also satisfies the com-mutative diagram (14) , then e F , ′ h and e F h differ by a phantom map.Proof. Firstly, recall that e Q h induces an identification K a ≃ K a, Q × F t, Q / Z , and, by Corollary 4.2, we know e F h and e F , ′ h restrict to homotopic maps on F t, Q / Z in Ho( P ).Secondly, via the Serre class theory [Rud08, Proposition 4.23, 4.25] (see[Wan17b, A.2] for the relation between P and A ), one can deduce[ F , K t ] Ho( P ) is a finitely generated abelian group, for every finite CW -prespectrum F . Onthe other hand, since K a ∧ M Q is rational, the abelian group[ E , K a ∧ M Q ] Ho( P ) is always divisible, for any CW -prespectrum E . Hence the homomorphism[ F , K a ∧ M Q ] Ho( P ) → [ F , K t ] Ho( P ) is trivial, for any finite CW -prespectrum F . In particular, this implies all divis-ible elements of [ F , K a ] Ho( P ) are in the image of the homomorphism[ F , K rel ] Ho( P ) → [ F , K a ] Ho( P ) . Using the commutative diagram (14) again, we conclude e F h, ∗ = e F , ′ h, ∗ : [ F , K a ] Ho( P ) → [ F , Fib ( ch )] Ho( P ) , for every finite CW -prespectrum F , where Fib ( ch ) is the homotopy fiber of K t → K t ∧ M F . Thus, we have proved the lemma.We now compare e F h and t F h with the maps e and ch rel defined in [Wan17b,Sections 4 . . Lemma 4.7 (Theorem 4 . . . The following diagram commutesup to weak homotopy K t K rel C K a C K t Ω K t Ω K t, C F t, C / Z K t ch rel eiπ ch (16) Theorem 4.8.
There is a lifting e ♮h unique up to phantom maps such that e ♮h, ∗ = e ∗ : π ∗ ( K a C ) → π ∗ ( F t, C / Z ); e ♮h, ∗ | Tor = e ∗ | Tor : Tor[
L, K a C ] → [ L, F t, C / Z ]; (17)ch ⊗ R ◦ Im ◦ e ♮h = Bo ∈ Ho( P ) . (18) Proof.
Since K t, Q is rational, there is a unique infinite loop map t ♮h such that t ♮h, ∗ = ch rel ∗ : π ∗ ( K rel C ) → π ∗ (Ω K t, C ) . As the homotopy cofiber and fiber sequences are isomorphic in the stable ho-motopy category, choosing a filler of the following triangle K rel C Ω K t, C Ω K tt ♮h gives an infinite loop map e ♮h : K a C → F t, C / Z which makes the diagram (14) commute. Combining the diagram (14) with thefact that π ∗ ( K t ) = 0, for ∗ is odd—hence π ∗ ( K rel C ) → π ∗ ( K a C ) is onto, weobtain: e ♮h, ∗ = e ∗ : π ∗ ( K a C ) → π ∗ ( F t, C / Z ) . Notice, for ∗ is even, π ∗ ( F t, C / Z ) = 0.As for the uniqueness, we assume there is another lifting e ♭h such that e ♭h, ∗ = e ∗ = e ♮h, ∗ .
17n view of Lemma 4.5, we may assume t ♭h is the induced infinite loop map from K rel C to Ω K t, C . Let p ∗ be the homomorphism π ∗ (Ω K t, C ) → π ∗ ( F t, C / Z ) . Then the diagram (14) implies p ∗ ◦ t ♭h, ∗ = e ♭ ∗ ◦ π ∗ = e ♮ ∗ ◦ π ∗ = p ∗ ◦ t ♮h, ∗ . (19)Now observe there is an exact sequence0 → Hom( π ∗ ( K rel C ) , π ∗ (Ω K t )) → Hom( π ∗ ( K rel C ) , π ∗ (Ω K t, C )) → Hom( π ∗ ( K rel C ) , π ∗ ( F t, C / Z ))given by the short exact sequence0 → π ∗ (Ω K t ) → π ∗ (Ω K t, C ) → π ∗ ( F t, C / Z ) → . Since Hom( π ∗ ( K rel C ) , π ∗ (Ω K t )) = 0, the homomorphismHom( π ∗ ( K rel C ) , π ∗ (Ω K t, C )) → Hom( π ∗ ( K rel C ) , π ∗ ( F t, C / Z ))is actually injective. Hence, the maps t ♭h, ∗ = t ♮h, ∗ , in view of the equality (19),and t ♭h and t ♮h are homotopic in Ho( P ) (see [Wan17b, Lemma 2 . . e ♭h and e ♮h differ only by a phantom map.The assertion (17) has been shown in Corollary4.2, whereas the equality (18)follows from the definition of t ♮h and the fact that Bo is also an infinite loop mapas we havech ⊗ R , ∗ ◦ Im ∗ ◦ t ♮h, ∗ = ch ⊗ R , ∗ ◦ Im ∗ ◦ ch rel ∗ = Bo ∗ : π ∗ ( K rel C ) → π ∗ ( H odd R ) . The theorem above gives strong evidence for the conjecture that e can belifted to a map in the stable homotopy category. In fact, if one can show ch rel is an infinite loop map and hence ch rel = t ♮h ∈ Ho( P ), then e ♮h and e are weaklyhomotopic as all free elements in [ L, K a C ] come from [ L, K rel C ] when L is afinite CW -complex. Combining Theorem 4.8 with the commutative diagram (2), which is given bythe DWW index theorem, we have the following refined BL index theorem:
Theorem 5.1.
Given a compact smooth fiber bundle E → B , then the diagram ˜ K ( E, C )˜ K ( B, C ) [ E, F t, C / Z ][ B, F t, C / Z ] π ! tr ∗ BG ¯ e ♮ ¯ e ♮ (20)18 ommutes, where ¯ e ♮ is the composition ˜ K ( − , C ) → [ − , K a C ] e ♮ ∗ −→ [ − , F t, C / Z ] . Theorem 5.1 implies the following index theorem in terms of ¯ e APS : Theorem 5.2.
Let E → B be a smooth compact fiber bundle with the funda-mental group π ( E, ∗ ) finite, for every point ∗ ∈ E , then the diagram ˜ K ( E, C )˜ K ( B, C ) [ E, F t, C / Z ][ B, F t, C / Z ] π ! tr ∗ BG ¯ e APS ¯ e APS (21) commutes.Proof.
The assumption that the fundamental group π ( E, ∗ ) is finite, for everypoint ∗ ∈ E , implies the fundamental group π ( B, ∗ ) is also finite, for everypoint ∗ ∈ B , and it is also known that [ BG, K ( Q , i )] is trivial, for any finitegroup G and every i ∈ N , where BG is the classifying space of G (see [Web,Corollary 4 . G → GL( C ), the induced ho-momorphism [ X, BG ] → [ X, K a C ] always factors through Tor[ X, K a C ], where X is a topological space, as there are isomorphisms:[ BG, K a C ] ⊗ Q ≃ [ BG, K a C Q ] ≃ [ BG, Y i ∈ N K ( π i ( K a C ) ⊗ Q , i )] = 0 . The assertion then follows from the commutative diagram below:˜ K ( E, C )˜ K ( B, C ) Tor[ E, K a C ]Tor[ B, K a C ] [ E, F t, C / Z ][ B, F t, C / Z ] π ! ¯ e APS ¯ e APS tr ∗ BG tr ∗ BG e ∗ = e ♮h. ∗ e ∗ = e ♮h, ∗ (22)19 The Adams e -invariant In this section, we explain how the e -invariant [Wan17b, Section 4 .
1] general-izes the Adams e -invariant. Recall the Adams e -invariant can be obtained asthe lifting of the following diagram—there exists only one lifting as we have[ B Σ ∞ , K ( Q , i )] = 0, for every i ∈ N [Web, Corollary 4 . B Σ ∞ K t F t, Q / Z F t, C / Z ¯ e Adams
Applying the universal property of the plus construction, one gets a map e Adams : B Σ + ∞ → F t, Q / Z whose induced homomorphism e Adams , ∗ : π ∗ ( B Σ + ∞ ) → π ∗ ( F t, Q / Z )gives the Adams e -invariant up to sign [Qui76].Because [ B Σ ∞ , Ω K t, C ] is trivial, the following diagram must commute: B Σ + ∞ K a C F t, Q / Z K t F t, C / Z e Adams ι e C h where e C h is any lifting of the comparison map K a C → K t . In particular, thisshows the two homomorphisms[ L, B Σ + ∞ ] ι ∗ −→ [ L, K a C ] e C h, ∗ −−→ [ L, F t, C / Z ];[ L, B Σ + ∞ ] e Adams , ∗ −−−−−→ [ L, F t, Q / Z ] → [ L, F t, C / Z ] , are the same, for any finite CW -complex L . Furthermore, by Corollary 2.3, weknow e C h, ∗ = e ∗ in this case. Thus, we have proved the following theorem: Theorem 6.1. ( e ◦ ι ) ∗ = e Adams , ∗ : π ∗ ( B Σ + ∞ ) → π ∗ ( F t, C / Z ) . cknowledgment The results of the present paper improve some theorems of [Wan17a]. Theauthor would like to thank his supervisor Sebastian Goette for bring to hisattention this interesting research project. The author is also grateful toUlrich Bunke and Wolfgang Steimle for their expert advice and insightfulcomments on the author’s work. The author would also like to express hissincere thanks to Jørgen Olsen Lye for reading part of the early draft carefully.The project is financially supported by the DFG Graduiertenkolleg 1821”Cohomological Methods in Geometry”.
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