aa r X i v : . [ m a t h . K T ] J u l Chern classes on differential K -theory Ulrich Bunke ∗ November 7, 2018
Abstract
In this note we give a simple, model-independent construction of Chern classes asnatural transformations from differential complex K -theory to differential integralcohomology. We verify the expected behaviour of these Chern classes with respectto sums and suspension. Contents
Complex K -theory and integral cohomology H Z are generalised cohomology theorieswhich have a unique differential extensions ( ˆ K, R, I, a, R ) and ( d H Z , R, I, a, R ) with inte-gration. Moreover, these extensions are multiplicative in a unique way. We refer to [BS]for a description of the axioms for differential extensions of cohomology theories and aproof of these statements.The i ’th Chern class is a natural transformation of set-valued functors c i : K → H Z i ∗ NWF I - Mathematik, Universit¨at Regensburg, 93040 Regensburg, GERMANY,[email protected] In our previous work instead of ”differential cohomology” we used the term ”smooth cohomology”.We were convinced by D. Freed that differential cohomology is the better name.
1n the category of topological spaces. The product H Z ev := Q i ≥ H Z i is a functor withvalues in commutative graded rings. We consider subfunctor H Z ev, ∗ := 1 + Q i ≥ H Z i ⊆ Q i ≥ H Z i which takes values in the subgroup of units. The total Chern class c := 1 + c + c + · · · : K −→ H Z ev, ∗ is a natural transformation of group-valued functors.Let Ω ∗ cl ( . . . , K ∗ ) ⊆ Ω ∗ ( . . . , K ∗ ) denote the graded ring valued functors on smooth mani-folds of smooth differential forms with coefficients in K ∗ and its subfunctor of closed forms.We use the powers of the Bott element in K in order to identify the functorsΩ ( . . . , K ∗ ) ∼ = Ω ev ( . . . ) , Ω − ( . . . , K ∗ ) ∼ = Ω odd ( . . . ) . We therefore have natural transformations a : Ω odd → ˆ K , R : ˆ K → Ω evcl , where a only preserves the additive structure, while R is multiplicative.We consider the symmetric formal power series in infinitely many variables˜ ch := X i ≥ ( e x i − ∈ Q [[ x , x , . . . ]] . We write ch i for the homogeneous component of degree i . Then there are polynomials C i ∈ Q [ s , s , . . . ]of degree i (where s i has degree i ) such that C i ( ch , . . . , ch i ) = σ i is the i th elementary symmetric function in the x i . The polynomial C i induces a naturaltransformation C i : Ω ev → Ω i which maps the even form ω = ω + ω + ω + . . . , ω k ∈ Ω k ( M ) to C i ( ω ) := C i ( ω , . . . , ω i ) ∈ Ω i ( M ) . The following theorem states that the Chern classes have unique lifts to the differentialextensions which are, in addition, compatible with the group structures.
Theorem 1.1
1. For every i ≥ there exists a unique natural transformation of set-valued functors on smooth manifolds ˆ c i : ˆ K → d H Z i uch that the following diagram commutes: Ω ev C i / / Ω i ˆ K R O O I (cid:15) (cid:15) ˆ c i / / d H Z iR O O I (cid:15) (cid:15) K c i / / H Z i (1)
2. The total class ˆ c = 1 + ˆ c + · · · : ˆ K → d H Z ev, ∗ preserves the group structure. Lifts of the Chern classes have previously been constructed in [Ber08]. The goal of thepresent paper is to give a much simpler, model-independent treatment. Further new, butnot very deep, points of the present theorem are the assertions about uniqueness and thesecond statement. Our method of proof is different from [Ber08]. It is in fact a speciali-sation of a general principle already used in [BS] and [Bun09] for the construction of liftsof natural transformations between cohomology functors to their differential refinements.In the next two paragraphs we connect the differential Chern classes on differential K -theory with previous constructions of differential Chern classes in specific geometric sit-uations.If V := ( V, h V , ∇ V ) is a hermitian vector bundle with connection over a manifold M , thenwe have the Cheeger-Simons classesˆ c CSi ( V ) ∈ d H Z i ( M )constructed in [CS85]. In the model of differential K -theory [BS07] the geometric bundleis a cycle for a differential K -theory class [ V ] ∈ ˆ K ( M ). We haveˆ c i ([ V ]) = ˆ c CSi ( V ) . An even geometric family E over M (see [Bun02] for this notion) gives rise to a Bismutsuperconnection A ( E ) on an infinite-dimensional Hilbert space bundle H ( E ) over M . Thissuperconnection A ( E ) = D ( E ) + ∇ H ( E ) + higher terms extends the family of Dirac operators D ( E ). If the kernel of D ( E ) is a vector bundle,then it has an induced metric h ker( D ( E )) and connection ∇ ker( D ( E )) obtained from ∇ H ( E ) byprojection. We thus get an induced geometric bundle H ( E ) = (ker( D ( E )) , h ker( D ( E )) , ∇ ker( D ( E )) )3nd can define the class ˆ c CSi ( H ( E )) ∈ d H Z i ( M ). One of the original goals of [Bun02],which was not quite achieved there, was to extend this construction to the general casewhere we do not have a kernel bundle. Under the assumption that index ( D ( E )) ∈ K ( M )belongs to the i ’th-step of the Atiyah-Hirzebruch filtration (i.e. vanishes after pull-backto any i − c i ( E ) ∈ d H Z i ( M ) such that I (ˆ c i ( E )) = c i ( index ( D ( E ))). On the other hand, the geometricfamily E represents a differential K -theory class [ E , ∈ ˆ K ( M ) in the model [BS07],and we have I ([ E , index ( D ( E )). The class ˆ c i ([ E , ∈ d H Z i ( M ) also satisfies I (ˆ c i ( E )) = c i ( index ( D ( E ))) and thus gives a second differential refinement of the i ’thChern class of the index of D ( E ). But in general the class ˆ c i ( E ) differs from ˆ c i ([ E , R (ˆ c i ( E )) = R ([ E , [2 i ] , R (ˆ c i ([ E , C i ( R ([ E , , where ω [2 i ] denotes the degree-2 i component of the form ω . In a sense, the present notegives the right answer to the problem considered in [Bun02].Finally we discuss odd Chern classes. In topology, the odd Chern classes c oddi : K − → H Z i are related with the even Chern classes by suspension˜ K (Σ M + ) c i +12 / / ∼ = (cid:15) (cid:15) g H Z i +1 (Σ M + ) ∼ = (cid:15) (cid:15) K − ( M ) c oddi / / H Z i ( M ) . In the smooth context the suspension isomorphism is replaced by the integration R along S × M → M . We have the following odd counterpart of Theorem 1.1. Theorem 1.2
For odd i ∈ N there are unique natural transformations ˆ c oddi : ˆ K − → d H Z i such that ˆ K ( S × M ) ˆ c i +12 / / R (cid:15) (cid:15) d H Z i +1 ( S × M ) R (cid:15) (cid:15) ˆ K − ( M ) ˆ c oddi / / d H Z i ( M ) commutes. The transformation in addition satisfies I ◦ ˆ c oddi = c oddi ◦ I . Note that in [Bun02] we index the Chern classes by their degree, where in the present note we adoptthe usual convention. π : W → B be a proper K -oriented map between manifolds. Then we have anUmkehr map π ! : K ∗ ( W ) → K ∗− n ( B ), where n = dim( W ) − dim( B ). An integral indextheorem is an assertion about the Chern classes c ∗ ( π ! ( x )), or c odd ∗ ( π ! ( x )) for x ∈ K ∗ ( W ),e.g. an expression of these classes in terms of the classes c ∗ ( x ) or c odd ∗ ( x ), respectively. Aprototypical example is given in [Mad09]. The construction of differential lifts of Chernclasses makes it possible to ask for geometric refinements of these kinds of results. Anexample of such a theorem related to the Pfaffian bundle will be discussed in a forthcomingpaper. Proof.
Let K ≃ Z × BU be a representative of the homotopy type of the classifyingspace of the functor K . We choose by [BS, Prop 2.1] a sequence of manifolds ( K k ) k ≥ together with maps x k : K k → K , κ k : K k → K k +1 such that1. K k is homotopy equivalent to an i -dimensional CW -complex,2. κ k : K k → K k +1 is an embedding of a closed submanifold,3. x k : K k → K is k -connected,4. x k +1 ◦ κ k = x k .Let u ∈ K ( K ) the universal class represented by the identity map K → K . By[BS, Prop. 2.6] we can further choose a sequence ˆ u k ∈ ˆ K ( K k ) such that I (ˆ u k ) = x ∗ k u and κ ∗ k ˆ u k +1 = ˆ u k for all k ≥
0. By [BS, Lem. 3.8] and 2 j − < k we have that H j − ( K k , R ) = 0. We consider the canonical natural transformation ι R : H Z ∗ → H R ∗ and the de Rham map Rham : Ω ∗ cl → H R ∗ . Since Rham is multiplicative we have ι R ( c i ( I (ˆ u k ))) = C i ( ch ( I (ˆ u k ))) = C i ( Rham ( R (ˆ u k ))) = Rham ( C i ( R (ˆ u k ))) . If we choose k ≥ i , then the diagram d H Z i ( K k ) I / / R (cid:15) (cid:15) H Z i ( K k ) ι R (cid:15) (cid:15) Ω icl ( K k ) Rham / / H R i ( K k )is cartesian. Hence for k ≥ i there exists a unique class ˆ z i,k ∈ d H Z i ( K k ) such that I (ˆ z i,k ) = c i ( I (ˆ u k )) , R (ˆ z i ) = C i ( R (ˆ u k )) . κ k ˆ z i,k +1 = ˆ z i,k . For k < i we define z i,k := ( κ ∗ k ◦ · · · ◦ κ ∗ i − ) z i, i .We now define the natural transformation ˆ c i . We start with the observation that if ˆ c i exists, then it satisfies ˆ c i (ˆ u k ) = ˆ z i,k . Let ˆ w ∈ ˆ K ( M ). By [BS, Prop. 2.6] we have K ( M ) ∼ = colim k [ M, K k ], and the underlyingclass I ( ˆ w ) ∈ K ( M ) can be written as I ( ˆ w ) = f ∗ x ∗ k u for some k and f : M → K k . Wechoose a form ρ ∈ Ω odd ( M ) such thatˆ w = f ∗ ˆ u k + a ( ρ ) . We consider a form ˜ ρ ∈ Ω odd ([0 , × M ) which restricts to ρ on { } × M and to 0 on { } × M . We get a class ˜ˆ w = pr ∗ M ˆ w + a ( ˜ ρ ) ∈ ˆ K ([0 , × M ). Note that˜ˆ w |{ }× M = f ∗ ˆ u k , ˜ˆ w |{ }× M = ˆ w . If ˆ c i exists, then we must have by naturality and the homotopy formula [BS, (1)]ˆ c i ( ˜ˆ w |{ }× M ) = f ∗ ˆ z i,k , ˆ c i ( ˜ˆ w |{ }× M ) − ˆ c i ( ˜ˆ w |{ }× M ) = a ( Z [0 , × M/M R (ˆ c i ( ˜ˆ w ))) . Furthermore, by the commutativity of the upper square in (1) we must require R (ˆ c i ( ˜ˆ w )) = C i ( R ( ˜ˆ w )) . Therefore we are forced to defineˆ c i ( ˆ w ) := f ∗ ˆ z i,k + a ( Z [0 , × M/M C i ( R ( ˜ˆ w ))) (2)We see that if ˆ c i exists, then it is automatically unique. Lemma 2.1
The definition of ˆ c i ( ˆ w ) by (2) is independent of the choices of ˜ ρ , ρ and f : M → K k .Proof. Let us start with a second choice ˜ ρ ′ and write ˜ˆ w ′ := pr ∗ M ˆ w + a ( ˜ ρ ′ ). Then wecan connect ˜ ρ with ˜ ρ ′ by a family of such forms, e.g. the linear path. This path can beconsidered as a form ¯ ρ on [0 , × [0 , × M . By construction ¯ ρ | [0 , ×{ j }× M is constant andhas no component in the direction of the first variable for j = 0 , R ( ˜ˆ w ′ ) | [0 , ×{ j }× M = 0 . (3)We set ¯ˆ w := pr ∗ M ˆ w + a ( ¯ ρ ) ∈ ˆ K ([0 , × [0 , × M ). By Stokes theorem we have d Z [0 , × [0 , × M/M C i ( R (( ¯ˆ w ))) = Z [0 , × M/M C i ( R ( ˜ˆ w ′ )) − Z [0 , × M/M C i ( R ( ˜ˆ w ))6these are the contributions of the faces { j } × [0 , × M ) since the integral over the othertwo faces [0 , × { j } × M vanishes by (3). Since a annihilates exact forms this impliesthat a ( Z [0 , × M/M C i ( R ( ˜ˆ w ))) = a ( Z [0 , × M/M C i ( R ( ˜ˆ w ′ ))) . Assume now that we have chosen a different ρ ′ . Then a ( ρ ′ − ρ ) = 0 so that by the exactnessaxiom [BS, (2)] there exists a class ˆ v ∈ ˆ K ( M ) with R (ˆ v ) = ρ ′ − ρ . Let ˆ e ∈ ˆ K ( S ) be a liftof the generator of K ( S ) ∼ = Z with R (ˆ e ) = dt . We consider the form ˜ σ ∈ Ω odd ([0 , × M )with no dt -component given by˜ σ |{ t }× M := Z [0 ,t ] × M/M R (ˆ e × ˆ v ) , where we identify S ∼ = R / Z and view the interval [0 , t ] as a subset of S . Then˜ σ |{ }× M = 0 , ˜ σ |{ }× M = ρ ′ − ρ , d ˜ σ = dt ∧ pr ∗ M R (ˆ v ) = R (ˆ e × ˆ v ) . We now consider ˜ˆ v := pr ∗ M ˆ w + pr ∗ M a ( ρ ) + a (˜ σ ) ∈ ˆ K ([0 , × M )and calculate modulo the image of d Z [0 , × M/M C i ( R (˜ˆ v )) ≡ Z S × M/M C i ( R ( pr ∗ M ( ˆ w )) + pr ∗ M dρ + R (ˆ e × ˆ v )) ≡ Z S × M/M C i ( R ( pr ∗ M ( ˆ w )) + R (ˆ e × ˆ v )) ≡ Z S × M/M C i ( R ( pr ∗ M ( ˆ w ) + ˆ e × ˆ v )) . It follows that
Rham ( Z [0 , × M/M C i ( R (˜ˆ v ))) = Rham ( Z S × M/M C i ( R ( pr ∗ M ( ˆ w ) + ˆ e × ˆ v )))= Z S × M/M
Rham ( C i ( R ( pr ∗ M ( ˆ w ) + ˆ e × ˆ v )))= Z S × M/M ι R ( c i ( I ( pr ∗ M ( ˆ w ) + ˆ e × ˆ v ))) . In other words,
Rham ( R [0 , × M/M C i ( R (˜ˆ v ))) is an integral class, and this implies a ( Z [0 , × M/M C i ( R (˜ˆ v ))) = 07y [BS, (2)].If ˜ ρ was the path connecting ρ with 0, then we construct the path ˜ ρ ′ from ρ ′ to 0 byconcatenating ˜ ρ with ˜ σ (in order to concatenate smoothly we can change ˜ ρ ). Then get˜ˆ w ′ := pr ∗ M ˆ w + a ( ˜ ρ ′ ) ∈ ˆ K ([0 , × M ) and a ( Z [0 , × M/M C i ( R ( ˜ˆ w ′ ))) = a ( Z [0 , × M/M C i ( R ( ˜ˆ w )))+ a ( Z [0 , × M/M C i ( R (˜ˆ v )))= a ( Z [0 , × M/M C i ( R ( ˜ˆ w )))This finishes the verification that our construction of c i is independent of the choice of ρ .Finally we verify that ˆ c i ( ˆ w ) is independent of the choice of f : M → K k . If we replace k by k + 1 and f by κ k ◦ f , then we obviously get the same result. For two choices f : M → K k and f ′ : M → K k ′ there exists k ′′ ≥ max { k, k ′ } such that κ k ′′ k ◦ f and κ k ′′ k ′ ◦ f ′ are homotopic. Here κ ji : K i → K j denotes for j > i the composition κ ji := κ j − ◦ · · · ◦ κ i .Therefore it remains to show that a choice f ′ : M → K k homotopic to f : M → K k givesthe same result for ˆ c i ( ˆ w ). Let H : [0 , × M → K k be a homotopy from f to f ′ . Thenwe use H in the construction of ˆ c i ( pr ∗ M ˆ w ) ∈ d H Z i ([0 , × M ). If we let ˆ c ′ i ( ˆ w ) denote theresult of the construction based on the choice of f ′ we have by the homotopy formulaˆ c ′ i ( ˆ w ) − ˆ c ′ i ( ˆ w ) = a ( Z R (ˆ c i ( pr ∗ M ˆ w ))) = a ( Z pr ∗ M C i ( ˆ w )) = 0 . ✷ Lemma 2.2
The construction of ˆ c i defines a natural transformation ˆ c i : ˆ K → d H Z i ofset-valued functors on smooth manifolds.Proof. Let g : N → M be a smooth map between manifolds. Let ˆ w ∈ ˆ K ( M ) andassume that we have constructed ˆ c i ( ˆ w ) using the choices of f : M → K k , ρ ∈ Ω odd ( M )and ˜ ρ ∈ Ω odd ([0 , × M ). Then we construct ˆ c i ( g ∗ ˆ w ) using the choices f ◦ g : N → K k and g ∗ ρ ∈ Ω odd ( N ), ( id × g ) ∗ ˜ ρ ∈ Ω odd ([0 , × N ). With these choices we have ( id × g ) ∗ ˜ˆ w = g g ∗ ˆ w ∈ ˆ K ([0 , × N ) and g ∗ ˆ c i ( ˆ w ) = g ∗ f ∗ ˆ z i,k + g ∗ a ( Z [0 , × M/M C i ( R ( ˜ˆ w )))= ( f ◦ g ) ∗ ˆ z i,k + a ( Z [0 , × M/M C i ( R (( id × g ) ∗ ˜ˆ w )))= ( f ◦ g ) ∗ ˆ z i,k + a ( Z [0 , × M/M C i ( R ( g g ∗ ˆ w )))= ˆ c i ( g ∗ ˆ w ) . This finishes the proof of Assertion 1 of Theorem 1.1.In order to show the second Assertion 2 we consider the natural transformationˆ B : ˆ K × ˆ K → d H Z ev given by ˆ B ( ˆ w, ˆ v ) := ˆ c ( ˆ w ) ∪ ˆ c (ˆ v ) − ˆ c ( ˆ w + ˆ v ) ∈ d H Z ev ( M ) , ˆ w, ˆ v ∈ ˆ K ( M ) . If we apply I we get I ( ˆ B ( ˆ w, ˆ v )) = I (ˆ c ( ˆ w ) ∪ ˆ c (ˆ v )) − I (ˆ c ( ˆ w + ˆ v ))= I (ˆ c ( ˆ w )) ∪ I (ˆ c (ˆ v )) − I (ˆ c ( ˆ w + ˆ v ))= c ( I ( ˆ w )) ∪ c ( I (ˆ v )) − c ( I ( ˆ w ) + I (ˆ v ))= 0 . Let C = 1 + C + C + · · · ∈ Q [[ s , s , . . . ]]. Then we have the identity C ( s + s ′ , s + s ′ , . . . ) = C ( s , s , . . . ) C ( s ′ , s ′ , . . . ) . Indeed, if ˜ ch = X i ≥ ( e x i − , then C ( ch , . . . ) = Y i ≥ (1 + x i ) . If we introduce another set of variables x ′ i and set ˜ ch ′ = P i ≥ ( e x ′ i − C ( ch + ch ′ , ch + ch ′ , . . . ) = Y i ≥ (1 + x i )(1 + x ′ i )= C ( ch , ch , . . . ) C ( ch ′ , ch ′ , . . . ) . We now calculate R ( ˆ B ( ˆ w, ˆ v )) = R (ˆ c ( ˆ w ) ∪ ˆ c (ˆ v )) − R (ˆ c ( ˆ w + ˆ v ))= R (ˆ c ( ˆ w )) ∪ R (ˆ c (ˆ v )) − R (ˆ c ( ˆ w + ˆ v ))= C ( R ( ˆ w )) ∧ C ( R (ˆ v )) − C ( R ( ˆ w ) + R (ˆ v ))= 0 .
9t follows that ˆ B factorises over the subfunctor H R odd /H Z odd ⊂ H R / Z odd ⊂ d H Z ev , where the inclusion is induced by a . Let ρ ∈ Ω odd ( M ) and consider ˜ ρ := t pr ∗ M ρ ∈ Ω odd ([0 , × M ). Then we haveˆ B ( ˆ w + a ( ρ ) , ˆ v ) − ˆ B ( ˆ w, ˆ v ) = ˆ B ( pr ∗ M ˆ w + a ( ˜ ρ ) , ˆ v ) |{ }× M − ˆ B ( pr ∗ M ˆ w + a ( ˜ ρ ) , ˆ v ) |{ }× M . Since ˆ B takes values in the homotopy invariant subfunctor H R odd /H Z odd we concludethat ˆ B ( ˆ w + a ( ρ ) , ˆ v ) = ˆ B ( ˆ w, ˆ v ). In a similar manner we see that ˆ B ( ˆ w, ˆ v + a ( ρ )) = ˆ B ( ˆ w, ˆ v ).Hence ˆ B has a factorisation over a natural transformation K × K → H R odd /H Z odd ⊂ H R / Z odd . Such a natural transformation between homotopy invariant functors on manifolds mustbe represented by a map of classifying spaces K × K → K ( R / Z , odd ) , where K ( R / Z , odd ) := W i ≥ K ( R / Z , i + 1) is a wedge of Eilenberg-MacLane spaces, i.e.by a class in B ∈ H odd ( K × K ; R / Z ). Since K and therefore K × K are even spaceswe know that H odd ( K × K ; Z ) = 0. It follows by the universal coefficient formula that H odd ( K × K ; R / Z ) ∼ = Hom ( H odd ( K × K ; Z ) , R / Z ) = 0. We see that B = 0 and thereforeˆ B = 0. This finishes the proof of Assertion 2 of Theorem 1.1. ✷ We now show Theorem 1.2. We let ˆ e ∈ K ( S ) be, as above, the unique element with R (ˆ e ) = dt , I (ˆ e ) = e ∈ K ( S ) the canonical generator, and ˆ e |∗ = 0 for a basepoint ∗ ∈ S .Then we define for odd i ∈ N and ˆ x ∈ ˆ K − ( M )ˆ c oddi (ˆ x ) := Z ˆ c i +12 (ˆ e × ˆ x ) . Note that I ( Z ˆ c i +12 (ˆ e × ˆ x )) = Z c i +12 ( e × I (ˆ x )) . We have a natural inclusion g H Z ∗ (Σ M + ) ⊂ H Z ∗ ( S × M ) as the subspace of classeswhose restriction to {∗} × M vanishes. Since e |∗ = 0 we see that e × I (ˆ x ) belongs to thissubspace. The restriction of R to this subspace coincides with the suspension isomorphism g H Z ∗ +1 (Σ M + ) ∼ → H Z ∗ ( M ), R ( e × x ) = x with inverse x e × x . Therefore Z c i +12 ( e × I (ˆ x )) = c oddi ( I (ˆ x )) .
10n this way we get a natural transformation which has the required property.Since R : ˆ K ( S × M ) → ˆ K − ( M ) is surjective it is clear that ˆ c oddi is unique. ✷ References [Ber08] Alain Berthomieu. A version of smooth K-theory adapted to the total Chernclass, 2008. arXiv.org:0806.4728[BS] U. Bunke and Th. Schick. Uniqueness of smooth extensions of generalized coho-mology theories. 2008. arXiv:0901.4423[BS07] Ulrich Bunke and Thomas Schick. Smooth K-theory, 2007. arXiv:0707.0046[Bun02] U. Bunke. Index theory, eta forms, and Deligne cohomology,
Memoirs of theAMS , 198 (5), 2009.[Bun09] Ulrich Bunke. Adams operations in smooth K-theory, 2009. arXiv:0904.4355[CS85] Jeff Cheeger and James Simons. Differential characters and geometric invariants.In
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