Remarks on flat and differential K-theory

In this note we prove some results in flat and differential $K$-theory. The first one is a proof of the compatibility of the differential topological index and the flat topological index by a direct computation. The second one is the explicit isomorphisms between Bunke-Schick differential $K$-theory and Freed-Lott differential $K$-theory.


Introduction
In this note we prove some results in flat and differential K-theory. While some of these results are known to the experts, the proofs given here have not appeared in the literature. We first prove the compatibility of the flat topological index ind t L and the differential topological index ind t FL by a direct computation, i.e., the following diagram commutes ([7, Proposition 8.10]) (1.1) Keywords: differential K-theory, topological index.
where i is the canonical inclusion, K −1 L (X; R/Z) is the geometric model of K-theory with R/Z coefficients and K FL (X) is Freed-Lott differential K-theory. The commutativity of (1.1) is a consequence of the compatibility of the differential analytic index ind a FL and the flat analytic index ind a L together with the differential family index theorem [7,Theorem 7.35]. The differential topological index ind t FL is defined to be the composition of an embedding pushforward and a projection pushforward. When defining the embedding pushforward, currential K-theory [7, §2.28] is used instead of differential K-theory due to the Bismut-Zhang current [2,Definition 1.3]. It is not clear whether currential K-theory should be regarded as a differential cohomology or a "differential homology" (see [6, §4.5] for a detailed discussion), so it may be clearer by looking at the direct computation.
The paper is organized as follows: Section 2 contains all the necessary background material, including the Freed-Lott differential K-theory, the differential topological index, the pairing between flat K-theory and Khomology, and Bunke-Schick differential K-theory. In Section 3 we prove the main results.
Acknowledgement. The author would like to thank Steve Rosenberg for valuable comments and suggestions, and Thomas Schick for his comments on the explicit isomorphisms between Bunke-Schick differential Ktheory and Freed-Lott differential K-theory.

Background material
2.1. Freed-Lott differential K-theory and the differential topological index In this section we review Freed-Lott differential K-theory and the construction of the differential topological index [7, §4, 5]. We refer the readers to [7] for the details.
The Freed-Lott differential K-group K FL (X) is the abelian group generated by quadruples E = (E, h, ∇, φ), where (E, h, ∇) → X is a complex vector bundle with a Hermitian metric h and a unitary connection ∇, and There is an exact sequence [7, (2.20)] [11], i is the canonical inclusion map, Let X → B and Y → B be fiber bundles of smooth manifolds with X compact. Let g T V X and g T V Y be metrics on the vertical bundles T V X → X and T V Y → Y respectively, and assume there are horizontal distributions T H X and T H Y . Let E = (E, h E , ∇ E , φ) ∈ K FL (X) and ι : X → Y be an embedding of manifolds. We assume the codimension of X in Y is even, and the normal bundle ν → X of X in Y carries a spin c structure. As in [7, §5] we assume for each b ∈ B, the map ι b : X b → Y b is an isometric embedding and is compatible with projections to B. Denote by S(ν) → X the spinor bundle associated to the spin c -structure of ν → X. We can locally choose a spin stricture for ν → X with spinor bundle S spin (ν). Then there exists a locally defined Hermitian line bundle L Note that the tensor product on the right is globally defined, and so is the Hermitian line bundle L(ν) → X defined by L(ν) := (L 1 2 (ν)) 2 . Let ∇ ν be a metric compatible connection on ν → X. It has a unique lift to a connection on S spin (ν), still denoted by ∇ ν . Choose a unitary connection ∇ L(ν) on L(ν) → X, which induces a connection on L 1 2 (ν). The tensor product of ∇ ν and the induced connection on L The embedding pushforward ι * : As noted in [7, p.926] the horizontal distributions of the fiber bundles is defined to correct this non-compatibility, and it satisfies the following transgression formula [7, (5.6)] The modified embedding pushforward ι mod * Definition 5.34] is defined by taking Y = S N × B for some even N and composing the embedding pushforward with the submersion pushforward π prod * defined in [7, Lemma 5.13], i.e., ind t FL := π prod * • ι mod * .

Pairing between flat K-theory and topological K-homology
Let X be an odd-dimensional closed spin c manifold, E = (E, h E , ∇ E , φ) ∈ δ K FL (X), and D X,E be the twisted Dirac operator on S(X) ⊗ E → X.

Bunke-Schick differential K-theory
In this subsection we briefly recall Bunke-Schick differential K-theory K BS , and refer to [4] for the details.

Compatibility of the topological indices
Note that every element E − F ∈ K FL (X) can be written in the form The existence of the connection ∇ G such that CS(∇ F ⊕ ∇ G , d) = 0, where d is the trivial connection on the trivial bundle X × C n → X, follows from [12, Theorem 1.8]. Here φ G := −φ F . Henceforth we assume an element of K FL (X; R/Z) is of the form E − [n]. These arguments also apply to elements in K −1 L (X; R/Z). Proposition 3.1. Let π : X → B be a fiber bundle with X compact and such that the fibers are of even dimension. The following diagram commutes.
). We prove that h = 0. By [7,Lemma 5.36] and the fact that ch K FL •i = 0 (see (2.1)), we have Since In the following, we write a ≡ b as a = b mod Z. By [7, (6.7)], we havē On the other hand, by [11, (49)], we havē Recall that the purpose of the modified embedding pushforward ι mod * is to correct the non-compatibility of the horizontal distributions T H (S N × B) and T H X. By (3.7) we may assume that the horizontal distributions T H (S N × B) and T H X are compatible by changing the one for X to be the restriction of the one for S N × B. Thus which implies that (3.6) is zero, and therefore h = 0.

Explicit isomorphisms between K BS and K FL
In this subsection we construct the explicit isomorphisms between Bunke-Schick differential K-group and the Freed-Lott differential K-group.
where, in the definition of f , E is the zero-dimensional geometric family associated to (E, h, ∇). Then f and g are well defined ring isomorphisms and are inverses to each other.
Proof. Note that it suffices to prove the statement under the assumption that ind a (E ) → B is actually given by a kernel bundle ker(D E ) → B in the definition of g. Indeed, by a standard perturbation argument every class in K BS has a representative where this is satisfied.
We prove that f and g are inverses to each other. Let (E, h, ∇, φ) be a generator of K FL (B). Then by [4,Corollary 5.5] again.
Since f is a ring homomorphism, the same is true for g. Thus f and g are ring isomorphisms and are inverses to each other.