Geometric Families Define Differential K-Theory

Abstract

The index-theoretic construction of differential K-theory by Bunke and Schick uses both a geometric family and a differential form as a cocycle data. We prove that geometric families alone can codify the differential K-theory.

1. Introduction

There has been considerable interest in differential K-theory in mathematics. Differential K-theory is a homotopy-theoretically consistent construction codified by complex vector bundles with connection. It incorporates a deep connection among topology, geometry, and analysis established by the Atiyah–Singer index theorem in the general form. It has applications to the classification of Ramond–Ramond fields in Type II superstring theory [1,2,3], algebraic K-theory [4], and equivariant $\eta$-invariants [5]. Differential K-theory is also one of the interesting applications of ∞-categorical machinery, which is nicely packaged by a certain type of ∞-sheaves of spectra on the site of smooth manifolds (see [6]).

The history of differential K-theory goes back to Karoubi [7], wherein he considered multiplicative K-theory, nowadays known as the flat differential K-theory. Lott [8] used Karoubi’s multiplicative K-theory to prove an index theorem therein using the Bismut superconnection and local index theory techniques. Both multiplicative K-theory and the index theorem were generalized as differential K-theory and as the differential index theorem by Freed and Lott [9]. To this date, various models of differential K-theory have been discovered and studied from different perspectives (see [6,9,10,11,12,13,14,15,16,17,18,19,20] for a survey). All of them are naturally isomorphic by satisfying a set of axioms for uniqueness due to Bunke and Schick [21], particularly to a model obtained by a homotopy-theoretically consistent manner (see [6,10]). In all these models, the wrong-way map in differential K-theory has been mostly considered from the following two approaches. First, as with Freed and Lott [9] (compare [8,17]), one can use geometric cocycles, such as vector bundles with connection and differential forms, to model differential K-theory. In this case, cocycles are intuitive and easy to work with, but construction of the differential index demands work. Alternatively, as with Bunke and Schick [11], one can work with a model whose cocycle consists of a geometric family and a differential form, wherein the local index theorem is already built in so that the construction of differential K-theoretic pushforward becomes easier. But actual work using such a model can be laborious.

An interesting aspect of the geometric model due to Freed–Lott [9] and Klonoff [12] is that, in their cocycle data, the differential forms are unnecessary for codifying differential K-theory. This is because of the Venice lemma by Simons and Sullivan [13] (compare [22,23]). Since the index-theoretic model by Bunke and Schick also has differential forms in its cocycle data, it is natural to ask whether differential K-theory can be codified solely with geometric families up to an equivalence defined by taming. In this paper, we answer this question within the context of the Bunke–Schick model.

A positive answer to the aforementioned question is suggested by Ho’s result [22] (Proposition 3.2, p. 98), which establishes an isomorphism between the Bunke–Schick index-theoretic model and the Freed–Lott–Klonoff geometric model. However, this result itself does not reveal necessary details about how the differential form data in the Bunke–Schick model becomes unnecessary. Surely the Venice lemma is a key ingredient here, but there are other important constituents to consider, such as the relationship between Bismut–Cheeger $\eta$-forms and Chern–Simons forms. Notice that Ho used the relationship between the Bunke $\eta$-form and the Chern–Simons form in key steps of his proof.

The advantage of the present work is that it works only in the Bunke–Schick model of differential K-theory to show that geometric families are enough to codify differential K-theory. In addition to the Venice lemma, we have used one of the results in Bunke and Schick [11], which shows how Bunke’s $\eta$-form and Bismut–Cheeger $\eta$-form are related (see Lemma 2 and Corollary 1 below). We also use a Lemma 1 that relates the Bismut–Cheeger $\eta$-form to a Chern–Simons form of a path of connections on a vector bundle modulo d-exact differential forms. While the Bunke–Schick model is one of the most effective for studying pushforwards, not many literatures deploy this model. We hope this work contributes to its wider adoption.

While Ho’s isomorphism theorem [22] (Proposition 3.2) already implies that differential forms are inessential by comparing the Bunke–Schick and Freed–Lott–Klonoff models, our contribution provides a different perspective in both technique and consequence. Technically, we work entirely within the Bunke–Schick model, systematically converting differential form data into geometric families via kernel bundle constructions, without invoking the model comparison isomorphism. This internal approach makes explicit the geometric mechanism by which forms are absorbed—namely through the interplay of the Venice lemma, the Bismut–Cheeger $\eta$-form, and kernel bundle projections. Consequently, our result clarifies how to perform concrete calculations in the Bunke–Schick framework and may facilitate extensions to equivariant and twisted settings where explicit geometric constructions are essential.

This paper is organized as follows: Section 2.1 reviews Freed–Lott–Klonoff model of differential K-theory and Section 2.2 reviews the Bunke–Schick model, with both giving a few relevant examples. Section 2.3.2 collects necessary facts on Bunke and Bismut–Cheeger $\eta$-forms. Section 3 states and proves our main theorem.

2. Preliminaries

In this section, we review the geometric model of differential K-theory by Freed–Lott [9] and Klonoff [12] as well as an index-theoretic model by Bunke–Schick [11]. After that, we review Bunke’s and Bismit and Cheeger’s $\eta$-forms and their relations to differential K-theory. While establishing notations and terminologies that are used throughout the paper, we collect results that will be used in Section 3.

2.1. Freed–Lott–Klonoff Differential K-Theory

Let X be a compact smooth manifold and $E\to X$ be a complex Hermitian vector bundle with a compatible connection ∇. The Chern character form associated to ∇ is defined by

$$\mathrm{ch}(\nabla ):=\mathtt{Tr}\exp \left({\displaystyle \frac{\sqrt{-1}}{2\pi }}{\nabla }^2\right)\in {\Omega }^{\mathrm{even}}\left(X\right).$$

(1)

It is a closed differential form whose de Rham cohomology class ${\left[\mathrm{ch}(\nabla )\right]}_{\mathrm{dR}}\in H_{\mathrm{dR}}^{\mathrm{even}}(X;\mathbb{C})$ is independent of the choice of ∇. If ${\nabla }^0$ and ${\nabla }^1$ are two connections on E, there is a Chern–Simons form $\mathrm{CS}(t\mapsto {\nabla }^t)\in {\Omega }^{\mathrm{odd}}\left(X\right)$, which interpolates the two connections in the following sense:

$$d\mathrm{CS}(t\mapsto {\nabla }^t)=\mathrm{ch}\left({\nabla }^0\right)-\mathrm{ch}\left({\nabla }^1\right).$$

(2)

One can write the Chern–Simons form explicitly. Let $p:[0,1]\times X\to X$ be the projection with the pullback bundle $p^{\ast }E$. Then,

$$\mathrm{CS}(t\mapsto {\nabla }^t):={\int }_{[0,1]\times X/X}\mathrm{ch}\left({\nabla }_t^E\right)\in {\Omega }^{\mathrm{odd}}\left(X\right),$$

(3)

where ${\nabla }_t^E$ is a connection on $p^{\ast }E$ interpolating connections ${\nabla }^0$ and ${\nabla }^1$ on E via $t\mapsto {\nabla }^t$. It is well-known that if we choose a different path, the difference is always d-exact (see Simons and Sullivan [13] (p. 583, Proposition 1.1) and compare [19] (p. 18, Lemma 3.12)). Thus, we may set $\mathrm{CS}({\nabla }^0,{\nabla }^1):=\mathrm{CS}(t\mapsto {\nabla }^t)mod\mathrm{Im}\left(d\right)$. The Freed–Lott–Klonoff differential K-group ${\widehat{K}}_{\mathrm{FLK}}^0\left(X\right)$ is the abelian group generated by the following generators and relations.

Definition 1. 

A  ${\widehat{K}}_{\mathit{FLK}}^0\left(X\right)$-generator  is a quadruple $\mathcal{E}=(E,h^E,{\nabla }^E,{\varphi }^E)$, where $(E,h^E,{\nabla }^E)\to X$ is a complex hermitian vector bundle over X with a $h^E$-compatible connection ${\nabla }^E$ and ${\varphi }^E\in {\Omega }^{\mathit{odd}}/Im\left(d\right)$.

Definition 2. 

The  ${\widehat{K}}_{\mathit{FLK}}^0\left(X\right)$-relations  are defined as $\mathcal{E}\sim \mathcal{F}$ if there is a complex hermitian vector bundle $(G,h^G,{\nabla }^G)\to X$ and a vector bundle isomorphism $\phi :E\oplus G\to F\oplus G$ covering the identity map on X such that $CS({\nabla }^E\oplus {\nabla }^F,{\phi }^{\ast }({\nabla }^F\oplus {\nabla }^G))={\varphi }^E-{\varphi }^F$.

Definition 3. 

An addition + between two equivalence classes of ${\widehat{K}}_{\mathit{FLK}}^0\left(X\right)$-generators is defined by

$$\left[(E,h^E,{\nabla }^E,{\varphi }^E)\right]+\left[(F,h^F,{\nabla }^F,{\varphi }^F)\right]:=\left[(E\oplus F,h^E\oplus h^F,{\nabla }^E\oplus {\nabla }^F,{\varphi }^E+{\varphi }^F)\right].$$

The addition + is well-defined and the set of all equivalence classes of ${\widehat{K}}_{\mathrm{FLK}}^0\left(X\right)$-generators forms a commutative monoid $(\mathfrak{M},+)$.

Definition 4. 

The  Freed–Lott–Klonoff differential K-group  of X, denoted by ${\widehat{K}}_{\mathit{FLK}}^0\left(X\right)$, is the Grothendieck group of the commutative monoid $(\mathfrak{M},+)$.

To avoid excessive notation, we shall write $\left[\mathcal{E}\right]$ for the image of the ${\widehat{K}}_{\mathrm{FLK}}^0\left(X\right)$-equivalence class $\left[\mathcal{E}\right]$ in the group ${\widehat{K}}_{\mathrm{FLK}}^0\left(X\right)$. Note that every element of ${\widehat{K}}_{\mathrm{FLK}}^0\left(X\right)$ is of the form $\left[\mathcal{E}\right]-\left[\mathcal{F}\right]$.

We can also represent ${\widehat{K}}_{\mathrm{FLK}}^0\left(X\right)$ using $\mathbb{Z}/2\mathbb{Z}$-graded vector bundles (or superbundles) (see [8] (p. 285)). A generator of ${\widehat{K}}_{\mathrm{FLK}}^0\left(X\right)$ is then a $\mathbb{Z}/2\mathbb{Z}$-graded complex vector bundle $E=E^{+}\oplus E^{-}$ with a hermitian metric $h^E=h^{E^{+}}\oplus h^{E^{-}}$ and a connection ${\nabla }^E={\nabla }^{E^{+}}\oplus {\nabla }^{E^{-}}$. Ungraded vector bundles may be thought of as superbundles that are purely even. We define $\mathrm{ch}\left({\nabla }^E\right):=\mathrm{ch}\left({\nabla }^{E^{+}}\right)-\mathrm{ch}\left({\nabla }^{E^{-}}\right)$. Given two connections ${\nabla }^E={\nabla }^{E^{+}}\oplus {\nabla }^{E^{-}}$ and ${\nabla }^{\tilde{E}}={\nabla }^{{\tilde{E}}^{+}}\oplus {\nabla }^{{\tilde{E}}^{-}}$ on the superbundle E, the Chern–Simons form $\mathrm{CS}({\nabla }^E,{\nabla }^{\tilde{E}})$ is defined for a path of connections that interpolates two pairs of connections simultaneously. It satisfies the transgression formula

$$d\mathrm{CS}({\nabla }^E,{\nabla }^{\tilde{E}})=\mathrm{ch}({\nabla }^E)-\mathrm{ch}({\nabla }^{\tilde{E}}),$$

which is similar to the Equation (2). We will not distinguish the type of Chern–Simons forms in the notation but it will be clear from the context.

2.2. Bunke–Schick Differential K-Theory

Let B be a compact manifold, possibly with boundary. The Bunke–Schick differential K-theory ${\widehat{K}}_{\mathrm{BS}}^0\left(B\right)$ is defined as the group completion of a semigroup of isomorphism classes of cycles modulo an equivalence relation. We begin with the definition of a cycle.

Definition 5 

([11] (p. 57, Definition 2.2)). A geometric family over B is a tuple

$$\mathcal{E}=(X,g^{T^v\pi },T^h\pi ,\mathcal{V},\mathrm{or})$$

where

1. 

$\pi :X\to B$ is a proper submersion with closed fibers.

2. 

$g^{T^v\pi }$ is a vertical Riemannian metric on the vertical bundle $T^v\pi \subset TX$ defined by $T^v\pi :=\ker (d\pi :TX\to {\pi }^{\ast }TB)$.

3. 

$T^h\pi$ is a horizontal distribution, i.e., a bundle $T^h\pi \subseteq TX$ such that $T^h\pi \oplus T^v\pi =TX$.

4. 

$\mathcal{V}$ is a family of Dirac bundles over X.

5. 

$\mathrm{or}$ is an orientation of $T^v\pi$.

• A family of Dirac bundles is given by a tuple

$$\mathcal{V}=(V,h^V,{\nabla }^V,c,z)$$

where

1. 

$(V,h^V,{\nabla }^V)$ is a hermitian vector bundle with connection over X.

2. 

$c:T^v\pi \otimes V\to V$ is a Clifford multiplication.

3. 

On the component where $\mathit{dim}\left(T^v\pi \right)$ has even dimension a $\mathbb{Z}/2\mathbb{Z}$-grading z of V

• with a requirement that the restriction of the family Dirac bundles to the fibers ${\pi }^{-1}\left(b\right)$, $b\in B$ gives Dirac bundles in the sense of [24] (p. 13, Definition 2.1.1).

Definition 6. 

A  cycle  of the differential K-theory ${\widehat{K}}_{\mathit{BS}}^0\left(B\right)$ is a pair $(\mathcal{E},\rho )$, where $\mathcal{E}$ is a geometric family over B and $\rho \in {\Omega }^{\mathit{odd}}\left(B\right)/\mathit{Im}\left(d\right)$.

As we consider even differential K-theory exclusively in our paper, we will assume for the cycle $(\mathcal{E},\rho )$ that $dim\left(T^v\pi \right)$ is even-dimensional and $\rho$ is an odd differential form.

We introduce the notion of the opposite geometric family, which is needed to define a representative for the additive inverse of ${\widehat{K}}_{\mathrm{BS}}^0\left(B\right)$-class.

Definition 7 

([11] (p. 58, Definition 2.3)). The opposite ${\mathcal{E}}^{\mathsf{o}\mathsf{p}}$ of a geometric family $\mathcal{E}$ is obtained by reversing the signs of the Clifford multiplication and the grading of the underlying Clifford bundles, as well as the orientation of the vertical bundles.

Next, we define the notion of isomorphism between the geometric families.

Definition 8 

([11] (p. 58, Section 2.1.7)). Let $\mathcal{E}$ and ${\mathcal{E}}^{\prime }$ be two geometric families over B. An isomorphism $\mathcal{E}\stackrel{\cong }{\to }{\mathcal{E}}^{\prime }$ consists of the following data:

• where

1. 

f is a diffeomorphism over B.

2. 

F is a bundle isomorphism over f.

3. 

f preserves the horizontal distribution, the vertical metric and the orientation.

4. 

F preserves the connection, Clifford multiplication, and the grading.

Definition 9 

([11] (p. 59, Definition 2.5)). Two cycles $(\mathcal{E},\rho )$ and $({\mathcal{E}}^{\prime },{\rho }^{\prime })$ are isomorphic if $\mathcal{E}\cong {\mathcal{E}}^{\prime }$ and $\rho ={\rho }^{\prime }$. We let $G\left(B\right)$ denote the set of isomorphism classes of cycles over B.

Definition 10. 

Let $\mathcal{E},{\mathcal{E}}^{\prime }$ be two geometric families over B. Then, the  sum  is defined as follows:

$$\mathcal{E}{\bigsqcup }_B{\mathcal{E}}^{\prime }=(X\bigsqcup X^{\prime },g^{T^v(\pi \bigsqcup {\pi }^{\prime })},T^h(\pi \bigsqcup {\pi }^{\prime }),\mathcal{V}\bigsqcup {\mathcal{V}}^{\prime },\mathit{or}\bigsqcup {\mathit{or}}^{\prime }),$$

where

1. 

$X\bigsqcup X^{\prime }$ is the disjoint union.

2. 

$\pi \bigsqcup {\pi }^{\prime }:X\bigsqcup X^{\prime }\to B$ is the proper submersion.

3. 

$g^{T^v(\pi \bigsqcup {\pi }^{\prime })}:=g^{T^v\pi }\bigsqcup g^{T^v{\pi }^{\prime }}$.

4. 

$T^h(\pi \bigsqcup {\pi }^{\prime }):=T^h\pi \bigsqcup T^h{\pi }^{\prime }$.

5. 

$\mathcal{V}\bigsqcup {\mathcal{V}}^{\prime }$ is the disjoint union of two Dirac bundles.

6. 

$\mathit{or}\bigsqcup {\mathit{or}}^{\prime }$ is the orientation of $T^v\pi \bigsqcup T^v{\pi }^{\prime }$.

Definition 11 

([11] (p. 59, 2.1.8)). The sum of two cycles $(\mathcal{E},\rho )$ and $({\mathcal{E}}^{\prime },{\rho }^{\prime })$ is defined by

$$(\mathcal{E},\rho )+({\mathcal{E}}^{\prime },{\rho }^{\prime }):=(\mathcal{E}{\bigsqcup }_B{\mathcal{E}}^{\prime },\rho +{\rho }^{\prime }).$$

Note that $\left(G\right(B),+)$ is an abelian semigroup with the identity element $0:=(Ø,0)$, where Ø is the empty geometric family.

If $\mathcal{E}$ is a geometric family over B, then we can form a family of $\mathbb{Z}/2\mathbb{Z}$-graded Hilbert spaces $H\left(\mathcal{E}\right):={\left(H\left({\mathcal{E}}_b\right)\right)}_{b\in B}$ where $H\left({\mathcal{E}}_b\right):=L^2(X_b,V{|}_{X_b})$ is the family of $L^2$-sections of the Dirac bundle. Then, the geometric family gives rise to a family of Dirac operators $D\left(\mathcal{E}\right):={\left(D\left({\mathcal{E}}_b\right)\right)}_{b\in B}$, where $D\left({\mathcal{E}}_b\right)$ is an odd unbounded selfadjoint operator on $H\left({\mathcal{E}}_b\right)$. A pretaming of $\mathcal{E}$ is a family $Q\left(\mathcal{E}\right):={\left(Q\left({\mathcal{E}}_b\right)\right)}_{b\in B}$ of odd selfadjoint operators $Q\left({\mathcal{E}}_b\right)$ on $H\left({\mathcal{E}}_b\right)$ given by a smooth fiberwise integral kernel $Q\in C^{\infty }(X{\times }_{B}X,V⊠V^{\ast })$.

Definition 12 

([11] (p. 60, 2.2.2)). A pretaming $Q\left(\mathcal{E}\right)$ of a geometric family $\mathcal{E}$ is a taming if $D\left({\mathcal{E}}_b\right)+Q\left({\mathcal{E}}_b\right)$ is invertible for all $b\in B$. A geometric family $\mathcal{E}$ together with a taming will be denoted by ${\mathcal{E}}_t$ and called a tamed geometric family. The opposite tamed family ${\mathcal{E}}_t^{\mathsf{o}\mathsf{p}}$ is given by the taming ${(-Q_b)}_{b\in B}$ of ${\mathcal{E}}^{\mathsf{o}\mathsf{p}}$.

By a suitable perturbation of the family (e.g., of the vertical metric $g^{T^v\pi }$ or the connection ${\nabla }^V$), we can arrange that $\dim \left(\ker \left(D\left({\mathcal{E}}_b\right)\right)\right)$ is independent of $b\in B$. Such perturbations exist under standard Fredholm-theoretic conditions (compact base, proper submersion with closed fibers, smooth geometric data). Once $\dim \left(\ker \left(D\left({\mathcal{E}}_b\right)\right)\right)$ is constant, the family ${\left(\ker \left(D\left({\mathcal{E}}_b\right)\right)\right)}_{b\in B}$ forms a $\mathbb{Z}/2\mathbb{Z}$-graded vector bundle $\ker \left(D\right(\mathcal{E}\left)\right)$ over B. In this way, one obtains the class $\left[\ker \left(D\left(\mathcal{E}\right)\right)\right]\in K^0\left(B\right)$. We define the index of a geometric family by (compare [24] (p. 2, 1.1.1.8) and [11] (p. 60, 2.2.2))

$$\mathtt{index}\left(\mathcal{E}\right):=\mathtt{index}\left(D\left(\mathcal{E}\right)\right)=\left[\ker \left(D\left(\mathcal{E}\right)\right)\right]\in K^0\left(B\right).$$

(4)

If the geometric family $\mathcal{E}$ admits a taming, then the associated family of Dirac operators admits an invertible compact perturbation and so $\mathtt{index}\left(\mathcal{E}\right)=0$.

Conversely, if $\mathtt{index}\left(\mathcal{E}\right)=0$ and the even part of the proper submersion $\pi$ is empty or has a component with $dim\left(T^v\pi \right)>0$, the geometric family admits a taming by [24] (p. 53, Lemma 2.2.6). This equivalence between $\mathtt{index}\left(\mathcal{E}\right)=0$ and the existence of taming holds under the assumptions that B is compact, $\pi :X\to B$ is a proper submersion with closed fibers, and all geometric structures are smooth. Thus, taming provides a geometric meaning of the index-zero condition for geometric families.

Bunke [24] introduced the $\eta$-form that measures the difference of the two tamings. Then Bunke and Schick [11] used the Bunke $\eta$-form to define the equivalence relation for the definition of K-group. See Section 2.3.1.

Definition 13 

([11] (p. 61, Definition 2.10)). Two cycles $(\mathcal{E},\rho )$ and $({\mathcal{E}}^{\prime },{\rho }^{\prime })$ are paired if there exists a taming ${(\mathcal{E}{\bigsqcup }_B{\mathcal{E}}^{\prime \mathsf{o}\mathsf{p}})}_t$ such that

$$\rho -{\rho }^{\prime }=\eta \left({\left(\mathcal{E}{\bigsqcup }_B{\mathcal{E}}^{\prime \mathsf{o}\mathsf{p}}\right)}_t\right).$$

We let ∼ denote the equivalence relation on $G\left(B\right)$ generated by the relation “paired.” Note that the relations “paired” and ∼ are compatible with the semigroup structure on $G\left(B\right)$ by [11] (p. 62, Lemma 2.12).

Definition 14 

([11] (p. 63, Definition 2.14)). The Bunke–Schick differential K-group ${\widehat{K}}_{\mathrm{BS}}^0\left(B\right)$ is defined as the group completion of the abelian semigroup $G\left(B\right)/\sim$.

We denote the K-class represented by the cycle $(\mathcal{E},\rho )$ as $[\mathcal{E},\rho ]\in {\widehat{K}}_{\mathrm{BS}}^0\left(B\right)$. The additive structure of $G\left(B\right)$ extends to ${\widehat{K}}_{\mathrm{BS}}^0\left(B\right)$ with the inverse of $[\mathcal{E},\rho ]$ given by $[{\mathcal{E}}^{\mathtt{o}\mathtt{p}},-\rho ]$ (see [11] (p. 63, Lemma 2.15)).

Remark 1. 

The assignment $B\to {\widehat{K}}_{\mathit{BS}}^0\left(B\right)$ extends to a contravariant functor from the category of smooth manifolds to the category of abelian groups (see [11] (p. 63, 2.3.2)).

Example 1 

([11] (p. 58, 2.1.4)). Let $\mathbf{E}=(E,h^E,{\nabla }^E,z^E)$ be a complex vector bundle over B with a Hermitian metric $h^E$ and a Hermitian connection ${\nabla }^E$, which are compatible with the involution $z^E$. There is a trivial submersion $\pi ={\mathcal{i}\mathcal{d}}_B:B\to B$ with zero-dimensional vertical bundle $T^v\pi$, a trivial vertical Riemannian metric $g^{T^v\pi }=0$, and the horizontal bundle $T^h\pi =TB$. Thus, we have a decomposition $TE=T^v\pi \oplus T^h\pi =0\oplus TB$. Furthermore, there is a trivial orientation of $T^v\pi$. Therefore, the tuple $\mathcal{V}:=(E,h^E,{\nabla }^E,0,z^E)$ can naturally be viewed as a family of Dirac bundles over B with the trivial Clifford multiplication. Thus, we can form a zero-dimensional geometric family $\mathcal{E}=(B,0,TB,\mathcal{V},\mathrm{or})$ over B associated with the data $\mathbf{E}$.

Remark 2. 

Example 1 suggests that for a superbundle E, we can identify the K-theory class $\left[(E,h^E,{\nabla }^E,{\varphi }^E)\right]$ in ${\widehat{K}}_{\mathit{FLK}}^0\left(B\right)$ as the K-theory class $[\mathcal{E},{\varphi }^E]$ in ${\widehat{K}}_{\mathit{BS}}^0\left(B\right)$, with the underlying proper submersion being the identity map on B. This is precisely the definition of the isomorphism constructed by Ho [22] (Proposition 3.2). So whenever a ${\widehat{K}}_{\mathit{FLK}}^0\left(B\right)$-generator is given, we shall view it as a zero-dimensional geometric family giving a ${\widehat{K}}_{\mathit{BS}}^0$-cycle.

Example 2 

([11] (p. 96, 5.3.1)). Let $\mathcal{E}$ be a geometric family over B and let $D\left(\mathcal{E}\right)$ be a family of Dirac operators on the bundle of Hilbert spaces $H\left(\mathcal{E}\right)\to B$. The geometry of $\mathcal{E}$ induces a connection ${\nabla }^H$ on this family (the connection part of the Bismut superconnection [25] (p. 333, Prop. 10.15)). Assuming that $\dim \left(\ker \left(D_b\right)\right)$ is constant, we can form a vector bundle $K:=\ker \left(D\right(\mathcal{E}\left)\right)$, and the projection of ${\nabla }^H$ to K gives a connection ${\nabla }^K$. Hence, we obtain a hermitian bundle $\mathbf{K}:=(K,h^K,{\nabla }^K)$ and the associated zero-dimensional geometric family $\mathcal{K}$.

Example 3 

([24] (p. 53, 2.2.2.1)). Let $\mathcal{E}$ be a zero-dimensional geometric family over B associated with $\mathbf{E}$ in Example 1. We have $E\to B\stackrel{\pi }{\to }B$ where ${\pi }^{-1}\left(b\right)=\left\{b\right\}$ and ${E|}_{{\pi }^{-1}\left(b\right)}{=E|}_{\left\{b\right\}}$. Hence, $H\left({\mathcal{E}}_b\right)=E_b$ and $H\left(\mathcal{E}\right)=E$. Note that $\ker \left(D\right(\mathcal{E}\left)\right)=H\left(\mathcal{E}\right)=E$ when $\mathcal{E}$ is a zero-dimensional geometric family over B since the Clifford multiplication is trivial (see [24] (p. 53, 2.2.2.1)). Hence, we have $\mathcal{i}\mathcal{n}\mathcal{d}\mathcal{e}\mathcal{x}\left(\mathcal{E}\right)=\left[E\right]:=\left[E^{+}\right]-\left[E^{-}\right]\in K^0\left(B\right)$.

2.3. $\eta$-Forms of Bunke and Bismut–Cheeger

The local index form $\Omega \left(\mathcal{E}\right)\in {\Omega }^{•}\left(B\right)$ is a differential form canonically associated with a geometric family $\mathcal{E}$. If a geometric family $\mathcal{E}$ admits a taming ${\mathcal{E}}_t$, then we have $\mathtt{index}\left(\mathcal{E}\right)=0$ and the local index form is exact. In this case, Bunke defined the $\eta$-form $\eta \left({\mathcal{E}}_t\right)$ [24] (p. 62, Definition 2.2.16); i.e., the differential of the Bunke’s $\eta$-form

$$d\eta \left({\mathcal{E}}_t\right)=\Omega \left(\mathcal{E}\right)$$

(5)

by [24] (p.58, Theorem 2.2.13). We also have the $\eta$-form of Bismut–Cheeger ${\eta }^{\mathrm{BC}}\left(\mathcal{E}\right)$ [26,27,28], which is a generalization of the $\eta$-invariant introduced by Atiyah–Patodi–Singer [29]. Freed and Lott [9] constructed the analytic index map for the differential K-theory ${\widehat{K}}_{\mathrm{FLK}}^0\left(X\right)$ using the Bismut–Cheeger $\eta$-form.

The relationship between the Bunke and Bismut–Cheeger $\eta$-forms allows us to rewrite a K-theory class in ${\widehat{K}}_{\mathrm{BS}}^0\left(B\right)$ with a canonical geometric family and Bismut–Cheeger $\eta$-form. The properties of the Bismut–Cheeger $\eta$-form and the K-theory functor (see Remark 1) show that the Bismut–Cheeger $\eta$-form serves the same role as the Chern–Simons form when $X=B$ and the underlying proper submersion is the identity. These observations are the key ingredients for proving the Venice lemma in ${\widehat{K}}_{\mathrm{BS}}^0\left(B\right)$. Since the technical details of the $\eta$-forms are quite complicated, we will focus on applying the properties of $\eta$-forms and the analytical details and proof will be referred to [11,24].

2.3.1. Bunke $\eta$-Form

(Compare [24] (2.2.3) and [11] (2.2.4)) Let $\mathcal{V}$ denote the Dirac bundle of a geometric family $\mathcal{E}$ with underlying fiber bundle $\pi :X\to B$ with closed fibers. Locally on X, we can assume that $T^v\pi$ has a spin structure and let $S\left(T^v\pi \right)$ be the associated spinor bundle. Then, we can write the family of Dirac bundles as $V=S\left(T^v\pi \right)\otimes \mathbf{W}$ for a twisting bundle $\mathbf{W}=(W,h^W,{\nabla }^W,z^W)$, which is a $\mathbb{Z}/2\mathbb{Z}$-graded hermitian vector bundle over X with connection. The form $\widehat{A}\left({\nabla }^{T^v\pi }\right)\wedge \mathrm{ch}\left({\nabla }^W\right)\in {\Omega }^{•}\left(X\right)$ is globally defined and the local index form is defined by the integration over the fiber:

$$\Omega \left(\mathcal{E}\right):={\int }_{X/B}\widehat{A}\left({\nabla }^{T^v\pi }\right)\wedge \mathrm{ch}\left({\nabla }^W\right).$$

(6)

Example 4. 

Let $\mathcal{E}$ be a zero-dimensional geometric family over B associated with $\mathbf{E}$ in Example 1. In this case, the fiber $X/B$ is trivial, $T^v\pi =0$, $W=E$, and $\widehat{A}\left({\nabla }^{T^v\pi }\right)\equiv 1$. Hence, $\Omega \left(\mathcal{E}\right)=\mathrm{ch}\left({\nabla }^E\right)$.

The local index form $\Omega \left(\mathcal{E}\right)$ is closed and represents a cohomology class $[\Omega \left(\mathcal{E}\right)]\in H_{\mathrm{dR}}^{\mathrm{even}}\left(B\right)$. We also have a de Rham representative of the topological index ${\mathrm{ch}}_{\mathrm{dR}}\left(\mathtt{index}\left(\mathcal{E}\right)\right)$ where ${\mathrm{ch}}_{\mathrm{dR}}:K\left(B\right)\to H_{\mathrm{dR}}^{\mathrm{even}}\left(B\right)$ is the Chern character map. Then, we have a version of the index theorem for families (compare Atiyah and Singer [30] and Bunke–Schick [11] (p. 61, Theorem 2.9)).

Theorem 1. 

${\mathrm{ch}}_{\mathit{dR}}\left(\mathcal{i}\mathcal{n}\mathcal{d}\mathcal{e}\mathcal{x}\left(\mathcal{E}\right)\right)=[\Omega \left(\mathcal{E}\right)]$.

2.3.2. Chern–Simons and Bismut–Cheeger $\eta$-Forms

Let $\mathcal{E}=(E,h^E,{\nabla }^E,{\varphi }^E)$ be a ${\widehat{K}}_{\mathrm{FLK}}^0\left(X\right)$-generator where $E\to X$ is a $\mathbb{Z}/2\mathbb{Z}$-graded hermitian vector bundle. Furthermore, let $\pi :X\to B$ be a proper submersion with a Riemannian structure, i.e., a pair consisting of a vertical metric $g^{T^v\pi }$ and a horizontal distribution $T^h\pi$ on $TX$.

Suppose that the map $\pi$ is ${\mathrm{spin}}^c$-oriented in the sense that $T^v\pi$ has a ${\mathrm{spin}}^c$-structure with characteristic hermitian line bundle $L^v\pi \to X$. Let $S\left(T^v\pi \right)$ be the associated spinor bundle. Let $\mathcal{H}$ denote the vector bundle on B whose fiber ${\mathcal{H}}_b$ at $b\in B$ is the space of smooth sections of $(S\left(T^v\pi \right)\otimes E){|}_{X_b}$. Let ${\nabla }^{T^v\pi }$ be the connection on $T^v\pi$ defined by the orthogonal projection of the Levi–Civita connection ${\nabla }^{TX}$ on $T^v\pi$. Let ${\widehat{\nabla }}^{T^v\pi }$ be the connection induced by ${\nabla }^{T^v\pi }$ and ${\nabla }^{L^v\pi }$.

Let $\mathsf{D}$ be the Dirac-type operator acting on ${\mathcal{H}}_b$ and assume that $\ker \left(\mathsf{D}\right)$ forms a smooth vector bundle on B. There is an induced $L^2$-metric $h^{\ker \left(\mathsf{D}\right)}$ and a compatible projected connection ${\nabla }^{\ker \left(\mathsf{D}\right)}$. Note that we have the $\eta$-form of Bismut–Cheeger ${\eta }^{\mathrm{BC}}$ (see [26,27,28]), which is defined explicitly by [9] (p. 17, Equations (3)–(9)). The $\eta$-form satisfies the following transgression formula (compare [9] (p. 17, Equations (3)–(10)))

$$d{\eta }^{BC}={\pi }_{\ast }\left(\mathrm{ch}\left({\nabla }^E\right)\right)-\mathrm{ch}\left({\nabla }^{\ker \left(\mathsf{D}\right)}\right).$$

Here, the pushforward ${\pi }_{\ast }:{\Omega }^{\mathrm{even}}\left(X\right)\to {\Omega }^{-n}\left(B\right)$ is defined by

$${\pi }_{\ast }\left(\varphi \right):={\int }_{X/B}\mathrm{Todd}\left({\widehat{\nabla }}^{T^v\pi }\right)\wedge \varphi .$$

We can also define the differential K-theory pushforward ${\pi }_{!}:{\widehat{K}}_{\mathrm{FLK}}^0\left(X\right)\to {\widehat{K}}^{-n}\left(B\right)$ by ([9] (p. 18, Definition 3–11))

$${\pi }_{!}\left(\mathcal{E}\right):=(\ker \left(\mathsf{D}\right),h^{\ker \left(\mathsf{D}\right)},{\nabla }^{\ker \left(\mathsf{D}\right)},{\pi }_{\ast }\left(\varphi \right)+{\eta }^{\mathrm{BC}}).$$

We have the following relationship between the Bismut–Cheeger $\eta$-form and the Chern–Simons form. It appears in several places in the literature (see Bunke–Ma [31] (Section 3.4.4), Ma-Zhang [32] (Section 2.4), as well as the survey by Bismut [33] (Section 2)). It can be readily seen as an immediate corollary of the fact that the differential K-theory pushforward is functorial and thus takes the identity map $\pi :B\to B$ to the corresponding identity map on ${\widehat{K}}_{\mathrm{FLK}}^0\left(B\right)$ (Compare Freed–Lott [9] (p. 919, Equation (3).11.)).

Lemma 1. 

Let $\mathbf{E}=(E,h^E,{\nabla }^E,z^E)$ where $E\to B$ be a hermitian vector bundle with involution $z^E$ and $(\ker \left(\mathsf{D}\right),h^{\ker \left(\mathsf{D}\right)},{\nabla }^{\ker \left(\mathsf{D}\right)})\to B$ be the associated kernel bundle. Then, there is an isomorphism $\phi :E\to \ker \left(\mathsf{D}\right)$ covering the identity map on B such that

$$-\mathit{CS}({\nabla }^E,{\phi }^{\ast }{\nabla }^{\ker \left(\mathsf{D}\right)})={\eta }^{\mathrm{BC}}\mathit{mod}\mathit{Im}d.$$

(7)

Furthermore, the $\eta$-forms of Bunke and Bismut–Cheeger are also related. Given a geometric family $\mathcal{E}$ over B, recall that we can construct a canonical geometric family $\mathcal{K}$ which is the zero-dimensional geometric family for the kernel bundle $\ker \left(D\right(\mathcal{E}\left)\right)$ over B (see Example 2). Then, we have

Lemma 2 

([11] (p. 97, 3.2.6)). ${\eta }^{\mathit{BC}}\left(\mathcal{E}\right)=\eta \left({\left(\mathcal{E}{\bigsqcup }_B{\mathcal{K}}^{\mathsf{o}\mathsf{p}}\right)}_t\right)$$mod$$\mathit{Im}\left(d\right)$.

In other words, the kernel bundle associated with the geometric family together with a taming allows one to interchange the Bunke and Bismut–Cheeger η-forms. Using the definition of K-relation in ${\widehat{K}}_{\mathrm{BS}}\left(B\right)$ (see Definition 13), we can express the relationship between two η-forms in the level of cycles. Note that the following Collorary can be used to relate differential K-theoretic pushforwards à la Bunke–Schick [11] and Freed–Lott [9] (compare [20] (p. 338)).

Corollary 1 

([11] (p. 97, Corollary 5.5)). $[\mathcal{E},0]=[\mathcal{K},{\eta }^{\mathit{BC}}\left(\mathcal{E}\right)]$.

Also, the Bismut–Cheeger η-form shares some properties of the Bunke η-forms.

Corollary 2 

([24] (p. 58, Lemma 2.2.12)).

1. 

${\eta }^{\mathit{BC}}\left({\mathcal{E}}_t^{\mathsf{o}\mathsf{p}}\right)=-{\eta }^{\mathit{BC}}\left({\mathcal{E}}_t\right)$.

2. 

${\eta }^{\mathit{BC}}\left({\mathcal{E}}_t{\bigsqcup }_B{\mathcal{E}}_t^{\prime }\right)={\eta }^{\mathit{BC}}\left({\mathcal{E}}_t\right)+{\eta }^{\mathit{BC}}\left({\mathcal{E}}_t^{\prime }\right)$.

3. Main Results

We begin with a review of the Venice lemma of Simons and Sullivan [13] (Proposition 2.6) (Compare [23] and [34] (Corollary 2)).

Lemma 3 

(Venice lemma). Let B be a smooth manifold and $\varphi \in {\Omega }^{\mathit{odd}}\left(B\right)$. There exist a rank N product bundle ${\mathbb{C}}^N$ over B, connections ${\nabla }_1$ and ${\nabla }_2$ on ${\mathbb{C}}^N$, and a path of connections joining ${\nabla }_1$ and ${\nabla }_2$ such that

$$\varphi =\mathit{CS}({\nabla }_1,{\nabla }_2)\mathit{mod}\mathit{Im}d.$$

(8)

We state and prove the main result of the paper.

Theorem 2. 

Every element in ${\widehat{K}}_{\mathit{BS}}\left(B\right)$ can be written as $[\tilde{\mathcal{E}},0]$ for some geometric family $\tilde{\mathcal{E}}$.

Proof. 

By the Corollary 1, we can write any $[\mathcal{E},\rho ]\in {\widehat{K}}_{\mathrm{BS}}\left(B\right)$ as $[\mathcal{K},\varphi ]$, where $\varphi ={\eta }^{\mathrm{BC}}+\rho$ and $\mathcal{K}$ is the zero-dimensional geometric family of the kernel bundle $\mathbf{K}=(K,h^K,{\nabla }^K)$ associated with $\mathcal{E}$ (see Example 2). Let $(K,h^K,{\nabla }^K,\varphi )$ be a ${\widehat{K}}_{\mathrm{FLK}}^0\left(B\right)$-generator. Then, by Lemma 3, we have

$$\varphi =\mathrm{CS}({\nabla }_1,{\nabla }_2)\mathrm{mod}\mathrm{Im}d,$$

where ${\nabla }_1$ and ${\nabla }_2$ are connections on ${\mathbb{C}}^N$ for some N determined by the proof of Lemma 3. Note that we can identify ${\mathbb{C}}^N$ as a hermitian superbundle with the standard metric h and the identity involution $\mathrm{Id}$. Let $\mathsf{D}$ be the Dirac operator associated with ${\mathbb{C}}^N$ and $\ker \left(\mathsf{D}\right)$ be the kernel bundle with the connection ${\nabla }_0={\nabla }^{\ker \left(\mathsf{D}\right)}$. By [13] (p. 583, Proposition 1.1), we may assume that the path joining ${\nabla }_1$ and ${\nabla }_2$ passes through ${\nabla }_0$ in the middle,

$$\varphi =\mathrm{CS}({\nabla }_1,{\nabla }_0)-\mathrm{CS}({\nabla }_2,{\nabla }_0)\mathrm{mod}\mathrm{Im}d.$$

Then, we can write $\mathrm{CS}({\nabla }_1,{\nabla }_0)=-{\eta }^{\mathrm{BC}}\left({\mathcal{E}}_1\right)\mathrm{mod}\mathrm{Im}d$ and $\mathrm{CS}({\nabla }_2,{\nabla }_0)=-{\eta }^{\mathrm{BC}}\left({\mathcal{E}}_2\right)$ mod Im d by using the Lemma 1, where ${\mathcal{E}}_1$ and ${\mathcal{E}}_2$ are zero-dimensional geometric families corresponding to $({\mathbb{C}}^N,h,{\nabla }_1,\mathrm{Id})$ and $({\mathbb{C}}^N,h,{\nabla }_2,\mathrm{Id})$, respectively. From the properties of the η-form (Corollary 2), we have

$$\varphi ={\eta }^{\mathrm{BC}}\left({\mathcal{E}}_1^{\mathtt{o}\mathtt{p}}\right)+{\eta }^{\mathrm{BC}}\left({\mathcal{E}}_2\right)={\eta }^{\mathrm{BC}}({\mathcal{E}}_1^{\mathtt{o}\mathtt{p}}\bigsqcup {\mathcal{E}}_2)\mathrm{mod}\mathrm{Im}d.$$

Let $\mathcal{F}={\mathcal{E}}_1^{\mathtt{o}\mathtt{p}}\bigsqcup {\mathcal{E}}_2$. By Corollary 1 again,

$$[\mathcal{F},0]=[\mathcal{L},{\eta }^{\mathrm{BC}}\left(\mathcal{F}\right)],$$

where $\mathcal{L}$ is the geometric family associated with the kernel bundle of $\mathcal{F}$. Then

$$\begin{array}{cc}\hfill [\mathcal{K},{\eta }^{\mathrm{BC}}\left(\mathcal{F}\right)]=& [\mathcal{K}{\bigsqcup }_B\mathcal{L},0]-[\mathcal{L},-{\eta }^{\mathrm{BC}}\left(\mathcal{F}\right)]\hfill \\ \hfill =& [\mathcal{K}{\bigsqcup }_B\mathcal{L},0]+[{\mathcal{L}}^{\mathtt{o}\mathtt{p}},{\eta }^{\mathrm{BC}}\left(\mathcal{F}\right)]\hfill \\ \hfill =& [\mathcal{K}{\bigsqcup }_B\mathcal{L},0]+([{\mathcal{L}}^{\mathtt{o}\mathtt{p}},0]+[\mathcal{L},0])+[{\mathcal{L}}^{\mathtt{o}\mathtt{p}},{\eta }^{\mathrm{BC}}\left(\mathcal{F}\right)]\hfill \\ \hfill =& [\mathcal{K}{\bigsqcup }_B\mathcal{L},0]+2[{\mathcal{L}}^{\mathtt{o}\mathtt{p}},0]+[\mathcal{F},0]\hfill \\ \hfill =& [\mathcal{K}{\bigsqcup }_B{\mathcal{L}}^{\mathtt{o}\mathtt{p}}{\bigsqcup }_B\mathcal{F},0].\hfill \end{array}$$

Here, in the second and the third equalities, we have used the inverses of the Bunke–Schick differential K-group (See the paragraph following Definition 14). In other words, $[\mathcal{E},\varphi ]=[\tilde{\mathcal{E}},0]$ where $\tilde{\mathcal{E}}:=\mathcal{K}{\bigsqcup }_B{\mathcal{L}}^{\mathtt{o}\mathtt{p}}{\bigsqcup }_B\mathcal{F}$ and this completes the proof of the theorem. □

4. Conclusions

We have established that every element in the Bunke–Schick differential K-theory ${\widehat{K}}_{\mathrm{BS}}^0\left(B\right)$ can be represented by a cycle of the form $[\mathcal{E},0]$, where $\mathcal{E}$ is a geometric family (Theorem 2). This demonstrates that geometric families alone suffice in codifying differential K-theory without requiring explicit differential form data in the cycle representation. Our proof systematically converts differential forms into Bismut–Cheeger η-forms via the Venice lemma and realizes these as geometric families using the kernel bundle construction. This geometric characterization may facilitate future investigations into twisted and equivariant differential K-theory.