Relative differential K-characters

We define a group of relative differential K-characters associated with a smooth map between two smooth compact manifolds. We show that this group fits into a short exact sequence as in the non-relative case. Some secondary geometric invariants are expressed in this theory.


Introduction
Cheeger and Simons [10] introduced the notion of differential characters to express some secondary geometric invariants of a principal G-bundle in the base space. This theory has been appearing more and more frequently in quantum field and string theories (see [7,15,13]). On the other hand, it was shown recently (see [4,16,17]) that K-homology of Baum-Douglas [5] is an appropriate arena in which various aspects of D-branes in superstring theory can be described.
In [8] we have defined with M.T. Benameur the notion of differential characters in K-theory on a smooth compact manifold. Our original motivation was to explain some secondary geometric invariants coming from the Chern-Weil and Cheeger-Simons theory in the language of K-theory. To do this, we have used the Baum-Douglas construction of K-homology. As a result, we obtained the eta invariant of Atiyah-Patodi-Singer as a R/Z-differential K-character, while it is a R/Q-invariant in the works of Cheeger and Simons. Recall that a geometric K-cycle of Baum-Douglas over a smooth compact manifold X is a triple (M, E, φ) such that: M is a closed smooth Spin c -compact manifold with a fixed Riemannian structure; E is a Hermitian vector bundle over M with a fixed Hermitian connection ∇ E and φ : M → X is a smooth map. Let C * (X) be the semi-group for the disjoint union of equivalence classes of K-cycles over X generated by direct sum and vector bundle modification [5]. A differential K-character on X is a homomorphism of semi-group ϕ : C * (X) → R/Z such that it is restriction to the boundary is given by the formula where ω is a closed form on X with integer K-periods [8], Ch(E) is the Chern form of the connection ∇ E and Td(M ) is the Todd form of the tangent bundle of M . This can be assembled into a group which is denoted byK * (X) and called the group of differential K-characters. We showed then that many secondary invariants can be expressed as a differential K-characters, and the group K * (X, R/Z) of K-theory of X with coefficients in R/Z [2] is injected inK * (X).
The aim of this work is to define the groupK * (ρ) of relative differential K-characters associated with a smooth map ρ : Y → X between two smooth compact manifolds Y and X following [9,12] and [13]. We show that this group fits into a short exact sequence as in the non-relative case. The paper is organized as follows: In Section 2, we define a group of relative geometric K-homology K * (ρ) adapted to this situation and study some of its properties. This generalizes the works of Baum-Douglas [6] for Y a submanifold of X. Section 3 is concerned with the definition and the study of the group K * (ρ) of relative differential K-characters. An odd relative group K −1 (ρ, R/Z) of K-theory with coefficients in R/Z is also defined here. We proof the following short exact sequence where Ω even 0 (ρ) is the group of relative differential forms (Definition 6) with integer K-periods. We show then that some secondary geometric invariants can be expressed in this theory.

Relative geometric K-homology
Let Y and X be smooth compact manifolds and ρ : Y → X a smooth map. In this section, we define the relative geometric K-homology K * (ρ) for the triple (ρ, Y, X). This construction generalizes the relative geometric K-homology group K * (X, Y ) of Baum-Douglas for Y being a closed submanifold of X. We recall the definition of the geometric K-homology of a smooth manifold following the works of Baum and Douglas. This definition is purely geometric. For a complete presentation see [5,6] and [17].
• M is a smooth Spin c -compact manifold which may have non-empty boundary ∂M , and with a fixed Riemannian structure; • E is a Hermitian vector bundle over M with a fixed Hermitian connection ∇ E ; • φ : M → X is a smooth map.
Denote that M is not supposed connected and the fibres of E may have different dimensions on the different connected components of M . Two K-chains (M, E, φ) and (M ′ , E ′ , φ ′ ) are said to be isomorphic if there exists a diffeomorphism ψ : M → M ′ such that: • ψ * E ′ ∼ = E as Hermitian bundles over M .
A K-cycle is a K-chain (M, E, ϕ) without boundary; that is ∂M = ∅. The boundary ∂(M, E, ϕ) of the K-chain (M, E, ϕ) is the K-cycle (∂M, E| ∂M , ϕ| ∂M ). The set of K-chains is stable under disjoint union. The vector bundleĤ onM is obtained by putting H 0 on B 0 (H) and H 1 on B 1 (H) and then clutching these two vector bundles along S(H) by the isomorphism σ.

Vector bundle modification
The K-chain (M ,Ĥ ⊗ ρ * E,φ = ρ • φ) is called the Bott K-chain associated with the K-chain (M, E, φ) and the Spin c -vector bundle H.
The boundary of the Bott K-chain (M ,Ĥ ⊗ρ * E,φ) associated with the K-chain (M, E, φ) and the Spin c -vector bundle H is the Bott K-cycle of the boundary ∂(M, E, φ) with the restriction of H to ∂M . Definition 2. We denote by C * (X) the set of equivalence classes of isomorphic K-cycles over X up to the following identifications: any Hermitian vector bundle H over M .
We can easily show that disjoint union then respects these identifications and makes C * (X) into an Abelian semi-group which splits into C 0 (X) ⊕ C 1 (X) with respect to the parity of the connected components of the manifolds in (the equivalence classes of) the K-cycles.
The above bordism relation induces a well defined equivalence relation on C * (X) that we denote by ∼ ∂ . The quotient C * (X)/ ∼ ∂ turns out to be an Abelian group for the disjoint union.
Definition 4 (Baum-Douglas). The quotient group of C * (X) by the equivalence relation ∼ ∂ is denoted by K * (X) and is called the geometric K-homology group of X. It can be decomposed into The K * is a covariant functor from the category of smooth compact manifolds and smooth maps to that of Abelian groups and group homomorphisms.
In the same way we can form a semi-group L * (X) out of K-chains (N , E, ψ), say with the same definition as C * (X) and the boundary where i : ∂N ֒→ N . This gives a well defined map The Hermitian structure of the complex vector bundle E| ∂N is the restricted one. The group of K-cochains with coefficients in Z denoted by L * (X) is the group of semi-group homomorphisms f from L * (X) to Z. On the group L * (X) there is a coboundary map defined by transposition The set of K-cocycles is the subset C * (X) of L * (X) of those K-cochains that vanish on boundaries, i.e. the kernel of δ. The set of K-coboundaries is the image of δ in L * (X).

The relative geometric group K * (ρ)
Let Y and X be smooth compact manifolds and ρ : Y → X a smooth map.
The set L * (ρ) of relative K-chains associated with the triple (ρ, Y, X) is by definition The boundary ∂ : L * +1 (ρ) → L * (ρ) is given by We will denote by C * (ρ) the set of relative K-cycles in L * (ρ), i.e., the kernel of ∂.
where ∐ is the disjoint union. We say that two relatives K-cycles(σ, τ ) and (σ ′ , τ ′ ) are bordant and we write (σ, where −x denotes the relative K-cycle x with the reversed Spin c -structure of the underlying manifold. Definition 5. The relative geometric K-homology group denoted by K * (ρ) is the quotient group The inverse of the K-cycle x is −x. The equivalence relation on the relative K-cycle (σ, τ ) preserves the dimension modulo 2 of the K-cycles σ and τ . Hence, there is a direct sum decomposition The construction of the group K * (ρ) is functorial in the sense that for a commutative diagram is compatible with the equivalence relation on the relative K-cycles and induces a homomorphism from K * (ρ) to K * (ρ ′ ). As in the homology theory, we have the long exact sequence for the triple (ρ, Y, X) The boundary map ∂ associates to a relative K-cycle (σ, τ ) the cycle τ whose image ρ * τ is a boundary in X and ς * (σ) = (σ, 0). The exactness of the diagram is an easy check.
Remark 1. The relative topological K-homology group K t * (ρ) can be constructed in the same way for normal topological spaces X and Y , and ρ : Y → X is a continuous map. Let K t * (X, Y ) be the relative topological K-homology group defined by Baum-Douglas in [6] for Y ⊂ X is a closed subset of a X. We can easily show that K t * (X, Y ) = K t * (ρ), where ρ is the inclusion of Y in X.

Relative differential K-characters
This section is concerned with the definition and the study the notion of relative differential K-characters [8]. This is a K-theoretical version of the works of [9,12] and [13].
Let X be a smooth compact manifold. The graded differential complex of real differential forms on the manifold X will be denoted by where d denotes the de Rham differential on X. Furthermore, we denote by Ω * 0 (X) the subgroup of closed forms on the manifold X with integer K-periods [8].
In the remainder of this section we fix ρ : Y → X a smooth map and we consider the complex We can view Ω * (ρ) as a subgroup of the the group Hom(L * (ρ), R) via integration such that j(ω, θ)(σ, τ ) = ω(σ) + θ(τ ).
(i) The set of K-periods of (ω, θ) is the subset of R image of the map j(ω, θ) restricted to C * (ρ).
Ω * 0 (ρ) is an Abelian group for the sum of differential forms.
Let f be a relative differential K-character for the smooth map ρ : Y → X. We deduce from Remark 2 that the pair of forms (ω, θ) associated to f in Definition 7 is unique. It will be denoted by δ 1 (f ). We thus have a group morphism , which is odd for the grading.

Relative R/Z-K-theory
Let X be a smooth manifold, E a Hermitian vector bundle on X and ∇ E a Hermitian connection on E. The geometric Chern form Ch(∇ E ) of ∇ E is the closed real even differential form given by The cohomology class of Ch(∇ E ) does not depend on the choice of the connection ∇ E [14]. Let ∇ E 1 and ∇ E 2 be two Hermitian connections on E. There is a well defined Chern-Simons form [14] Given a short exact sequence of complex Hermitian vector bundles on X and choose a splitting map s : E 3 → E 2 . Then i⊕ s : E 1 ⊕ E 3 → E 2 is an isomorphism. For ∇ E 1 , ∇ E 2 and ∇ E 3 are Hermitian connection on E 1 , E 2 and E 3 respectively, we set The form CS(∇ E 1 , ∇ E 2 , ∇ E 3 ) is independent of the choice of the splitting map s and Definition 8. Let X be a smooth manifold. A R/Z-K-generator of X is a quadruple where E is a complex vector bundle on X, h E is a positive definite Hermitian metric on E, ∇ E is a Hermitian connection on E, ω ∈ Ω odd (X) Im(d) which satisfies dω = Ch(∇ E ) − rank(E), where rank(E) is the rank of E.
An R/Z-K-relation is given by three R/Z-K-generators E 1 , E 2 , E 3 , along with a short exact sequence of Hermitian vector bundles Definition 9 ( [14]). We denote by M K(X, R/Z) the quotient of the free group generated by the R/Z-K-generators and R/Z-K-relation E 2 = E 1 + E 3 . The group K −1 (X, R/Z) is the subgroup of M K(X, R/Z) consisting of elements of virtual rank zero.
The elements of K −1 (X, R/Z) can be described by Z/2Z-graded cocycles [14], meaning quad- Im(d) and satisfies dω = Ch(∇ E ) = Ch(∇ E + ) − Ch(∇ E − ). We consider now two smooth compact manifolds Y and X. Let ρ : Y → X be a smooth map and let the exact sequence obtained from the one in p. 4 and after identification of the groups K * (Y, R/Z) and Hom(K * (Y ), R/Z) following Proposition 4 of [8].

Application
Let G be an almost connected Lie group and M be a smooth compact manifold. Let π : Y → M be a compact principal G-bundle with connection ∇. We denote by I * (G) the ring of symmetric multilinear real functions on the Lie algebra of G which are invariant under the adjoint action of G [11]. Let Ω be the curvature of ∇. For any P ∈ I * (G), there is a well defined closed form P (Ω) on M . The pullback π * P (Ω) is an exact form on Y [11]. For P ∈ I * (G), let T P (∇) be such that π * P (Ω) = dT P (∇). The form ω = (π * P (Ω), dT P (∇)) is a closed form in the complex (Ω * (π), δ). The relative differential K-character f ω has a trivial δ 1 and defines consequently an element of the group K −1 (π, R/Z). This gives an additive map from I * (G) to K −1 (π, R/Z) which can be looked as a home of secondary geometric invariants of the principal G-bundle with connection (M, Y, ∇) analogous to the Chern-Weil theory.