On the functoriality of Lott’s secondary analytic index

Ulrich Bunke^{∗}

November 27, 2001

Contents

1 Introduction 0

2 Secondary K-groups and analytic push-forward 1

3 The analytic torsion form of iterated fibre bundles 3

4 Verification of Theorem 1.1 5

1 Introduction

In [3] for a smooth manifold M J. Lott has defined a secondary K-group K̄_{R}^{0}(M) of local systems of R-modules, where R is some ring with a fixed representation ρ : R → End(ℂ^{n}). Given such a local system the corresponding element of K̄_{R}^{0}(M) encodes an explicit trivialization of the characteristic classes of the flat bundle obtained by complexifying the local system. If p : E → B is a smooth fibre bundle with closed fibres, then he defines a push-forward p_{!} : K̄_{R}^{0}(E) → K̄_{R}^{0}(B). This construction involves the real analytic torsion form of [1]. We will review Lott’s construction in Section 2.

In [4] X. Ma has studied the behavior of the real analytic torsion form under iterated fibre bundles. We review his result in Section 3.

2 SECONDARY K-GROUPS AND ANALYTIC PUSH-FORWARD 1

On the one hand in Lott’s paper [3] the question whether the push-forward behaves functorially with respect to iterated fibre bundles was left open. On the other hand, Ma’s work [4] provides the main and most complicated step towards a verification of this functoriality property. Unfortunately, this application is not discussed in [4]. The motivation for the present note is to fill this gap in the literature and to verify that Ma’s result indeed implies that the push-forward of Lott is functorial.

∗Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, GERMANY, Email:bunke@uni-math.gwdg.de

We now describe this assertion in detail. Let p : E → B be a fibre bundle with closed fibre Z which has the structure of an iterated fibre bundle. So we assume that there are fibre bundles p_{1} : E → E_{2} with fibre Z_{1} and p_{2} : E_{2} → B with fibre Z_{2} such that Z → Z_{2} is a fibre bundle with fibre Z_{1}. We will verify the following:

Theorem 1.1 We have p_{!} = (p_{2})_{!} ∘ (p_{1})_{!} as maps from K̄_{R}^{0}(E) to K̄_{R}^{0}(B).

The details of the proof are contained in Section 4.

Note that the second part of Lott’s paper [3] discusses another secondary index theory for flat duality bundles. Here instead of the real analytic torsion form the eta form plays an important role. The corresponding push-forward is not functorial with respect to iterated fibre bundles. This can be seen already on the level of the primary index theorem. The reason is that the equivalence relation which used to define the L- and L̄-groups is too small. Together with X. Ma in [2] we modify Lott’s definition such that we obtain a functorial push-forward in this case, too.

2 Secondary K-groups and analytic push-forward

This section reviews Sections 2.1 and 2.2 of [3]. Suppose that R is a right-Noetherian ring which is right-regular, i.e. every finitely generated right-R-module has a finite resolution by finitely generated projective right-R-modules. We fix a representation ρ : R → End(ℂ^{n}) such that ℂ^{n} becomes a flat left R-module. If V is a right-R-module, then V_{ℂ} := V ⊗_{R,ρ} ℂ^{n} is called its complexification.

Let M be a connected smooth manifold. If F is a local system (a locally constant sheaf) of finitely generated right-R-modules, then F_{ℂ} (the fiberwise complexification) is the sheaf of parallel sections of a flat complex vector bundle which we denote by (F_{ℂ}, ∇^{F_{ℂ}}).

Let (E, ∇^{E}) be a flat complex vector bundle. If we choose a hermitian metric h^{E} (we consider h^{E} as a section of the flat bundle Hom_{ℂ}(E, E^{∗})) on E, then we can define the characteristic forms

ω(∇^{E}, h^{E}) := (h^{E})^{−1}∇^{Hom(E,E^{∗})}h^{E}
c_{k}(∇^{E}, h^{E}) = (2πı)^{− \frac{k−1}{2}} 2^{−k}Tr ω(∇^{E}, h^{E})^{k}
c(∇^{E}, h^{E}) = ∑_{j=0}^{∞} \frac{1}{j!}c_{2j+1}(∇^{E}, h^{E}) .

The form c(∇^{E}, h^{E}) is closed and represents the characteristic class c(∇^{E}) ∈ H_{dR}^{odd}(M) of the flat vector bundle (E, ∇^{E}) which is independent of the choice of h^{E}.

The Abelian group K̂_{R}^{0}(M) is generated by triples f = (F, h^{F_{ℂ}}, η), where

1. F is a local system of finitely generated right-R-modules,

2. h^{F_{ℂ}} is a hermitian metric of the corresponding flat complex vector bundle (F_{ℂ}, ∇^{F_{ℂ}}), and

3. η ∈ Ω^{ev}(M)/image(d),

subject to the following relations : If

0 → F_{1} → F_{2} → F_{3} → 0

is an exact sequence of local systems of finitely generated right-R-modules, h^{(F_{i})_{ℂ}} are hermitian metrics, η_{i} ∈ Ω^{ev}(M)/image(d), and we form f_{i} := (F_{i}, h^{(F_{i})_{ℂ}}, η_{i}), then f_{2} ∼ f_{1} + f_{3} if

η_{2} = η_{1} + η_{3} + T (C, h^{C}) ,

where T (C, h^{C}) is the analytic torsion form associated to the exact complex of flat complex vector bundles (C, h^{C})

0 → (F_{1})_{ℂ} → (F_{2})_{ℂ} → (F_{3})_{ℂ} → 0

equipped with the metric h^{C} induced by h^{(F_{i})_{ℂ}}. For the details of the definition of the torsion form T (C, h^{C}) we refer to [1], Sec. 2(f), or [3], A.3. The appearance of the torsion form in the equivalence relation is explained by the relation

dT (C, h^{C}) = ∑_{i=1}^{3}(−1)^{i}c((F_{i})_{ℂ}, h^{(F_{i})_{ℂ}}) .

Lott shows that

f ↦ c(∇^{F_{ℂ}}, h^{F_{ℂ}}) − dη

extends to a map c′ : K̂_{R}^{0}(M) → Ω^{odd}(M), and he defines

K̄_{R}^{0}(M) := ker(c′) .

The assignment M ↦ K̄_{R}^{0}(M) yields a homotopy invariant contravariant functor from the category of smooth manifolds to Abelian groups.

We now consider a smooth fibre bundle p : E → B with compact fibre Z. If F is a locally constant sheaf of finitely generated right-R-modules, then we can form the sheaves R^{i}p_{∗}F on B (the higher derived images of F) which are again locally constant sheaves of finitely generated right-R-modules.

If we choose a fiberwise Riemannian metric g^{Z} (i.e. a metric on the vertical bundle T Z := ker(dp) ), then we can compute (R^{i}p_{∗}F)_{ℂ} = R^{i}p_{∗}(F_{ℂ}) (this equality holds because ℂ^{n} is a flat R-module) using the fiberwise de Rham complex twisted with F_{ℂ}. The metric g^{Z} and a hermitian metric h^{F_{ℂ}} induce L^{2}-scalar products. Identifying the stalk (R^{i}p_{∗}(F)_{ℂ})_{b} (which is the fibre of the flat complex vector bundle (R^{i}p_{∗}F)_{ℂ}) with harmonic forms we obtain metrics h^{(R^{i}p_{∗}F)_{ℂ}}.

We further choose a horizontal distribution T^{H}E (i.e. a complementary bundle to T Z such that T Z ⊕ T^{H}E = T E). It induces a connection ∇^{T Z} on the vertical bundle. Let e(T Z, ∇^{T Z}) be the associated Euler form. We refer to [1] for the definition of the real analytic torsion form T (T^{H}E, g^{Z}, h^{F_{ℂ}}) ∈ Ω^{ev}(B).

In [3] Lott defines the push-forward p_{!} : K̄_{R}^{0}(M) → K̄_{R}^{0}(B) by the assignment:

(F, h^{F_{ℂ}}, η) ↦ ∑_{i}(−1)^{i}(R^{i}p_{∗}(F), h^{(R^{i}p_{∗}F)_{ℂ}}, 0) + (0, 0, ∫_{Z} e(T Z, ∇^{T Z}) ∧ η − T (T^{H}E, g^{Z}, h^{F_{ℂ}})) ,

where ∫_{Z} means integration over the fibre of p : E → B. Lott proves that p_{!} is well-defined and independent of the choices of T^{H}E and g^{Z}.

3 The analytic torsion form of iterated fibre bundles

In this section we review the result of [4]. Let p : E → B be a fibre bundle with fibre Z which has the structure of an iterated fibre bundle. We assume that there are fibre bundles p_{1} : E → E_{2} with fibre Z_{1} and p_{2} : E_{2} → B with fibre Z_{2} such that Z → Z_{2} is a fibre bundle with fibre Z_{1}. For notational purposes we shall sometimes write E_{1} := E when we consider E as the total space of the bundle p_{1} : E → E_{2}. By T Z, T Z_{1}, and T Z_{2}, we denote the corresponding vertical bundles. We choose vertical Riemannian metrics g^{Z}, g^{Z_{1}}, g^{Z_{2}}.

Furthermore, we choose horizontal bundles T^{H}E, T^{H}E_{1}, and T^{H}E_{2}, for p, p_{1}, p_{2}. We obtain connections ∇^{T Z}, ∇^{T Z_{1}}, and ∇^{T Z_{2}}. We identify T Z with T Z_{1} ⊕ p_{1}^{∗}T Z_{2} (using T^{H}E_{1}) and obtain another connection ^{0}∇^{T Z} := ∇^{T Z_{1}} ⊕ p_{1}^{∗}∇^{T Z_{2}} on T Z. By ẽ(T Z, ∇^{T Z},^{0}∇^{T Z}) we denote the corresponding transgression of the Euler form such that

dẽ(T Z, ∇^{T Z},^{0}∇^{T Z}) = e(T Z,^{0}∇^{T Z}) − e(T Z, ∇^{T Z}) .

The main result of X. Ma is the following formula

T (T^{H}E, g^{Z}, h^{F_{ℂ}}) − ∫_{Z_{2}} e(T Z_{2}, ∇^{T Z_{2}}) ∧ T (T^{H}E_{1}, g^{Z_{1}}, h^{F})

− ∑_{i}(−1)^{i}T (T^{H}E_{2}, g^{Z_{2}}, h^{(R^{i}(p_{1})_{∗}F)_{ℂ}})

+ ∫_{Z} ẽ(T Z, ∇^{T Z},^{0}∇^{T Z}) ∧ c(∇^{F_{ℂ}}, h^{F_{ℂ}})
= S (mod image(d)) ,

where S is a higher analytic torsion invariant associated to the Leray spectral sequence of the family of fibre bundles (Z_{b} → (Z_{2})_{b})_{b∈B}.

We now describe S in detail. The spectral sequence (E_{r}, d_{r}) is associated to the composition (p_{2})_{∗} ∘ (p_{1})_{∗}. Its second term is E_{2}^{p,q} = R^{p}(p_{2})_{∗}R^{q}(p_{1})_{∗}(F), and it converges to GrR^{p+q}p_{∗}(F). Since ℂ^{n} is a flat R-module complexification commutes with taking the spectral sequence. In particular, we obtain flat complex vector bundles (E_{r}^{p,q})_{ℂ}. The differentials d_{r} of the spectral sequence induce corresponding bundle homomorphisms such that we obtain complexes of flat complex vector bundles

… →^{d_{r}} (E_{r}^{p,q})_{ℂ} →^{d_{r}} (E_{r}^{p+r,q+1−r})_{ℂ} →^{d_{r}} …

with cohomology (E_{r+1}^{p,q})_{ℂ}. The metric h^{(R^{q}(p_{1})_{∗}F)_{ℂ}} induces metrics h^{(E_{2})_{ℂ}^{p,q}}. Now we obtain inductively metrics on the cohomology groups h^{(E_{r+1}^{p,q})_{ℂ}}. In order to save notation we introduce the complexes of flat bundles D_{r} for r ≥ 2 as follows. The degree k-component of D_{r} is given by D_{r}^{k} := ⊕_{p+q=k}(E_{r}^{p,q})_{ℂ}. The family of differentials d_{r} induce a differential d_{D_{r}} : D_{r}^{k} → D_{r}^{k+1}. By h^{D_{r}} we denote the metric induced by the family of metrics h^{(E_{r})_{ℂ}^{p,q}}.

Let

V : … →^{d_{i−1}} V_{i} →^{d_{i}} V_{i+1} →^{d_{i+1}} …

be a finite complex of flat complex vector bundles equipped with hermitian metrics h^{V_{i}} (we write h^{V} for the whole collection). We further choose hermitian metrics on the flat cohomology bundles h^{H^{i}} (and we again write h^{H} for this collection). We form the short exact sequences

C_{i} : 0 → ker(d_{i}) → V_{i} → image(d_{i}) → 0
L_{i} : 0 → image(d_{i−1}) → ker(d_{i}) → H^{i} → 0 ,

where all spaces have induced hermitian metrics h^{C_{i}}, h^{L_{i}}. We define

T (V, h^{V}, h^{H}) := ∑_{i}(−1)^{i} (T (C_{i}, h^{C_{i}}) + T (L_{i}, h^{L_{i}})) ,

where T (C_{i}, h^{C_{i}}) and T (L_{i}, h^{L_{i}}) denote again the torsion forms of complexes of vector bundles defined in [1], Sec. 2(f).

If V is a flat complex vector bundle with a filtration 0 ⊂ V_{0} ⊂ V_{1} ⊂ … ⊂ V_{n} = V by flat sub bundles, then we consider the short exact sequences

A_{i} : 0 → V_{i} → V_{i+1} → Gr_{i+1}(V) := V_{i+1}/V_{i} → 0 .

If we further choose hermitian metrics h^{V} (inducing h^{V_{i}} by restriction) and h^{Gr_{i}(V)}, then we can define metrics h^{A_{i}} and

T (V, Gr(V), h^{V} , h^{Gr(V)}) := ∑_{i} T (A_{i}, h^{A_{i}}) .

We can now define

S := ∑_{r=2}^{∞} T (D_{r}, h^{D_{r}}, h^{D_{r+1}}) − ∑_{k=0}^{∞}(−1)^{k}T ((R^{k}p_{∗}F)_{ℂ}, ⊕_{p+q=k}(E_{∞}^{p,q})_{ℂ}, h^{(R^{k}p_{∗}F)_{ℂ}}, h^{⊕_{p+q=k}(E_{∞}^{p,q})_{ℂ}}) .

In the last term we use the natural identification Gr_{k−p}(R^{k}p_{∗}F)_{ℂ} →^{≅} (E_{∞}^{p,q})_{ℂ}.

4 Verification of Theorem 1.1

Let x ∈ K̄_{R}^{0}(E) be represented by ∑_{α} ϵ_{α}f_{α}, where ϵ_{α} ∈ ℤ, f_{α} := (F_{α}, h^{F_{α,ℂ}}, η_{α}), and ∑_{α} ϵ_{α}dη_{α} = ∑_{α} ϵ_{α}c(∇^{F_{α,ℂ}}, h^{F_{α,ℂ}}). Let us write out a representative of (p_{1})_{!}(x). We obtain

∑_{α} ϵ_{α} (∑_{q}(−1)^{q}(R^{q}(p_{1})_{∗}F_{α}, h^{(R^{q}(p_{1})_{∗}F_{α,ℂ}}, 0)

+ (0, 0, ∫_{Z_{1}} e(T Z_{1}, ∇^{T Z_{1}}) ∧ η_{α} − T (T^{H}E_{1}, g^{Z_{1}}, h^{F_{α,ℂ}}))) .

A representative of (p_{2})_{!} ∘ (p_{1})_{!}(x) is given by

∑_{α} ϵ_{α} (∑_{p,q}(−1)^{p+q}(R^{p}(p_{2})_{∗}R^{q}(p_{1})_{∗}F_{α}, h^{(E_{α,2})_{ℂ}^{p,q}}, 0)

+(0, 0, ∫_{Z_{2}} e(T Z_{2}, ∇^{T Z_{2}}) ∧ (∫_{Z_{1}} e(T Z_{1}, ∇^{T Z_{1}}) ∧ η_{α} − T (T^{H}E_{1}, g^{Z_{1}}, h^{F_{α,ℂ}})))

− ∑_{q}(−1)^{q}(0, 0, T (T^{H}E_{2}, g^{Z_{2}}, h^{(R^{q}(p_{1})_{∗}F_{α})_{ℂ}}))) .

The index α attached to (E_{α,2})_{ℂ}^{p,q} indicates that this flat vector bundle is associated to F_{α} (see Section 3). We must show that this expression represents the same element as

∑_{α} ϵ_{α} (∑_{i}(−1)^{i}(R^{i}p_{∗}F_{α}, h^{(R^{i}p_{∗}F_{α})_{ℂ}}, 0) + (0, 0, ∫_{Z} e(T Z, ∇^{T Z}) ∧ η_{α} − T (T^{H}E, g^{Z}, h^{F_{α,ℂ}}))) .

We first compare the terms involving the forms η_{α}. Indeed we have

∑_{α} ϵ_{α} (∫_{Z_{2}} e(T Z_{2}, ∇^{T Z_{2}}) ∧ (∫_{Z_{1}} e(T Z_{1}, ∇^{T Z_{1}}) ∧ η_{α}) − ∫_{Z} e(T Z, ∇^{T Z}) ∧ η_{α})

= ∑_{α} ϵ_{α} ∫_{Z} dẽ(T Z, ∇^{T Z},^{0}∇^{T Z}) ∧ η_{α}

= ∑_{α} ϵ_{α} ∫_{Z} ẽ(T Z, ∇^{T Z},^{0}∇^{T Z}) ∧ dη_{α}
= ∑_{α} ϵ_{α} ∫_{Z} ẽ(T Z, ∇^{T Z},^{0}∇^{T Z}) ∧ c(∇^{F_{α,ℂ}}, h^{F_{α,ℂ}}) (mod image(d)) .

Thus it remains to show that

∑_{α} ϵ_{α} (∑_{p,q}(−1)^{p+q}(R^{p}(p_{2})_{∗}R^{q}(p_{1})_{∗}F_{α}, h^{(E_{α,2})_{ℂ}^{p,q}}, 0)

−(0, 0, ∫_{Z_{2}} e(T Z_{2}, ∇^{T Z_{2}}) ∧ T (T^{H}E_{1}, g^{Z_{1}}, h^{F_{α,ℂ}}))

− ∑_{q}(−1)^{q}(0, 0, T (T^{H}E_{2}, g^{Z_{2}}, h^{(R^{q}(p_{1})_{∗}F_{α})_{ℂ}}))

− ∑_{p}(−1)^{p}(R^{p}p_{∗}(F_{α}), h^{(R^{p}p_{∗}F_{α})_{ℂ}}, 0)

+(0, 0, T (T^{H}E, g^{Z}, h^{F_{α,ℂ}})

+ ∫_{Z} ẽ(T Z, ∇^{T Z},^{0}∇^{T Z}) ∧ c(∇^{F_{α,ℂ}}, h^{F_{α,ℂ}})))

represents the trivial element in K̂_{R}^{0}(B).

Let

V : … →^{d_{i−1}} V_{i} →^{d_{i}} V_{i+1} →^{d_{i+1}} …

be a finite complex of local systems of finitely generated right-R-modules over B. We fix hermitian metrics h^{(V_{i})_{ℂ}} (we write h^{V_{ℂ}} for the whole collection). Since ℂ^{n} is a flat R-module we can interchange the operation of complexification and of taking fiberwise cohomology. We let H^{i} denote the flat complex vector bundle obtained from the complexification of the cohomology sheaves H^{i}(V). We further choose hermitian metrics h^{H^{i}} (and we write h^{H} for this collection). We consider the short exact sequences

C_{i} : 0 → ker(d_{i}) → V_{i} → image(d_{i}) → 0
L_{i} : 0 → image(d_{i−1}) → ker(d_{i}) → H^{i}(V) → 0 ,

where the corresponding complexes of flat complex vector bundles (C_{i})_{ℂ}, (L_{i})_{ℂ} have induced hermitian metrics h^{(C_{i})_{ℂ}}, h^{(L_{i})_{ℂ}}. In K̂_{R}^{0}(B) we have

∑_{i}(−1)^{i}(V_{i}, h^{(V_{i})_{ℂ}}, 0) = ∑_{i}(−1)^{i}(H^{i}(V), h^{H^{i}}, −T (C_{i}, h^{(C_{i})_{ℂ}}) − T (L_{i}, h^{(L_{i})_{ℂ}}))
= ∑_{i}(−1)^{i}(H^{i}(V), h^{H^{i}}, 0) − (0, 0, T ((V)_{ℂ}, h^{(V)_{ℂ}}, h^{H_{ℂ}})) .

Using this we compute in K̂_{R}^{0}(B)

∑_{p,q}(−1)^{p+q}(R^{p}(p_{2})_{∗}R^{q}(p_{1})_{∗}F_{α}, h^{(E_{α,2})_{ℂ}^{p,q}}, 0)

= ∑_{p,q}(−1)^{p+q}(E_{α,3}^{p,q}, h^{(E_{α,3})_{ℂ}^{p,q}}, 0) − (0, 0, T (D_{α,2}, h^{D_{α,2}}, h^{D_{α,3}}))
= …
= ∑_{p,q}(−1)^{p+q}(E_{α,∞}^{p,q}, h^{(E_{α,∞})_{ℂ}^{p,q}}, 0) − ∑_{r=2}^{∞}(0, 0, T (D_{α,r}, h^{D_{α,r}}, h^{D_{α,r+1}})) .

Let now V be a local system of finitely generated right-R-modules which is filtered by local systems of submodules 0 ⊂ V_{0} ⊂ V_{1} ⊂ … ⊂ V_{n} = V. We fix a hermitian metric h^{V_{ℂ}} which induces metrics h^{(V_{i})_{ℂ}}. Furthermore, we fix metrics h^{Gr_{i}(V)_{ℂ}}. In K̂_{R}^{0}(B) we have

(V, h^{V_{ℂ}}, 0) = ∑_{i}(Gr_{i}(V), g^{Gr_{i}(V)_{ℂ}}, 0) − (0, 0, T (V_{ℂ}, Gr(V)_{ℂ}, h^{V_{ℂ}}, h^{Gr(V)_{ℂ}})) .

Using this observation we further compute

∑_{p+q=k}(−1)^{k}(E_{∞}^{p,q}, h^{(E_{α,∞})_{ℂ}^{p,q}}, 0)

= ∑_{i}(Gr_{i}(R^{k}p_{∗}F_{α}), h^{Gr_{i}(R^{k}p_{∗}F_{α})_{ℂ}}, 0)
= (R^{k}p_{∗}F_{α}, h^{(R^{k}p_{∗}F_{α})_{ℂ}}, 0) + (0, 0, T ((R^{k}p_{∗}F_{α})_{ℂ}, ⊕_{p+q=k}(E_{α,∞}^{p,q})_{ℂ}, h^{(R^{k}p_{∗}F_{α})_{ℂ}}, h^{Gr(R^{k}p_{∗}F_{α})_{ℂ}})) .

Thus it remains to show that

∑_{α} ϵ_{α} (− ∑_{r=2}^{∞} T (D_{α,r}, h^{D_{α,r}}, h^{D_{α,r+1}})

+ ∑_{k=0}^{∞}(−1)^{k}T ((R^{k}p_{∗}F_{α})_{ℂ}, ⊕_{p+q=k}(E_{α,∞}^{p,q})_{ℂ}, h^{(R^{k}p_{∗}F_{α})_{ℂ}}, h^{Gr(R^{k}p_{∗}F_{α})_{ℂ}})

− ∫_{Z_{2}} e(T Z_{2}, ∇^{T Z_{2}}) ∧ T (T^{H}E_{1}, g^{Z_{1}}, h^{F_{α,ℂ}})

− ∑_{p}(−1)^{p}T (T^{H}E_{2}, g^{Z_{2}}, h^{(R^{p}(p_{1})_{∗}F_{α})_{ℂ}})

+T (T^{H}E, g^{Z}, h^{F_{α,ℂ}})

+ ∫_{Z} ẽ(T Z, ∇^{T Z},^{0}∇^{T Z}) ∧ c(∇^{F_{α,ℂ}}, h^{F_{α,ℂ}}))

is an exact form. But it is exactly the result of X. Ma that for all α the corresponding summand is eaxt. This finishes the verification. □

References

[1] J. M. Bismut and J. Lott. Flat vector bundles, direct images, and higher real analytic torsion. Journal f the AMS, 8(1995), 291–363.

[2] U. Bunke and X. Ma. Index and secondary index theory for flat bundles with duality. Preprint math.DG/0106068.

[3] J. Lott. Secondary analytic indices. Regulators in analysis, geometry and number theory. Birkhäuser, Progr. Math., 171, 231-293, 2000.

[4] X. Ma. Functoriality of real analytic torsion form. IHES-Preprint M/99/03, 1999. Submitted.