1.1. INTRODUCTION 7

This definition has the drawback of being not natural with respect to pull-back.

In fact, if f : B → B is a smooth map, then in general

f∗indexDel,Q(Egeom,0)

=

indexDel,Q(f∗Egeom,0)

since equality would hold by an accidental choice.

It is one of the main objectives of the paper [23] to show how this approach

can be modified in order to define a natural indexDel,Q(Egeom) ∈ HDel(B, ∗ Q).

1.1.4. Integral secondary invariants.

1.1.4.1. The group K(B) admits a natural decreasing filtration

··· ⊆ Kn(B) ⊆ Kn−1 ⊆ ··· ⊆ K0(B) = K(B) .

By definition x ∈ Kn(B), if

f∗x

= 0 for all n − 1-dimensional CW -complexes A

and continuous maps f : A → B. In the present paper we index the Chern classes

by their degrees. The odd-degree classes are related to the even degree ones by

suspension (see 3.1.4.1).

Let k, m ∈ N be such that k = 2m or k = 2m − 1. If x ∈ Kk(B), then cl(x) = 0

for all l k. This implies that

ck

Q

(x) =

(−1)m−1(m

− 1)!chk(x) .

Thus if x is the index of a geometric family Egeom,0, then

[(−1)m−1(m

−

1)!Ωk(Egeom,0)]

= dR(ck

Q

(x)) .

In particular, this multiple of the local index form has integral periods.

1.1.4.2. This leads to a relative secondary invariant as follows. If Egeom,1 is another

family with index x, then we have

(−1)m−1(m

−

1)!Ωk(Egeom,1)

−

(−1)m−1(m

−

1)!Ωk(Egeom,0)

=

(−1)m−1(m

−

1)!dηk−1(Ft)

.

Observe that for any u ∈ K(B) the rational class

(−1)m−1(m

− 1)!chk−1(u)

has integral periods. Therefore using Corollary 2.2.19, after fixing Egeom,0, we can

define the relative invariant with values in

AB−1(B)/AB−1(B, k k

d = 0, Z) such that

it associates to the family Egeom,1 the class [(−1)m−1(m − 1)!ηk−1(Ft)].

1.1.4.3. Let us turn this into an absolute invariant right now. It takes values in the

integral Deligne cohomology HDel(B) k which has similar structures as its rational

counterpart 1.1.3.3.

If Egeom is a geometric family with index(Egeom) ∈ Kk(B), then we want to

define a natural class ˆk(Egeom) c ∈ HDel(B)

k

such that

Rˆk(Egeom) c

=

(−1)m−1(m

−

1)!Ωk(Egeom)

and v(ˆk(Egeom)) c = ck(index(Egeom)). We will use the fact that a

class c ∈ HDel(B) k is determined by its holonomy H(c) : Zk−1(B) → R/Z, where

Zk−1(B) denotes the group of smooth singular k − 1-cycles on B.

Consider a cycle z ∈ Zk−1(B). The trace of z is the union of the images of the

singular simplices belonging to z. This trace admits a neighborhood U which up

to homotopy looks like an at most k − 1-dimensional CW -complex. By assumption

index(Egeom)|U = 0 so that (Egeom)|U admits a taming (Egeom)|U,t. In Subsection

4.1.4 we show that there exists a unique class ˆk(Egeom) c ∈ HDel(B) k such that

H(ˆk(Egeom))(z) c =

[(−1)m−1(m

− 1)!

z

ηk−1((Egeom)|U,t)]R/Z