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
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