8 1

for all z ∈ Zk−1(B). Note that the right-hand side is independent of the choice of

the taming by the observation 1.1.4.2.

1.1.4.4. If x ∈ Kk(B), then in fact chk(x) itself has integral periods. In fact, ck(x)

is divisible by (m − 1)!. However the reason for this is more complicated.

Let us assume that B has the structure of a CW -complex with filtration Bk−1 ⊆

Bk ⊆ ··· ⊆ B by skeletons. Then the restriction x|Bk−1 is trivial. Let us consider

a trivialization of the restriction to

Bk−1

of a geometric object which represents x.

Then we could ask for an extension of this trivialization to

Bk.

In the present paper we represent x as the index of a geometric family Egeom.

Then the trivialization is a taming of the restriction of this family to

Bk−1

(or better

some open neighborhood). We ask for an extension of the taming to

Bk.

In this

situation there is an obstruction theory with an obstruction class

ok

∈

Hk(B,

Z).

This class depends on the choice of the restriction of the taming to

Bk−2,

and if

ok

vanishes, then there exists and extension of the taming from

Bk−2

to

Bk.

By a result of Kervaire [38] we know that the image of

ok

in rational cohomology

is equal to

(−1)mchk(x).

So the geometric reason for the integrality of the periods

of chk(x) is the existence of the taming of the restriction of Egeom to

Bk−1.

Thus

one could again ask for an absolute invariant

ˆk

o ∈ HDel(B)

k

such that

v(ˆk)

o =

ok

and

Rˆk

o

=

(−1)mΩk(Egeom).

1.1.4.5. It seems not to be very natural to construct a global object like

ˆk

o from data

defined in a neighborhood of

Bk−1.

In fact, from the geometric point of view even

to fix a CW -structure on B is not natural. Thus we will introduce another picture

of a trivialization of the index associated to a geometric family. The main notion is

that of a tamed k −1-resolution. In the present paper we discuss two sorts of tamed

resolutions 3.3.4 and 3.3.12. To either sort of a tamed k − 1-resolution we associate

integral Deligne cohomology classes in HDel(B)

k

with the expected properties and

study their dependence on the choices. Unfortunately it is to technical to state the

precise result here in the introduction. We refer to Section 4.1 for further details.

1.1.4.6. It turns out that for k = 0, 1, 2, 3 the Deligne cohomology class associated

to a tamed k − 1-resolution of Egeom is (up to sign) equal to the class ˆk(Egeom). c

Thus our analytic obstruction theory provides an alternative construction of that

class.

For these small degrees the integral Deligne cohomology classifies well-known

geometric objects, namely continuous Z-valued functions, smooth R/Z-valued func-

tions, hermitian line bundles with connection, and geometric gerbes.

In fact, such objects have been associated previously to geometric families. For

even-dimensional geometric families we have the index as Z-valued function and

the determinant line bundle with Quillen metric and Bismut-Freed connection. To

odd-dimensional families we associate the spectral asymmetry, i.e. the η-invariant,

and the index gerbe (recently constructed by Lott [37]). In Section 4.2 we show

that ˆk(Egeom) c indeed classifies the expected object.

1.1.5. Dirac operators, boundaries, and corners.

1.1.5.1. A large part of the present paper is devoted to the local index theory for

geometric families with corners. The main motivation for developing this theory

was its use in the definition and analysis of tamed resolutions. But in the form

presented in Part 2 it seems to be interesting and useful in its own right.