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.
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