(1) The paper sets up a language to deal with Dirac operators on manifolds
with corners of arbitrary codimension. In particular we develop a precise
theory of boundary reductions.
(2) We introduce the notion of a taming of a Dirac operator as an invertible
perturbation by a smoothing operator. Given a Dirac operator on a man-
ifold with boundary faces we use the tamings of its boundary reductions
in order to turn the operator into a Fredholm operator. Its index is an
obstruction against extending the taming from the boundary to the inte-
rior. In this way we develop an inductive procedure to associate Fredholm
operators to Dirac operators on manifolds with corners and develop the
associated obstruction theory.
(3) A central problem of index theory is to calculate the Chern character of
the index of a family of Dirac operators. Local index theory uses the heat
semigroup of an associated superconnection in order to produce differen-
tial forms representing this Chern character. In this paper we develop a
version of local index theory for families of Dirac operators on manifolds
with corners. The resulting de Rham representative of the Chern char-
acter is a sum of the local index form and η-form contributions from the
boundary faces. If the index of the family vanishes and we have chosen
a taming, then local index theory in addition gives a transgression form
whose differential trivializes this Rham representative. This transgression
form plays an important role in the construction of secondary invariants.
(4) Assume that the K-theoretic index of a family of Dirac operators (on a
family of closed manifolds) vanishes on all i−1-dimensional subcomplexes
of the parameter space. The obstruction against increasing i by one is an
i-dimensional integral cohomology class. One of the main goals of this
paper is to use the additional information given by local index theory in
order to refine this obstruction class to a class in i-th integral Deligne
cohomology. As a byproduct we get a lift of the i-th Chern class of the
index of a family of Dirac operators to Deligne cohomology.
(5) In low degrees ≤ 3 integral Deligne cohomology classifies well-known geo-
metric objects like Z-valued functions, U(1)-valued smooth functions, her-
mitean line bundles with connections and geometric gerbes. Such objects
have been previously associated to families of Dirac operators. We verify
that these constructions are compatible with our definitions.
Received by the editor 20.9.2006.
2000 Mathematics Subject Classification. Primary 58j28, Secondary 55S35.