Abstract

(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.

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