1.1. INTRODUCTION 9

Before we give a more detailed overview we shall express one warning right

now. A manifold with corners for us is mainly a combinatorial object. It allows

to talk about faces and product structures. But the index theory is in fact a

L2-index

theory over the extended manifold. Here the extension is obtained by

glueing cylinders to all faces in order to obtain a manifold without singularities like

boundaries or corners. The operators will be extended in a translation invariant

manner over the cylinders.

1.1.5.2. Consider a generalized Dirac operator on a manifold with boundary. As-

sume that we have collar neighborhood of the boundary where the operator is

translation invariant. We now glue a cylinder in order to complete the manifold

and extend the operator naturally.

It turns out that the operator is Fredholm (in the sense that zero does not

belong to the essential spectrum) if and only if the boundary reduction of the Dirac

operator is invertible. This condition is of course not fulfilled in general. But it is

true that the index of the boundary reduction vanishes by the bordism invariance

of the index. Therefore we can find a taming of the boundary reduction. This

taming can be lifted to the cylinder and then extended as a bounded operator to

the whole manifold using a cut-off function.

We use this lift of the taming in order to perturb the original Dirac operator.

We call this a boundary taming. The resulting boundary tamed operator is now

Fredholm and we can consider its index. In this case our index theorem 2.2.13 is

just a version of the classical Atiyah-Patodi-Singer [3] theorem.

In fact all this works for families and yields a version of the results of Bismut

and Cheeger [10], [11]. Our approach is very close to that of Melrose-Piazza [41]

who just consider a slightly different way of lifting the taming in order to produce

the Fredholm operator.

1.1.5.3. If the boundary has several components, then there is a complication which

already appears in the case of the Dirac operator on the interval [0, 1]. If we con-

sider all boundary components as one boundary face, then we can proceed as before.

But the indices of the boundary reductions of the Dirac operator to the individ-

ual components do not necessarily vanish separately. So if we wish to consider

the components separately, then we are stuck. This problem becomes even more

complicated in the presence of corners of higher codimension.

1.1.5.4. Let us now assume that we have a corner of codimension two. Then locally

there are two boundary components which meet in this corner. The corner now

appears as a boundary of the boundary components. In order to find a Fredholm

perturbation we first must choose a taming of the reduction of the Dirac operator

to the corner. This induces Fredholm perturbations of the boundary reductions

as explained above. Assuming that their indices vanish we can choose tamings of

the boundary reductions and finally get a Fredholm perturbation of the original

operator.

For corners of higher codimension we proceed in a similar inductive manner. In

each intermediate step where we encounter a Fredholm perturbation whose index

is an obstruction to proceed further.

1.1.5.5. In order to get some control on this procedure we develop an obstruction

theory. We introduce the notion of an admissible face decomposition. In general

a face in this decomposition may have several connected components, which we

call atoms. Admissibility first of all requires that each face must be embedded.