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
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. 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. 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. 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
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. 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.
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