10 1
So e.g. the one-eck, a two-dimensional manifold with one corner point and one
boundary component, is excluded. Furthermore, it is required that a face of higher
codimension which meets a face of lower codimension is contained in the latter.
The combinatorial datum of the inclusion relation between faces is used to
define the face complex, a chain complex over Z. Its cohomology receives the
obstructions against continuing the inductive process of choosing tamings. For
details we refer to Subsection 2.1.7.
1.1.5.6. What we said above extends to families. For a boundary tamed family of
compatible Dirac operators on a family of manifolds with corners we prove a local
index theorem 2.2.18.
The boundary faces of a boundary tamed family are tamed. In Subsection 2.2.4
we extend the construction of the η-forms to tamed geometric families with corners.
Then the index theorem has the expected form:
The de Rham representative of the Chern character of the index is the sum of
the local index form and the η-forms of the boundary faces.
1.1.5.7. Above we used the notion of the boundary reduction in a sloppy way. In
fact, we must form iterated boundary reductions in order to reduce to corners of
higher codimension. The difficulty is that geometric families may have non-trivial
automorphisms.
A boundary face of a geometric family is a well-defined isomorphism class of
geometric families. Since in our framework a manifold with corners comes with a
distinguished collar there is an essentially canonical way of restricting Dirac bun-
dles. If we take this canonical restriction of the Dirac bundle we call the resulting
geometric family a canonical model of the boundary face. Recall that we want
to lift a taming of the boundary reduction to a boundary taming of the original
operator. In order to do this we must be precise with the identifications.
But consider now a corner of codimension two. Then we have two local bound-
ary components meeting in this corner. Therefore we have two ways of reducing
the Dirac operator to the corner. Already the induced orientations of the corner
are opposite. In fact it turns out that the isomorphism classes of geometric families
obtained in these two ways are opposite to each other. It is an important observa-
tion that if we take canonical models in these two ways, then there is a preferred
(orientation and grading reversing) isomorphism between them (see Lemma 2.1.42).
This will be important for the following reason. Let us consider the two bound-
ary faces separately. Then we want to fix tamings of their boundaries. If possible
we then extend these boundary tamings (of the boundary faces) to tamings. We
want to say that if the chosen tamings on the boundaries of the boundary faces
coincide, then all choices together induce a boundary taming of the original family.
In order to make this comparison we need the preferred isomorphism.
In Section 2.1 we develop a precise language in order to deal with this kind of
problems.
1.1.5.8. Note that the problem already appears for the simplest kind of Dirac oper-
ators. Consider a Riemannian spin manifold. Then we have a well-defined isomor-
phism class of the spinor bundle. If one takes into account that the spinor bundle
has a real structure, then the isomorphisms between different representatives of the
spinor bundle are determined up to sign.
Assume that the manifold is odd-dimensional. If the manifold has a bound-
ary, then we have an induced spin structure on the boundary. Furthermore, the
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