By Ulrich Bunke
This paper units up a language to accommodate Dirac operators on manifolds with corners of arbitrary co-dimension. particularly the writer develops an exact conception of boundary rate reductions. the writer introduces the idea of a taming of a Dirac operator as an invertible perturbation by way of a smoothing operator. Given a Dirac operator on a manifold with boundary faces the writer makes use of the tamings of its boundary savings with a view to flip the operator right into a Fredholm operator. Its index is an obstruction opposed to extending the taming from the boundary to the inner. during this approach he develops an inductive approach to affiliate Fredholm operators to Dirac operators on manifolds with corners and develops the linked obstruction thought
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Extra info for Index theory, eta forms, and Deligne cohomology
Example text
By (γ) we denote the parallel transport in V along the path γ. Then we define for φ ∈ G∗ Wj,Tki (a,r) Γ(a, r)(φ) := U ◦ (γ(a,r) ) ◦ φ . It is now easy to check, that Γ is an isomorphism of Dirac bundles as required. Note that once we have chosen the models for Ni,geom and Nj,geom this construction of the isomorphism is completely canonical. If we interchange the roles of i and op j, then the same construction gives an isomorphism (G , Γ ) : Ukj,geom → Uki,geom . op op Its opposite (G , Γ ) : Ukj,geom → Uki,geom turns out to be the inverse of (G, Γ).
H ) on Qh . ¯ for all i ∈ I1 (M ) as follows. Let T1 := T0 + 1. We now define functions φi on M ¯ . We can decompose M ¯ \ int(T M ¯) ¯ We first fix these functions on M \ T1 M 1 k as a union of closed subsets Vj := T1 ∂j M × (−∞, −T1 ] , j ∈ Ik (M ), which meet along common boundaries. Consider now i ∈ I1 (M ). Let k > 0 and j ∈ Ik (M ). If Vj ∩ ∂i M × (−∞, −T1 ) = ∅, then we set φi|Vj = 0. Otherwise there exists a unique h ∈ {1, . . , k} such that the normal variable rh on Vj = T1 ∂j M × (−∞, −T1 ]k coincides with the normal variable to ∂i M .
13. At the present stage the distinction between the isomorphism class ∂j M and its models looks unnecessary complicated. In fact, two models of ∂j M are isomorphic by a unique isomorphism. But later we will consider faces with additional structures like Dirac bundles which allow for non-trivial automorphisms. Then this distinction will be unavoidable. 15 where we will introduce the notion of a distinguished model of a boundary face of a geometric manifold. The main point of the discussion there is how to find a canonical lift of an isomorphism of the underlying manifolds-with-corner models of the boundary face to an isomorphism of the induced Dirac bundles.