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I am reading the thesis of Chris Kottke (https://dspace.mit.edu/bitstream/handle/1721.1/60193/681923895-MIT.pdf) and I would need some help to try to understand intuitively why he makes the choice of the scattering geometry framework for its index theory proof via K-theory.

The thing is that I don’t know much about K-theory, so I would like to know if it is possible to get more or less what he means when he says : « The symbolic structure of the scattering calculus has a very simple interpretation in terms of topological K-theory, permitting us to utilize a powerful families index theorem derived from [7], which is proved in section 2.1. » (p.26) and « Among calculi of pseudodifferential operators on noncompact manifolds, the scattering calculus is particularly simple since its boundary symbols are local, and hence give well-defined elements in the compactly supported topological K-theory of the scattering cotangent bundle. In particular, this allows for the index to be computed by a reduction to the Atiyah-Singer index theorem for compact manifolds ([7], [6]). » (p.29)

More specifically I’m wondering about what « local » exactly means for boundary (or scattering) symbols.

It would be too long to explain the whole technical background, but if you do not know about scattering geometry, the idea is that we consider locally the basis of vector fields $x^2\partial_x, x\partial_{y_i}$ on a manifold with boundary $X$ (with $x$ a boundary defining function and $y_i$ coordinates on the boundary). Then any differential geometry object is defined again (sc-tangent bundle, sc-connections, and finally sc-differential operators). But my question is actually about what it means to be local for symbols and how it is related to the K-theory of the cotangent bundle, intuitively.

Many thanks.

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    $\begingroup$ For starters, you should look at the paragraph below (1.38) in the thesis and compare with the discussion below (1.25). My (limited) understanding is this: For the standard $\psi$do calculus (say, classical, on a closed manifold), checking ellipticity is easy because one only needs to look a the principal symbol, which is a function. For the b-calculus (and several other calculi) checking ellipticity requires to additionally invert the normal operator homomorphism. For the scattering calculus, life is easier again: the normal operator can be written in terms of a function. [...] $\endgroup$
    – Jan Bohr
    Commented Nov 28 at 16:25
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    $\begingroup$ [...] I have heard this being referred to as the scattering calculus being fully symbolic, and my best guess is that this what the author wants to express with symbols being local (rather than a possibly non-local normal operator). $\endgroup$
    – Jan Bohr
    Commented Nov 28 at 16:27

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