By Calvin C. Moore
Foliated areas glance in the neighborhood like items, yet their worldwide constitution is usually no longer a product, and tangential differential operators are correspondingly extra complicated. within the Nineteen Eighties, Alain Connes based what's referred to now as noncommutative geometry. one of many first effects was once his generalization of the Atiyah-Singer index theorem to compute the analytic index linked to a tangential (pseudo)-differential operator and an invariant transverse degree on a foliated manifold, by way of topological information at the manifold and the operator. This booklet provides a whole evidence of this gorgeous end result, generalized to foliated areas (not simply manifolds).
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Extra resources for Global Analysis on Foliated Spaces
6) p1 p closed leaf p2 p2 ı2 ı2 ı1 ı1 In these two views of the same vicinity of p, the line in the middle represents the closed leaf; curves 1 , ı1 lie in the xz plane; and 2 ; ı2 lie in the yz plane. Curves 1 ; 2 belong to the same leaf; so do curves ı1 ; ı2 . Schematically the snake below the closed leaf is moving in the x direction, whereas the snake above the closed leaf is moving in the y direction. This is important for the resulting holonomy property, as we shall see. The notion of foliated space is strictly more general than that of a foliated manifold.
Possibly empty/ closed set A X . X; Y / with h D f on some neighborhood of A. Proof. 10. X; Y /. In the construction of the maps fgk g, add the additional condition that gk D f on A. In the induction assume that every map in Ᏻ agrees with f on some neighborhood II. FOLIATED SPACES 41 of A. 10 allows the same argument to proceed. 14. X; ޒp N /, let A be a closed subset of X , and suppose that f is tangentially smooth on some neighborhood of A. x/ for x 2 A. X; ޒp N /. (3) For each t , H . ; t / is arbitrarily close to f on compact subsets.
The theorem then provides a formula for the average of these local indices, the average being taken over all leaves. This averaging process is by no means straightforward and requires a whole subsequent chapter, Chapter IV, to explain. The framework of locally traceable operators provides a convenient bridge to the work of Atiyah  on the index theorem for covering spaces. Let X be a manifold (not necessarily compact) and let Xz ! X be a covering space with fundamental domain U and covering group .
Global Analysis on Foliated Spaces by Calvin C. Moore