By Dan Burghelea, Richard Melrose, Alexander S. Mishchenko, Evgenij V. Troitsky

This ebook contains a suite of unique, refereed examine and expository articles on elliptic points of geometric research on manifolds, together with singular, foliated and non-commutative areas. the themes lined contain the index of operators, torsion invariants, K-theory of operator algebras and L^{2}-invariants. the consequences offered during this ebook, that's mostly encouraged and encouraged by way of the Atiyah-Singer index theorem, will be of curiosity to graduates and researchers in mathematical physics, differential topology and differential analysis.

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Sur les feuilletages de Lie. C. R. Acad. Sci. Paris. Ser. A-B 272 (1971), A999–A1001. [12] A. Haeﬂiger. Some remarks on foliations with minimal leaves. J. Diﬀ. Geom. 15 (1980), 269–284. [13] A. Haeﬂiger. Foliations and compactly generated pseudogroups. In Foliations: Geometry and Dynamics (Warsaw, 2000), pages 275–295. World Sci. Publishing, River Edge, NJ, 2002. L. Heitsch and C. Lazarov. A Lefschetz theorem for foliated manifolds. Topology 29 (1990), 127–162. A. Kordyukov. Transversally elliptic operators on G-manifolds of bounded geometry.

For each b ∈ G, the restriction pr1 : Mb → M is a covering map. Indeed, we have Mb ≡ Γb \(G × X), where Γb = {γ ∈ Γ | γb = bγ} = Γ ∩ Gb . The leaves of the foliation Fb = pr∗1 F on Mb are described as La = {([a, y], a−1 ba) | y ∈ X} , a∈G, with La1 = La2 if and only if Γb a1 = Γb a2 . Therefore the leaves of F are the ﬁbers of the natural map Mb ≡ Γb \(G × X) → Γb \G , ([a, y], a−1 ba) → Γb a . Take a C ∞ global representation φ : M × G → M of Φ deﬁned by φ([a, x], g) = [ag, x] . We have Fix(φ ) = {([a, x], g) ∈ M × G | [ag, x] = [a, x]} .

42 2 Characteristic forms and vector ﬁelds . . . . . . . . . . . . . . . . . . 45 3 Euler and coEuler structures . . . . . . . . . . . . . . . . . . . . . . 48 4 Complex representations and cochain complexes . . . . . . . . . . . . 51 5 Analytic torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6 Milnor–Turaev and dynamical torsion . . . . . . . . . . . . . . . .