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The restriction of ωΛ to smooth local transversals is the precise interpretation of Λ as a holonomy invariant measure on local transversals. By non-commutative integration theory [9], the holonomy invariant transverse measure Λ defines a trace TrΛ on the twisted foliation von Neumann algebra W ∗ (M, F , T F ∗ ). Consider also the corresponding supertrace TrsΛ , equal to ± TrΛ , depending on whether the even-odd bigrading is preserved or interchanged. 5, we have Cc∞ (S) ⊂ W ∗ (M, F , T F ∗ ); here, each k ∈ Cc∞ (S) is identified to the family of operators on the leaves whose Schwartz kernels are the restrictions of k, and moreover TrΛ (k) = Tr k(x, x) ωM (x) , M TrsΛ (k) = Trs k(x, x) ωM (x) .

I. Mal’cev. On a class of homogeneous spaces. Transl. Amer. Math. Soc. 39 (1951), 276–307. [19] P. Molino. G´eom´etrie globale des feuilletages Riemanniens. Proc. Nederl. Acad. A1 85 (1982), 45–76. A. A. Kordyukov [20] B. M¨ umken. On tangential cohomology of Riemannian foliations. Amer. J. Math. 128 (2006), 1391–1408. [21] A. Nestke and P. Zuckermann. The index of transversally elliptic complexes. Rend. Circ. Mat. Palermo 34, Suppl. 9 (1985), 165–175. [22] K. Nomizu. On the cohomology of compact homogeneous spaces of nilpotent lie groups.

48 4 Complex representations and cochain complexes . . . . . . . . . . . . 51 5 Analytic torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6 Milnor–Turaev and dynamical torsion . . . . . . . . . . . . . . . . . 57 7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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