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

This ebook includes a set of unique, refereed study and expository articles on elliptic facets of geometric research on manifolds, together with singular, foliated and non-commutative areas. the subjects lined contain the index of operators, torsion invariants, K-theory of operator algebras and L^{2}-invariants. the implications provided during this e-book, that's mostly encouraged and motivated by means of the Atiyah-Singer index theorem, could be of curiosity to graduates and researchers in mathematical physics, differential topology and differential analysis.

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3 This ω has no relation with the Kamber–Tondeur form ω(∇, b) considered in the previous section. 4 For a closed trajectory the map whose homotopy class is considered is θ ˆ : R/T Z → M . 48 D. Burghelea and S. Haller Proposition 1. 1. Given a vector ﬁeld X which satisﬁes (H) and (L) arbitrary close in the C r topology for any r ≥ 0 there exists a vector ﬁeld Y which agrees with X on a neighborhood of the rest points and satisﬁes (H), (L), (MS) and (NCT). 2. Given a vector ﬁeld X which satisﬁes (H) and (L) arbitrary close in the C 0 topology there exists a vector ﬁeld Y which agrees with X on a neighborhood of the rest points and satisﬁes (H), (EG), (L), (MS) and (NCT).

65 1. Introduction For a ﬁnitely presented group Γ denote by Rep(Γ; V ) the algebraic set of all complex representations of Γ on the complex vector space V . For a closed base pointed manifold (M, x0 ) with Γ = π1 (M, x0 ) denote by RepM (Γ; V ) the algebraic closure of RepM 0 (Γ; V ), the Zariski open set of representations ρ ∈ Rep(Γ; V ) so that H ∗ (M ; ρ) = 0. The manifold M is called V -acyclic iﬀ RepM (Γ; V ), or equivalently e characterRepM 0 (Γ; V ), is non-empty.

It is easy to see that χΛ (F ) = vol(G) · χΓ (X) , where χΓ (X) is the Γ-Euler characteristic of the covering manifold X of X deﬁned by Atiyah [6]. By Atiyah’s Γ-index theorem [6], we have χΓ (X) = χ(X), where χ(X) is the Euler characteristic of X. Lefschetz Distribution of Lie Foliations 33 There is a C ∞ global representation φ : M × G → M of the structural transverse action Φ, deﬁned by φ([x, a], g) = [x, ag] . This φ is a free action. 1) on the whole of G. In particular, if χ(X) = 0, then dim H(F ) = ∞ for any homomorphism h : Γ → G.