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

This ebook comprises 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 subjects coated comprise the index of operators, torsion invariants, K-theory of operator algebras and L^{2}-invariants. the consequences offered during this booklet, that is mostly encouraged and prompted by means of the Atiyah-Singer index theorem, can be of curiosity to graduates and researchers in mathematical physics, differential topology and differential analysis.

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**Example text**

Because the endomorphism Φ∗g of H(F ) corresponds to the operator Π ◦ φ∗g in H(F ) by the leafwise Hodge isomorphism, the composite Π ◦ Pf is independent (i) (i) of the choice of φ. Moreover Tr Pf = Tr(Π ◦ Pf ). 2 can be combined to form a global distribution Tridis (F ) on G; in this notation, F refers to the foliation endowed with the given transverse Lie structure, which indeed is determined by the foliation when the leaves are dense. Each Tridis (F ) is called a distributional trace of F , and deﬁne the Lefschetz distribution of F by the formula (−1)i Tridis (F ) .

Hence Fix(φ ) = {([a, x], a−1 γa) | x ∈ X, α(γ)x = x, a ∈ G} . γ∈Γ Lefschetz Distribution of Lie Foliations 35 We see that if Fix(φ ) ∩ Mb = ∅, then one can assume that b = γ ∈ Γ and α(γ) has a ﬁxed point in X. In this case, Fix(φ ) Mγ = {([a, x], a−1 γa) | x ∈ X, α(γ)x = x, a ∈ G} . A point ([a, x], a−1 γa) ∈ Fix(φ ) ∩ Mγ is simple if and only if x is a simple ﬁxed point of α(γ); in this case, we have ([a, x], a−1 γa) = sign det(α(γ)∗ − id : Tx X → Tx X) , which is denoted by α(γ) (x). Assume that, for any γ ∈ Γ\{e}, all the ﬁxed points of the diﬀeomorphism α(γ), denoted by x1 (γ), x2 (γ), .

Nevertheless, since [X ν , Y ν ] − [X, Y ]ν ∈ X(F ) for all X, Y ∈ g, it follows that the distribution deﬁned by W is completely integrable. Thus there is a C ∞ foliation G on M × G so that T G = W. It is easy to check that the leaves of G are the sets M(x,g) . Let pr1 and pr2 denote the ﬁrst and second factor projections of M × G onto M and G, respectively. 2. For each leaf M of G, we have the following: (i) the restriction pr1 : M → M is a covering map; and (ii) pr2 restricts to a ﬁber bundle map of M to some orbit of the adjoint action of G on itself.