In this paper we consider Γ → ˜ M → M, a Galois covering with boundary and ˜ D/, a Γ-invariant generalized Dirac operator on ˜M. We assume that the group Γ is of polynomial growth with respect to a word metric. By employing the notion of noncommutative spectral section associated to the boundary operator ˜ D/ 0 and the b-calculus on Galois coverings with boundary, we develop a higher Atiyah-Patodi-Singer index theory. Our main theorem extends to such Γ-Galois coverings with boundary the higher index theorem of Connes-Moscovici.

We present a detailed proof of the existence-theorem for noncommutative spectral sections (see the noncommutative spectral flow, unpublished preprint, 1997). We apply this result to various index-theoretic situations, extending to the noncommutative context results of Booss– Wojciechowski, Melrose–Piazza and Dai–Zhang. In particular, we prove a variational formula, in K ðC n r ðGÞÞ; for the index classes associated to 1-parameter family of Dirac operators on a G-covering with boundary; this formula involves a noncommutative spectral flow for the boundary family. Next, we establish an additivity result, in K ðC * n r ðGÞÞ; for the index class defined by a Dirac-type operator associated to a closed manifold M and a map r: M-BG when we assume that M is the union along a hypersurface F of two manifolds with boundary M Mþ,F M: Finally, we prove a defect formula for the signature-index classes of two cut-and-paste equivalent pairs ðM1; r1: M1-BGÞ and ðM2; r2: M2-BGÞ; where M1 Mþ, ðF;f1Þ M; M2 Mþ, ðF;f2Þ M and f jADiffðFÞ: The formula involves the noncommutative spectral flow of a suitable 1-parameter family of twisted signature operators on F: We give applications to the problem of cut-and-paste invariance of Novikov’s higher signatures on closed oriented manifolds.

We also added a brief epilogue, essentially “What there wasn’t time for. ” Although the focus of the conference was on noncommutative geometry, the topic discussed was conventional commutative motivations for the circle of ideas related to the Novikov and Baum-Connes conjectures. While the article is mainly expository, we present here a few new results (due to the two of us). It is interesting to note that while the period from 80’s through the mid-90’s has shown a remarkable convergence between index theory and surgery theory (or more generally, the classification of manifolds) largely motivated by the Novikov conjecture, most recently, a number of divergences has arisen. Possibly, these subjects are now diverging, but it also seems plausible that we are only now close to discovering truly deep phenomena and that the difference between these subjects is just one of these. Our belief is that, even after decades of mining this vein, the gold is not yet all gone. As the reader might guess from the title, the focus of these notes is not quite on the Novikov conjecture itself, but rather on a collection of problems that are suggested by heuristics, analogies and careful consideration of consequences. Many of the related conjectures are false, or, as far as we know, not directly mathematically related to the original conjecture; this is a good thing: we learn about the subtleties of the original problem, the boundaries of the associated phenomenon, and get to learn about other realms of mathematics.

Let � be a finitely generated discrete group. In this paper we establish vanishing results for rho-invariants associated to (i) the spin Dirac operator of a spin manifold with positive scalar curvature and fundamental group �; (ii) the signature operator of the disjoint union of a pair of homotopy equivalent oriented manifolds with fundamental group �. The invariants we consider are more precisely theAtiyah–Patodi–Singer ( APS) rho-invariant associated to a pair of finite dimensional unitary representations 1; 2 W � ! U.d/, the L 2-rho-invariant of Cheeger–Gromov, the delocalized eta-invariant of Lott for a non-trivial conjugacy class of � which is finite. We prove that all these rho-invariants vanish if the group � is torsion-free and the Baum–Connes map for the maximal group C*-algebra is bijective. This condition is satisfied, for example, by torsion-free amenable groups or by torsion-free discrete subgroups of SO.n; 1 / and SU.n; 1/. For the delocalized invariant we only assume the validity of the Baum–Connes conjecture for the reduced C*-algebra. In addition to the examples above, this condition is satisfied e.g. by Gromov hyperbolic groups or by cocompact discrete subgroups of SL.3; C/. In particular, the three rho-invariants associated to the signature operator are, for such groups, homotopy invariant. For the APS and the Cheeger–Gromov rho-invariants the latter result had been established by Navin Keswani. Our proof reestablishes this result and also extends it to the delocalized eta-invariant of Lott. The proof exploits in a fundamental way results from bordism theory as well as various generalizations of the APS-index theorem; it also embeds these results in general vanishing phenomena for degree zero higher rho-invariants (taking values in A=ŒA; A � for suitable C*-algebras A). We also obtain precise information about the eta-invariants in question themselves, which are usually much more subtle objects than the rho-invariants.

Abstract. We discuss the behaviour of the signature index class of closed foliated bundles under the operation of cutting and pasting. Along the way we establish several index theoretic results: we define Atiyah-Patodi-Singer ( ≡ APS) index classes for Dirac-type operators on foliated bundles with boundary; we prove a relative index theorem for the difference of two APS-index classes associated to different boundary conditions; we prove a gluing formula on closed foliated bundles that are the union of two foliated bundles with boundary; we establish a variational formula for APS-index classes of a 1-parameter family of Dirac-type operators on foliated bundles (this formula involves the noncommutative spectral flow of the boundary family). All these formulas take place in the K-theory of a suitable cross-product algebra. We then apply these results in order to find sufficient conditions ensuring the equality of the signature index classes of two cut-and-paste equivalent foliated bundles. We give applications to the question of when the Baum-Connes higher signatures of closed foliated bundles are cut-and-paste invariant.

Abstract. We construct algebras of pseudodifferential operators on a continuous family groupoid G that are closed under holomorphic functional calculus, contain the algebra of all pseudodifferential operators of order 0 on G as a dense subalgebra, and reflect the smooth structure of the groupoid G, when G is smooth. As an application, we get a better understanding on the structure of inverses of elliptic pseudodifferential operators on classes of non-compact manifolds. For the construction of these algebras closed under holomorphic functional calculus, we develop three methods: one using semi-ideals, one using