By Stouffer E. B.

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Let M be a compact oriented surface of genus g and with k > 1 boundary components (all necessarily circles), as in Figure 4, which shows the case g = 1 and k = 3. Regard M as a Z/k-manifold. Compute the Z/k-index of the Euler characteristic operator on M . 24. Construct complete Riemannian metrics g on R2 for which ∗ K∗ (Corb (X)), X = (R2 , g) is not isomorphic to K∗ (pt), and give an example of an application to index theory on X. (Hint: The Coarse Baum-Connes Conjecture is valid for the open cone on a compact metrizable space Y .

Also show that if dim ker Dx and dim ker Dx∗ remain constant, so that x → ker Dx and x → ker Dx∗ deﬁne vector bundles ker D and ker D∗ over X, then Ind D = [ker D] − [ker D∗ ] in K0 (C(Y )) ∼ = K 0 (Y ). ) 4. 1. Crossed Products and Invariants of Group Actions. If a (locally compact) group G acts on a locally compact space X, one can form the transformation group C ∗ -algebra or crossed product C ∗ (G, X) or C0 (X) G. The deﬁnition is easiest to explain when G is discrete; then C ∗ (G, X) is the universal C ∗ -algebra generated by a copy of C0 (X) and unitaries ug , g ∈ G, subject to the relations that (3) ug uh = ugh , ug f u∗g = g · f for g, h ∈ G, f ∈ C0 (X).

This foliation does not have an invariant transverse measure, since such a measure would correspond to a π-invariant measure of H\G ∼ = S 1 , which does not exist. However, the discussion above computes the index of the leafwise Dirac operator on (V, F ) and shows it is non-zero. 3. C ∗ -Algebras and Z/k-Index Theory. 10. A Z/k-manifold is a smooth compact manifold with boundary, M n , along with an identiﬁcation of ∂M with a disjoint union of k copies of a ﬁxed manifold βM n−1 . It is oriented if M is oriented, the boundary components have the induced orientation, and the identiﬁcations are orientation-preserving.