By Lazarsfeld R.
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The neighborhood of a point on βM is a cone on k copies of B n−1 joined along S n−2 , as illustrated in Figure 5. If M is an oriented Z/2-manifold, then MΣ is a manifold, but is not orientable, because of the way the two copies of βM have been glued together. ) So an oriented Z/k-manifold of dimension 4n does not have a signature in the usual sense. But it does have a signature mod k, just as a non-orientable manifold has a signature for Z/2-cohomology. ) The signature of a Z/k-manifold was deﬁned by Sullivan , who showed that MΣ has a fundamental 30 JONATHAN ROSENBERG Figure 5.
If there is an invariant transverse measure µ with [ω], [Cµ ] > 0, then F has a set of closed leaves of positive µ-measure. If there is an invariant transverse measure µ with [ω], [Cµ ] < 0, then F has a set of (conformally) hyperbolic leaves of positive µ-measure. If all the leaves are (conformally) parabolic, then [ω], [Cµ ] = 0 for every invariant transverse measure. ω Proof. By Chern-Weil theory, the de Rham class of 2π represents the Euler class of F . 5, [ω], [Cµ ] is the µ-average of the L2 -Euler characteristic of the leaves.
Consider the Dirac operator DV with coeﬃcients in the universal Cr∗ (π)-bundle VM . As we remarked earlier, the bundle VM has a natural ﬂat connection. If we use this connection to deﬁne DV , then Lichnerowicz’s identity (2) will still hold with DV in place of D, since there is no contribution from the curvature of the bundle. Thus κ > 0 implies Ind DV = A(u∗ ([D])) = 0. Thus if A is injective, we can conclude that u∗ ([D]) = 0 in KOn (Bπ). 7. See  for details. 9 (Gromov-Lawson). A closed aspherical manifold cannot admit a metric of positive scalar curvature.
Barth-type theorem for branched coverings of projective space by Lazarsfeld R.