FIBERED CUSP VERSUS d- INDEX THEORY

We prove that the indices of fibered-cusp and d-Dirac operators on a spin manifold with fibered boundary coincide if the associated family of Dirac operators on the fibers of the boundary is invertible. This answers a question raised by Piazza. Under this invertibility assumption, our method yields an index formula for the Dirac operator of horn-cone and of fibered horn metrics. Let X be a compact manifold whose boundary is the total space of a locally trivial fiber bundle φ : ∂X → Y of closed manifolds. Let x : X → R+ be a defining function for ∂X and denote by X the interior of X. The fibered cusp (or Φ-) tangent bundle TX is a smooth vector bundle on X defined in terms of the above data by its global sections: C∞(X, TX) :={V ∈ C(X,TX);V|∂X is tangent to the fibers of φ, 〈dx, V 〉 ∈ x2C∞(X)}. When restricted to X, the Φ tangent bundle is canonically isomorphic to the usual tangent bundle TX. By definition, a fibered cusp metric gΦ is the restriction to X of a Euclidean metric in the bundle TX, smooth down to the boundary of X. Let gd := x gΦ be the conformally equivalent d-metric. Such metrics appear naturally in a variety of geometric situations. Example 1. Let (X, g) be a complete hyperbolic manifold of finite volume. Then outside a convex set, X is isometric to the disjoint union of a finite number of “cusps”, i.e., cylinders [0,∞)×M with metric dt + ehM , where hM is flat. Compactify X by setting X := X t ({∞} ×M). This space becomes a smooth manifold with boundary if we impose that e−t be a boundary-defining function. By the change of variables x = e−t, the metric becomes a d-metric for the trivial boundary fibration M → {pt}. More generally, a locally symmetric space X with Q-rank 1 cusps is diffeomorphic, outside a compact set, to [R,∞) ×M where M is the total space of a fibration φ :M → Y with a canonical connection; moreover, X has a natural Riemannian metric which near infinity takes the form dt + φgY + e gZ Date: July 24, 2006. Partially supported by the CERES contract 4-187/2004 and by a CNCSIS contract (2006). 1