An index formula for the intersection Euler characteristic of an infinite cone
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Mathematische Zeitschrift
An index formula for the intersection Euler characteristic of an infinite cone Ursula Ludwig1 Received: 14 January 2019 / Accepted: 18 September 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract The aim of this article is to establish an index formula for the intersection Euler characteristic of a cone. The main actor is the model Witten Laplacian on the infinite cone. First, we study its spectral properties and establish a McKean-Singer type formula. We also give an explicit formula for the zeta function of the model Witten Laplacian. In a second step, we apply local index techniques to the model Witten Laplacian. By combining these two steps, we express the absolute and relative intersection Euler characteristic of the cone as a sum of two terms, a term which is local, and a second term which is the Cheeger invariant. Keywords Analytic torsion · Conic singularities · Witten deformation Mathematics Subject Classification 58J52
1 Introduction The setting of this article is the following: Let (L, g T L ) be a smooth connected compact Riemannian manifold (without boundary) of dim L ≥ 1, which in the following we will call the link manifold. We denote by cL := [0, ∞) × L /(0,x)∼(0,y) (1.1) the cone over L, by 0 the cone tip and by C := cL \{0} R>0 × L. We denote by r ∈ [0, ∞) the radial coordinate on cL. We equip the cone C with the conic metric g T C := dr 2 + r 2 g T L .
(1.2)
Let ρ L : π1 (L) → U (q) be a representation of the fundamental group of the link L. We denote by (FL , ∇ FL , g FL ) the flat bundle associated to the representation Hermitian vector ρ L . The flat Hermitian bundle FL , ∇ FL , g FL over L can be extended in a trivial way to a flat Hermitian vector bundle over C, we denote by F, ∇ F , g F this extension. We denote ∗ ∗ by FL the flat bundle dual to FL and by F its extension to C.
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Ursula Ludwig [email protected] Universität Duisburg-Essen, Fakultät für Mathematik, 45117 Essen, Germany
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Set n := dim C = dim L + 1. In the whole paper we will assume that the Witt condition is satisfied, i.e. either n is even, or n is odd and H
n−1 2
(L, FL ) = H n−1 (L, FL∗ )∗ = 0. 2
(1.3)
An important topological invariant for singular spaces is the intersection homology introduced by Goresky and MacPherson in [23,24]. An important consequence of the Witt condition (1.3) is, that the intersection homology with lower middle perversity coincides with the intersection homology with upper middle perversity with coefficients in the local system F ∗ . We will simply speak of the intersection homology in the following. We denote by I H• (cL, F ∗ ) the intersection homology (with compact supports) of the cone as defined by Goresky and MacPherson [23,24], we also speak of the absolute intersection homology. By I H• (cL, L, F ∗ ) we denote the relative intersection homology in the sense of Goresky and MacPherson; recall that by this we mean the relative intersection homology of the open finite cone relative to a collared
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