Some geometric results on K-theory with $${\mathbb{Z}}/k{\mathbb{Z}}$$ Z / k Z -coefficients
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Some geometric results on K‑theory with ℤ∕kℤ ‑coefficients Adnane Elmrabty1
© Instituto de Matemática e Estatística da Universidade de São Paulo 2020
Abstract We establish some geometric results on K-theory with coefficients in ℤ∕kℤ . The first one is a new proof of the Atiyah–Patodi–Singer mod k index theorem (Math Proc Camb Philos Soc 79:71–99, 1976) in the case of Dirac operators, i.e. in a geometric situation. The second one is a Grothendieck–Riemann–Roch theorem for ℤ∕kℤ-K-theory. Keywords ℤ∕kℤ-K-theory · 𝜂-Invariant · Chern character · Direct image · Spectral flow Mathematics Subject Classification 19L10 · 58J28 · 58J30
1 Introduction In this paper, we prove some results in the topological K-theory with ℤ∕kℤ -coefficients K −1 ℤ∕kℤ [4, Section 5]. We first give a geometric proof of the Atiyah–Patodi–Singer mod k index theorem [5, (8.4)] for Dirac operators, that is, the main trick to derive a topological formula for the rho-invariant of [5]. In order to describe this proof, we will briefly recall some constructions in K −1 ℤ∕kℤ. Let X be an odd-dimensional closed Spinc manifold. Let (E, F, 𝛼) be a triple, where E, F are complex vector bundles over X and 𝛼 ∶ kE → kF is an isomorphism. It determines a natural element in K −1 (X, ℤ∕kℤ) [4, Proposition(5.5)]. We will use SF(DE , DF , 𝛼)(∈ ℤ∕kℤ) and Ind(E, F, 𝛼) to denote, respectively, the mod k spectral flow associated to (E, F, 𝛼) and the image of (E, F, 𝛼) under the direct image map K −1 (X, ℤ∕kℤ) → ℤ∕kℤ . We refer the reader to [5, Sections7,8] and [4, Section 5] for more details. Recall from [5, (8.4)] that Communicated by Ugo Bruzzo. * Adnane Elmrabty [email protected] 1
Department of Mathematics, Ibn Tofaïl University, Kenitra, Morocco
13
Vol.:(0123456789)
São Paulo Journal of Mathematical Sciences
SF(DE , DF , 𝛼) = Ind(E, F, 𝛼).
(1)
As it is well known, Atiyah and Singer [6] have a K-theoretic proof of their index theorem for closed manifolds. In such a proof, one transforms the problem, through direct image constructions in K-theory, to a sphere and then applies the Bott periodicity theorem on the sphere to establish the result. It is thus quite natural to ask whether this strategy can be used to prove (1). The purpose of this paper is to present such a proof, of which the Bismut–Zhang localization formula for eta-invariants [13, (3.65)] plays an essential role. It turns out immediately that one can refine (1) to a mod 2k index theorem for a class of real vector bundles. Second we will explicit a Grothendieck–Riemann–Roch theorem for ℤ∕kℤ-K-theory. Our motivation came from the triviality of the well-known GRR theorem [27, Corollary 2] for ℝ∕ℤ-K-theory on restriction to torsion elements and, in particular, those arising from unitary representations of the fundamental group [4]. To describe 𝜋 our result, consider a smooth fiber bundle Q ↪ Z → Y whose fiber Q is a nonzero even-dimensional closed Spinc manifold and whose base Y is a smooth compact manifold. Let (E, F, 𝛼) be a generator of K −1 (Z, ℤ∕kℤ) . Assume that the index bundle
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