Graded brauer groups and K-theory with local coefficients

Abstract

implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ GRADED BRAUER GROUPS AND K-THEORY WITH LOCAL COEFFICIENTS by P. DONOVAN at Sydney and M. KAROUBI at Strasbourg (1) The Bott periodicity theorem shows that the real K-theory, KO, and the complex K-theory, KU, are generalised cohomology theories graded by Zg and Zg respectively. Our aim is to define a (c K-theory with local coefficients 5? K^X) (K denotes either KO or KU) which shall generalize the usual groups K^X), yzeZg or TzeZg. The ordinary cohomology with local coefficients H^X, a) is defined for (n, o^eZxH^X.Z^). At least when X is a connected finite CW-complex, KO^X) is defined for aeZgXH^X, Z^xH^X, Zg) and KU^X) is defined for oceZ^x H^X, Z^) x Tors^X, Z)), and these may be called K-theory with local coefficients. The sets which index K-theory appear here naturally as cc graded Brauer groups " associated with the space X; these groups were essentially studied by Serre [7] and Wall [17]. These graded Brauer groups, together with another less important set (§ 2), seem to index all reasonable <c K-theories with local coefficients 3?. One motivation for this work is that it gives a complete satisfactory c< Thorn isomorphism 5? theorem in K-theory. More precisely, if V is a real vector bundle on X, the KO-theory of its Thorn space is isornorphic to KO^X), a~1 =(W(V), ^i(V), w^(V)), where rf(V) ==dirn(V) mod 8 and where the ^(V) are the first two Stiefel-Whitney classes of V. However not all oc are of this form. GBrO(X), the real graded Brauer group of X, is in fact the direct sum of Zg with an extension of H^X, Zg) by H^X, Zg) if X is a connected finite GW-cornplex (2). (The extension splits set-theoretically only.) If oc, oc'eGBrO(X), there is a product KO^X^KO^X) ^KO^X); this is constructed by means of Fredholm operators in Hilbert space. So (DKO^X) is a graded ring; GBrU(X), the complex graded a Brauer group of X, has analogous properties.

T-Duality for Torus Bundles with H-Fluxes via Noncommutative Topology

“…When δ is torsion, twisted K-theory had earlier been considered by Karoubi and Donovan [16]. When δ = 0, twisted K-theory reduces to ordinary K-theory (with compact supports).…”

Algebraic K-theory, K-regularity, and 𝕋 $\mathbb {T}$ -duality of 𝒪 ∞ $\mathcal {O}_{\infty }$ -stable C ∗-algebras

“…A crucial insight of Rosenberg in [54] is that certain bundles of compact operators K on locally compact spaces can be used to model twisted K-theory that was introduced in [23]. More precisely, given any pair (E, h) with E locally compact one can construct a noncommutative stable C * -algebra CT(E, h), whose topological K-theory is the twisted K-theory of the pair (E, h).…”

The Hasse Invariant of a Vector Bundle

“…Added in proof. A proof that KAz(X) ->■ H2(X; Z2) is onto which avoids the use of the Bott Periodicity Theorem is given in [6]. They show that End is naturally split by the map \j: KAz(X) -*■ KFP(X), where j is induced by the map of PO(n) to 0(n2) sending ±a to a ® a. Lemma 3.11 shows that for spheres, A^4z and KFP © H2 agree, and hence by 7.1 of [5], they agree for all finite complexes.…”

“…In this case it can be read off from the diagram of Theorem 3.1 that the Clifford bundle of L is a bundle of graded endomorphism algebras of a graded module bundle, which is then used to construct the Thorn class of L. Donovan and Karoubi in studying this orientability question have done independently much of this same work, applying it further to define L-theory with local coefficients [6], [7].…”