The period-index problem for twisted topologicalK–theory

Abstract

We introduce and solve a period-index problem for the Brauer group of a topological space. The period-index problem is to relate the order of a class in the Brauer group to the degrees of Azumaya algebras representing it. For any space of dimension d , we give upper bounds on the index depending only on d and the order of the class. By the Oka principle, this also solves the period-index problem for the analytic Brauer group of any Stein space that has the homotopy type of a finite CW-complex. Our methods use twisted topological K -theory, which was first introduced by Donovan and Karoubi. We also study the cohomology of the projective unitary groups to give cohomological obstructions to a class being represented by an Azumaya algebra of degree n. Applying this to the finite skeleta of the Eilenberg-MacLane space K(Z/ℓ, 2), where ℓ is a prime, we construct a sequence of spaces with an order ℓ class in the Brauer group, but whose indices tend to infinity.19L50, 16K50, 57T10, 55Q10, 55S35 IntroductionThis paper gives a solution to a period-index problem for twisted topological K -theory. The solution should be viewed as an existence theorem for twisted vector bundles.Let X be a connected CW-complex. An Azumaya algebra A of degree n on X is a noncommutative algebra over the sheaf C of complex-valued functions on X such that A is a vector bundle of rank n 2 and the stalks are finite dimensional complex matrix algebras M n (C). Examples of Azumaya algebras include the sheaves of endomorphisms of complex vector bundles and the complex Clifford bundles Cl(E) of oriented evendimensional real vector bundles E . The Brauer group Br(X) classifies topological Azumaya algebras on X up to the usual Brauer equivalence: A 0 and A 1 are Brauer equivalent if there exist vector bundles E 0 and E 1 and an isomorphismof sheaves of C -algebras. Define Br(X) to be the free abelian group on isomorphism classes of Azumaya algebras modulo Brauer equivalence.The period-index problem for twisted topological K -theory 13 Proposition 2.5 If A is an Azumaya algebra of degree n with class α, then A ∼ = End(E) for some α-twisted vector bundle E of rank n. The α-twisted sheaf is unique up to tensoring with untwisted line bundles. Proof See [16, Theorem 1.3.5] or [35, Section 3]. Proposition 2.6 Tensoring with E * , the dual of E , induces an equivalence of categories Vect α → Vect A . Proof See [16, Theorem 1.3.7] or [35, Section 3]. Proposition 2.7 If A and B are Brauer-equivalent Azumaya algebras over X , then Vect A and Vect B are equivalent categories.Proof This follows from the previous two propositions.If X is a finite CW-complex, the twisted topological K -group KU 0 (X) α may be identified with the Grothendieck group of left A-modules.Proposition 2.8 If X is compact and Hausdorff, and if A is an Azumaya algebra on X with class α (so that α is torsion), then K A 0 (X) ∼ = KU 0 (X) α . This isomorphism is uniquely defined up to the natural action of H 2 (X, Z) on the left.Proof This follows from [9, Section 3.1]. In fact, they show that KU ...

Godeaux–Serre varieties and the étale index

“…Because the composition [3]. This already shows that the étale index is different, in general, from the period.…”

“…It is a major open problem in the theory of Azumaya algebras, even when X D Speck, to determine the possible pairs .per.˛/;ind.˛// for˛2 Br K et .X/ where X is fixed, or where the dimension of X is fixed. For an introduction to what is known and for further references, see [3,Section 1].…”

The topological period–index problem over 8‐complexes, I

“…This paper is a sequel to , in which Antieau and Williams initiated the study of the topological period–index problem. Given a path‐connected topological space , let be the topological Brauer group defined in , whose underlying set is the Azumaya algebras modulo the Brauer equivalence: and are called Brauer equivalent if there are vector bundles and such that The multiplication is given by tensor product. Remark In its full generality, Brauer group can be defined for any locally ringed topos.…”

“…Let denote the order of as an element of the group , then Serre also showed that , for all such that there is a ‐torsor over associated to in the way described above. Let denote the greatest common divisor of all such , then in particular we have Furthermore, Antieau and Williams showed the following Theorem Let be a locally ringed connected topos and let . There exists a representative of such that the prime numbers dividing and coincide.…”

“…Examples have been known such that , so the bound is sharp if it holds. The conjecture has been proved in a few low‐dimensional cases, which are summarized in . Very little is known in high dimensions.…”