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 ...
By comparing the Postnikov towers of the classifying spaces of projective unitary groups and the differentials in a twisted Atiyah-Hirzebruch spectral sequence, we deduce a lower bound on the topological index in terms of the period, and solve the topological version of the period-index problem in full for finite CW complexes of dimension less than 6. Conditions are established that, if they were met in the cohomology of a smooth complex 3-fold variety, would disprove the ordinary period-index conjecture. Examples of higher-dimensional varieties meeting these conditions are provided. We use our results to furnish an obstruction to realizing a period-2 Brauer class as the class associated to a sheaf of Clifford algebras, and varieties are constructed for which the total Clifford invariant map is not surjective. No such examples were previously known.
We show that, in general, over a regular integral noetherian affine scheme X of dimension at least 6, there exist Brauer classes on X for which the associated division algebras over the generic point have no Azumaya maximal orders over X. Despite the algebraic nature of the result, our proof relies on the topology of classifying spaces of algebraic groups.
We study a version of the James model for the loop space of a suspension in unstable A 1 -homotopy theory. We use this model to establish an analog of G.W. Whitehead's classical refinement of the Freudenthal suspension theorem in A 1 -homotopy theory: our result refines F. Morel's A 1 -simplicial suspension theorem. We then describe some E 1 -differentials in the EHP sequence in A 1 -homotopy theory. These results are analogous to classical results of G.W. Whitehead's. Using these tools, we deduce some new results about unstable A 1 -homotopy sheaves of motivic spheres, including the counterpart of a classical rational non-vanishing result.
We show that in locally-ringed connected topoi the primes dividing the period and index of a Brauer class coincide. The result applies in particular to Brauer classes on connected schemes, algebraic stacks, topological spaces and to the projective representation theory of profinite groups.
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