The topological period-index problem over 6-complexes

Abstract

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.

Godeaux–Serre varieties and the étale index

“…This shows that we can compute ind.˛/ D ind G .˛/ for˛2 Br K et .X/. It follows from an argument of David Saltman that ind.˛C .X / / D ind.˛/, since X is regular and noetherian; see [4,Proposition 7.2].…”

“…But, when one starts cutting the dimension down, non-flat bundles do appear, at least topologically. For instance, by using the ideas of [4], one can show that on a smooth projective 4-fold X, the topological index of a period p class, with p a prime greater than 3, is at most p 3 . Thus, the arbitrary period and topological index behavior exhibited by the theorem fails if we restrict the dimension of the varieties we consider.…”

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.…”

“…There is an obvious topological analog of Conjecture , which is proposed by Antieau and Williams in and referred to as ‘straw man’, or the topological period–index conjecture: Conjecture If X is a ‐dimensional finite CW complex, and , then …”

“…There exists a connected finite CW complex X of dimension 6 equipped with a class α ∈ Br(X) for which per(α) = n and ind(α) = 2 (n)n 2 . Theorem 1.5 would be a special case of the following Conjecture 1.6 [1]. Let X be a finite 2d-dimensional CW-complex, and let α ∈ Br(X) have period m = p r1 1 · · · p r k k .…”