The topological period-index conjecture

Index Theory and the Baum-Connes conjecture

These notes are based on lectures on index theory, topology, and operator algebras at the "School on High Dimensional Manifold Theory" at the ICTP in Trieste, and at the Seminari di Geometria 2002 in Bologna. We describe how techniques coming from the theory of operator algebras, in particular $C^*$-algebras, can be used to study manifolds. Operator algebras are extensively studied in their own right. We will focus on the basic definitions and properties, and on their relevance to the geometry and topology of manifolds. The link between topology and analysis is provided by index theorems. Star…

The Topological Period-Index Conjecture for spin$^c$ 6-manifolds

The Topological Period-Index Conjecture is an hypothesis which relates the period and index of elements of the cohomological Brauer group of a space. It was identified by Antieau and Williams as a topological analogue of the Period-Index Conjecture for function fields. In this paper we show that the Topological Period-Index Conjecture holds and is in general sharp for spin$^c$ 6-manifolds. We also show that it fails in general for 6-manifolds.

On the Gray index conjecture for phantom maps

We study the Gray index of phantom maps, which is a numerical invariant of phantom maps. It is conjectured that the only phantom map with infinite Gray index between finite-type spaces is the constant map. We disprove this conjecture by constructing a counter example. We also prove that this conjecture is valid if the target spaces of phantom maps are restricted to simply connected finite complexes. As an application of the counter example we show that $\SNT^{\infty}(X)$ can be non-trivial for some space $X$ of finite type.

The period-index problem for twisted topological K-theory

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 al…

Hedetniemi's conjecture from the topological viewpoint

This paper is devoted to studying a topological version of the famous Hedetniemi conjecture which says: The $\mathbb Z/2$-index of the Cartesian product of two $\mathbb Z/2$-spaces is equal to the minimum of their $\mathbb Z/2$-indexes. We fully confirm the version of this conjecture for the homological index via establishing a stronger formula for the homological index of the join of $\mathbb Z/2$-spaces. Moreover, we confirm the original conjecture for the case when one of the factors is an $n$-sphere. Analogous results for $\mathbb Z/p$-spaces are presented as well. In addition, we answer a…

关于拓扑Hedetniemi猜想的一个弱形式

On a Weak Form of the Topological Hedetniemi Conjecture

针对推广拓扑Hedetniemi猜想在非素数幂阶循环群、非广义四元数群G下不成立的结论，本文证明反例中复形与其乘积的拓扑指数差值可任意大，即该猜想弱形式对这类群最强意义下不成立。