Topological Period–Index Conjecture

Definitions and mathematical statement

Let $X$ be a finite CW complex. A complex Azumaya algebra bundle of degree $n\geq 1$ on $X$ is a locally trivial bundle of unital complex algebras with fibre $M_n({\mathbb{C}})$. Write $\operatorname{Br}(X)$ for the topological Brauer group of Morita equivalence classes of such bundles, with group operation induced by tensor product. For $\alpha\in\operatorname{Br}(X)$, define its topological period and index by \[ \operatorname{per}_{\mathrm{topo}}(\alpha)=\operatorname{ord}_{\operatorname{Br}(X)}(\alpha),\qquad \operatorname{ind}_{\mathrm{topo}}(\alpha)=\gcd\left\{n\geq 1: \text{an Azumaya algebra bundle of degree }n\text{ represents }\alpha\right\}. \] The period is finite, and the zero class has period and index $1$. No connectedness assumption is supplied; in the degree condition above, the fibre is $M_n({\mathbb{C}})$ throughout $X$.

Supplied conjectural formulation. For every integer $d\geq 1$, every finite CW complex $X$ of dimension $2d$, and every $\alpha\in\operatorname{Br}(X)$, put $m=\operatorname{per}_{\mathrm{topo}}(\alpha)$. The proposed assertion is \[ \left(d\neq 3\ \text{or}\ m\not\equiv 2\pmod 4\right) \quad\Longrightarrow\quad \operatorname{ind}_{\mathrm{topo}}(\alpha)\mid m^{d-1}. \] The supplied statement makes no assertion for $d=3$ and $m\equiv 2\pmod 4$; its phrase "except possibly" does not assert failure for every class in that range or supply a replacement bound.

Unresolved qualifications. The source does not specify the range of $d$; the display uses the positive-integer interpretation, and treatment of $d=0$ requires clarification. The dimension condition is retained as $\dim X=2d$, without replacing it by an upper bound. The source's $\operatorname{per}(\alpha)$ in the exceptional condition is interpreted as the same topological period used elsewhere. The supplied dated curation calls this an unresolved all-dimensions "straw-man" formulation with partial results; this draft preserves that qualification and does not establish its present validity or status.

Short English statement

The supplied conjecture asks whether the topological index of every Brauer class on a finite CW complex of dimension $2d$ divides the $(d-1)$-st power of its period. It excludes from the assertion the possible exceptional case $d=3$ with period congruent to $2$ modulo $4$. The supplied partially solved label and literature assessment require independent review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002878$, entitled Topological Period--Index Conjecture; exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2015 and names no individual proposer.
• Supplied report citation: Cohomology of Finite Groups: Interactions and Applications (2016), Oberwolfach Reports, DOI 10.4171/owr/2015/24, pp. 1317--1384. https://doi.org/10.4171/owr/2015/24. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The supplied 2016 citation year and 2015 proposed-year/DOI information are retained without resolving their relationship.
• Antieau--Williams, The period-index problem for twisted topological K-theory. Supplied literature reference: https://doi.org/10.1112/jtopol/jtt042.
• Antieau--Williams, The topological period-index conjecture. Supplied literature reference: https://arxiv.org/abs/2003.10539.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem partially solved and states that the low-dimensional cases and the sharp period-two dimension-three exception are established, that general prime-dependent bounds are known, and that the exact all-dimensions straw-man formulation remains unresolved. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified, explaining that mathematical content was preserved while notation, typography, or omitted definitions were repaired for publication. This is a supplied editorial label, not mathematical verification performed for this draft.