The paper is devoted to studying the Postnikov tower of classification space of compact Lie groups. To each connected CW complex \(X\) one associates a Brauer group \(\mathrm{Br}(X)\) consisting of equivalent classes of Azumaya algebras with respect to the equivalence relation: \(\mathcal A_0 \sim \mathcal A_1\) is and only if there exist some vector bundles \(\mathcal E_0\) and \(\mathcal E_1\) such that \(\mathcal A_0 \otimes \mathrm{End}(\mathcal E_0) \equiv \mathcal A_1 \otimes \mathrm{End}(\mathcal E_1)\). Because of the composition of Bockstein homomorphisms \(H^1(X;PU_r) \to H^2(X;\mathbb S^2 \to H^3(X;\mathbb Z)\) for the exact sequence \(1 \to \mathbb S^1 \to U_r \to PU_r \to 1\) of unitary groups and the homotopy-exact sequence \( 0 \to \mathbb Z \to\mathbb C \overset\exp\longrightarrow \mathbb C^\times \cong \mathbb S^1 \to 1 \), the Brauer group is included in the torsions \(H^3(X;\mathbb Z)_{\mathrm{tor}}\). Following J.-P. Serre, for finite CW complexes \(X\), \(\mathrm{Br}(X) = H^3(X;\mathbb Z)_{\mathrm{tor}}\). The period of any \(\alpha = \langle \mathcal A\rangle \in H^3(X;\mathbb Z)_{\mathrm{tor}}\) is the period \(\mathrm{per}(\alpha)\) of torsion element \(\alpha\), the index \(\mathrm{ind(}\alpha)\) is the greatest common divisor (GCD) of all \(r\) such that \(\mathrm{per}(\alpha) | r\).
The topological period-index conjecture of Antieau-William states that for any \(2d\)-dimensional finite CW complex \(X\), and any class \(\alpha\), \(\mathrm{ind}(\alpha)|\mathrm{per}(\alpha)^{d-1}\). It was shown by the same authors that the conjecture fails for 6-dimensional finite CW complexes by constructing a complex such that \(\mathrm{per}(\alpha) = n\) but \(\mathrm{ind}(\alpha) = \mathrm{GCD}(2,n). n^2\).
The main result of the paper is Theorem 1.7 stating that the topological period-index conjecture fails in dimension 8: \(\mathrm{ind}(\alpha) |\mathrm{GCD}(2,n).\mathrm{GCD}(3,n).n^3\) and if \(X\) is the 8th skeleton of the space \(X=K(\mathbb Z/n,2)\) and \(\alpha\) is the restriction of the fundamental class \(\beta_n\in H^2(K(\mathbb Z/n,2), \mathbb Z)\), then \(\mathrm{ind}(\alpha) = \mathrm{GCD}(3,n).n^3\), if \(n\) is odd and \(\mathrm{ind}(\alpha) =\mathrm{GCD}(3,n).n^2 | \mathrm{ind}(\alpha)\), if \(n\) is even.