Nikolaus Conference 2022

Let \(G\) be a finite group, \(L\) - a \(G\)-lattice and \(\alpha \in H^2(G,L)\) - a special element, i.e., a cohomology class such that its restriction to every non-trivial subgroup of \(G\) is non-zero. In 1970 Vasquez proved that there exists a sublattice \(L'\) of \(L\) such that \(p_*(\alpha)\) is special, where \(p \colon L \to L/L'\) is the natural homomorphism, and the \(\mathbb{Z}\)-rank of \(L/L'\) is less than or equal to a natural number which depends on \(G\) only. In the talk we will consider a situation in which we additionally demand from \(L\) and \(L/L'\) to have an almost complex structure (for \(L\) which is defined as \(J \in \operatorname{End}_{\mathbb{R}G}(\mathbb{R} \otimes_{\mathbb{Z}} L)\) such that \(J^2 = -1\)). This is a joint work with Anna Gąsior and Andrzej Szczepański.