Unitarity of the multiplicative group of an integral group ring

A. A. Bovdi

Sbornik. Mathematics, Tome 47 (1984) no. 2, pp. 377-389

Résumé

A homomorphism $f$ of a group $G$ into the multiplicative group of the ring of integers is called, in algebraic topology, an orientation homomorphism of the group $G$. If $x=\sum_{g\in G}\alpha_g g$ is an element of the integral group ring $ZG$, we will let $x^f$ denote the element $\sum_{g\in G}\alpha_g f(g)g^{-1}$. An element $x$ of the multiplicative group $U(ZG)$ is called $f$-unitary if the inverse $x^{-1}$ coincides with $x^f$ or $x^{-f}$. The collection of all $f$-unitary elements of the group $U(ZG)$ form a subgroup $U_f(ZG)$. If $U_f(ZG)=U(ZG)$, the group $U(ZG)$ is said to be $f$-unitary. Our study of the group $~U_f(ZG)$ is motivated by its appearance in algebraic topology, and was suggested by S. P. Novikov. The main result of this article consists of necessary conditions, given in terms of the kernel $\operatorname{Ker}f$ and an element $b$ such that $G=\langle\operatorname{Ker}f,b\rangle$, for the group $U(ZG)$ to be $f$-unitary. We also consider to what extent these conditions are sufficient. Bibliography: 3 titles.