Yang index of the deleted product

For any $\kappa\ge 1$ a $\kappa$-dimensional polyhedron $Y_\kappa$ is constructed such that the Yang index of its deleted product $Y^*_\kappa$ equals $2\kappa$. This answers a question of Izydorek and Jaworowski (1995). For any $\kappa\ge 1$ a $2\kappa$-dimensional closed manifold $M$ with involution is constructed such that $\operatorname{index} M=2\kappa$, but $M$ can be mapped into a $\kappa$-dimensional polyhedron without antipodal coincidence.