Coincidence theorems for involutions

Ščepin (1974) and Izydorek and Jaworowski (1995, 1996) showed that for each k and n such that 2k > n there exists a contractible k-dimensional simplicial complex Y and a continuous map ϑ:Sn → Y without the antipodal coincidence property, i.e., ϑ(x) /ne ϑ(−x) for all x ϵ Sn. On the other hand, if 2k ⩽ n then every map ϑ:Sn → Y to a k-dimensional simplicial complex has an antipodal coincidence point. In this paper it is shown that, with some minor modifications, these results remain valid when Sn and the antipodal map are replaced by any normal space and an involution with color number n + 2.