Simple spirals on double Warsaw circles

Let D be a symmetric double Warsaw circle in the plane E2 and let r be the antipodal rotation of D. In [Fund. Math. 144 (1994) 1–9], Mańka proved there is a spiral P in E2 limiting on D such that r extends to a fixed-point-free homeomorphism of the continuum D∪P into itself. It follows that D∪P lies in a uniquely arcwise connected continuum without the fixed-point property. Mańkaʹs spiral P has an increasing amplitude condition that distinguishes it from simple spirals. Suppose S is a simple spiral in Euclidean 3-space E3 limiting on D and a map f of D∪S into D∪S is an extension of r. We prove that f has a fixed point. This theorem remains true when r is replaced by any period 2 homeomorphism of D onto D. However, there exist a homeomorphism g of D onto itself and a simple spiral T in E2 limiting on D such that g extends to a fixed-point-free homeomorphism h of D∪T into itself. We use the extension h to show a uniquely arcwise connected continuum defined by Holsztyński [Fund. Math. 64 (1969) 289–312] does not have the fixed-point property.