Simply connected open 3-manifolds with rigid genus one ends

Abstract

We construct uncountably many simply connected open 3-manifolds with genus one ends homeomorphic to the Cantor set. Each constructed manifold has the property that any self homeomorphism of the manifold (which necessarily extends to a homeomorphism of the ends) fixes the ends pointwise. These manifolds are complements of rigid generalized Bing-Whitehead (BW) Cantor sets. Previous examples of rigid Cantor sets with simply connected complement in $R^{3}$ had infinite genus and it was an open question as to whether finite genus examples existed. The examples here exhibit the minimum possible genus, genus one. These rigid generalized BW Cantor sets are constructed using variable numbers of Bing and Whitehead links. Our previous result with \v{Z}eljko determining when BW Cantor sets are equivalently embedded in $R^{3}$ extends to the generalized construction. This characterization is used to prove rigidity and to distinguish the uncountably many examples.