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 BingWhitehead (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, 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.

COBISS.SI-ID: 16861529

For every finitely generated abelian group G, we construct an irreducible open 3-manifold M_G whose end set is homeomorphic to a Cantor set and with end homogeneity group of M_G isomorphic to G. The end homogeneity group is the group of self-homeomorphisms of the end set that extend to homeomorphisms of the 3-manifold. The techniques involve computing the embedding homogeneity groups of carefully constructed Antoine type Cantor sets made up of rigid pieces. In addition, a generalization of an Antoine Cantor set using infinite chains is needed to construct an example with integer homogeneity group. Results about local genus of points in Cantor sets and about geometric index are also used.

COBISS.SI-ID: 17071961

Let f \colon Z \to Z be a self-map on the topological space Z. Generalizing the well-known factorization of a map into the composite of a homotopy equivalence and a Hurewicz fibration we prove that f is homotopy equivalent to a self-fibration g \colon W \to W.

COBISS.SI-ID: 16943705