Inequivalent Cantor Sets in $R^{3}$ Whose Complements Have the Same Fundamental Group

TL;DR

This paper constructs infinitely many inequivalent Cantor sets in three-dimensional space with complements sharing the same fundamental group, addressing an open problem and impacting the understanding of 3-manifold topology.

Contribution

It demonstrates the existence of uncountably many inequivalent Cantor sets with identical complement fundamental groups, using local genus analysis and manifold decomposition techniques.

Findings

Existence of infinitely many inequivalent Cantor sets with the same fundamental group.

Construction of nonhomeomorphic open 3-manifolds sharing the same fundamental group.

Application to uncountably many nonhomeomorphic 3-manifolds with a given fundamental group.

Abstract

For each Cantor set C in $R^{3}$, all points of which have bounded local genus, we show that there are infinitely many inequivalent Cantor sets in $R^{3}$ with complement having the same fundamental group as the complement of C. This answers a question from Open Problems in Topology and has as an application a simple construction of nonhomeomorphic open 3-manifolds with the same fundamental group. The main techniques used are analysis of local genus of points of Cantor sets, a construction for producing rigid Cantor sets with simply connected complement, and manifold decomposition theory. The results presented give an argument that for certain groups G, there are uncountably many nonhomeomorphic open 3-manifolds with fundamental group G.