Neighborhoods of sphere-like continua

AbstractLet X be an Sk-like continuum in Euclicean space En. It is shown that if the embedding of X satisfies the small loops condition and 1⩽k⩽n−4, then X has arbitrarily small neighborhoods which are homeomorphic to Sk×Bn−k. It follows that if X and Y are Sk-like continua in Sn, 1⩽k⩽n−4, and both satisfy the small loops condition, then X and Y have homeomorphic complements if and only if they have the same shape