Neighborhoods of \(S^ 1\)-like continua in 4-manifolds (Q1313093)

We study the problem of determining which compact subsets of 4-manifolds have close neighborhoods that collapse to 1-dimensional spines. As is explained in [the third author, Topology Appl. 31, No. 1, 83-97 (1989; Zbl 0669.57005)], the study of this problem is motivated by the desire to understand engulfing of 2-dimensional polyhedra in piecewise linear 4- manifolds. The technology of 4-manifold topology does not seem to be well enough developed for us to characterize such compacta completely. We restrict our attention, therefore, to the case in which the neighborhood collapses to a copy of the circle, \(S^1\). In that case the fundamental groups which arise are infinite cyclic, so that we can apply the \(\mathbb{Z}\)- theory of Freedman and Quinn [\textit{M. H. Freedman}, Proc. Int. Congr. Math., Warszawa 1983, Vol. 1, 647-663 (1984; Zbl 0577.57003); \textit{M. H. Freedman} and \textit{F. S. Quinn}, Topology of 4-manifolds, Princeton Math. Ser. 39 (1990; Zbl 0705.57001)]. Our main theorem characterizes those compact subsets of 4-manifolds that have arbitrarily close neighborhoods with spines homeomorphic to \(S^1\). Theorem 1. Suppose \(X\) is a compact subset of the orientable 4-manifold \(M^4\). Then \(X\) has arbitrarily close neighborhoods homeomorphic to \(S^1\times B^3\) if and only if (1) \(X\) has the shape of some \(S^1\)-like continuum, and (2) \(X\) satisfies the inessential loops condition.