Neighborhoods of compacta in 4-manifolds
Let X be a compact subset of the interior of a PL 4-manifold M. Suppose that X satisfies the inessential loops condition (ILC), i.e. for every neighborhood \(U\subset M\) of X there exists a neighborhood \(V\subset U\) of X such that each loop in V-X which is homotopically inessential in V is also inessential in U-X. The main result of the paper asserts that under these conditions, (i) X has arbitrarily close neighborhoods whose components are 4-balls provided \(Fd(X)=0\); or (ii) X has arbitrarily close neighborhoods whose spines are homeomorphic to \(S^ 1\) provided \(Sh(X)=Sh(S^ 1)\). Note that this is a generalization of M. H. Freedman's 4-dimensional cellularity criterion (see the reviewer's paper: ``A criterion for cellularity in a topological 4-manifold'' [Proc. Am. Mat. Soc. 100, 564-566 (1987; Zbl 0623.57009)] for a correct statement and proof). Corollaries include two complement theorems and a weak flatness theorem for compacta in \(S^ 4\). The paper is remarkably well written.
compact subset of interior of a PL 4-manifold
scientific article; zbMATH DE number 4096386
scientific article