On the proper Steenrod homotopy groups, and proper embeddings of planes into 3-manifolds

Abstract

Standard algebraic invariants of proper homotopy type are discussed. These do not naturally fit into long exact sequences. Groups of proper homotopy classes of proper maps of Euclidean spaces and open annuli which do naturally form a long exact sequence are defined, and a diagram relating these groups to the standard algebraic invariants of proper homotopy type is given. The structures defined are used to compare several notions of essentiality for proper maps. Some results and examples are given for proper maps of spaces into manifolds of dimension 2 2 and 3 3 . These results are used to add information to a theorem of Brown and Feustel about properly embedding planes in 3 3 -manifolds.