Embeddings of Locally Connected Compacta

Let A' be a ^-dimensional compactum and /: X -» M" a map into a piecewise linear n-manifold. n > k + 3. The main result of this paper asserts that if X is locally (2k ^-connected and / is (2k n + l)-connected, then / is homotopic to a CE equivalence. In particular, every ^--dimensional, /-connected, locally /--connected compactum is CE equivalent to a compact subset of R2*~r as long as r < k 3. Introduction. Let X be a ^-dimensional compactum. In this paper we study the problem of finding an embedding of X into Euclidean H-space R". Specifically, we wish to investigate conditions under which we can embed X in R" when n is less than the classical dimension 2k + 1. In case X is a manifold, it has long been known that such improvements are possible. Whitney [Wh] showed that every smooth rc-dimensional manifold embeds in R". His techniques were later generalized to codimension three embedding theorems for piecewise linear (PL) manifolds by Irwin [Ir] and Hudson [Hd] who showed that if the manifold is /--connected, then it is possible to PL embed it in R2k~r. The situation for polyhedra is different. In order to accomplish the embedding into a lower-dimensional space it is necessary to identify certain contractible sets to points and this changes the homeomorphism type of the polyhedron. Thus the map produced is not a topological embedding, but rather a simple homotopy equivalence. So the appropriate theory for polyhedra is a theory of embedding up to simple homotopy type and this theory was worked out by Stallings [St]. It follows from his main theorem that every fc-dimensional, /--connected polyhedron is simple homotopy equivalent to a subpolyhedron of R2*"'. The purpose of this paper is to prove a theorem like that of Stallings, but for more general compacta rather than polyhedra. In moving from polyhedra to compacta it is necessary to add an additional hypothesis: We must assume that the polyhedra are locally /--connected as well as globally /--connected. In addition, we must use an appropriate generalization of the (strictly polyhedral) concept of simple homotopy equivalence. We substitute CE equivalence. This is the natural thing to do because Received by the editors January 3, 1985. 1980 Mathematics Subject Classification. Primary 57Q35, 54F35, 54C10, 57N15; Secondary 57N25, 57N60.