Embeddings of locally connected compacta

Abstract

Let X X be a k k -dimensional compactum and f : X → M n f:X \to {M^n} a map into a piecewise linear n n -manifold, n ⩾ k + 3 n \geqslant k + 3 . The main result of this paper asserts that if X X is locally ( 2 k − n ) (2k - n) -connected and f f is ( 2 k − n + 1 ) (2k - n + 1) -connected, then f f is homotopic to a CE equivalence. In particular, every k k -dimensional, r r -connected, locally r r -connected compactum is CE equivalent to a compact subset of R 2 k − r {{\mathbf {R}}^{2k - r}} as long as r ⩽ k − 3 r \leqslant k - 3 .