L.: Embeddings up to homotopy of twocones in Euclidean space

Abstract. We say that a finite CW-complex X embeds up to homotopy in a sphere Sn+1 if there exists a subpolyhedron K ⊂ Sn+1 having the homotopy type of X. The main result of this paper is a sufficient condition for the existence of such a homotopy embedding in a given codimension when X is a simply-connected two-cone (a two-cone is the homotopy cofibre of a map between two suspensions). We give different applications of this result: we prove that if X is a two-cone then there are no rational obstructions to embeddings up to homotopy in codi-mension 3. We give also a description of the homotopy type of the boundary of a regular neighborhood of the embedding of a two-cone in a sphere. This enables us to construct a closed manifold M whose Lusternik-Schnirelmann category and cone-length are not affected by removing one point of M