Embeddings up to homotopy of two-cones in euclidean space

Abstract

We say that a finite CW-complex X X embeds up to homotopy in a sphere S n + 1 S^{n+1} if there exists a subpolyhedron K ⊂ S n + 1 K\subset S^{n+1} having the homotopy type of X 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 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 X is a two-cone then there are no rational obstructions to embeddings up to homotopy in codimension 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 M whose Lusternik-Schnirelmann category and cone-length are not affected by removing one point of M M .