Embeddings up to homotopy of two-cones in euclidean space

Lambrechts, Pascal; Stanley, Don; Vandembroucq, Lucile

doi:10.1090/S0002-9947-02-03030-1

Embeddings up to homotopy of two-cones in euclidean space
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by Pascal Lambrechts, Don Stanley and Lucile Vandembroucq PDF

Trans. Amer. Math. Soc. 354 (2002), 3973-4013 Request permission

Abstract:

We say that a finite CW-complex $X$ embeds up to homotopy in a sphere $S^{n+1}$ if there exists a subpolyhedron $K\subset S^{n+1}$ 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 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 $M$ whose Lusternik-Schnirelmann category and cone-length are not affected by removing one point of $M$.

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