Embeddings up to homotopy of two-cones in euclidean space

Lambrechts, Pascal; Stanley, Don; Vandembroucq, Lucile

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

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ISSN 1088-6850(online) ISSN 0002-9947(print)

Embeddings up to homotopy of two-cones in euclidean space

Authors: Pascal Lambrechts, Don Stanley and Lucile Vandembroucq
Journal: Trans. Amer. Math. Soc. 354 (2002), 3973-4013
MSC (2000): Primary 57R40, 55P25, 55Q25, 55M30
Posted: June 10, 2002
MathSciNet review: 1926862
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Abstract: We say that a finite CW-complex 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 . The main result of this paper is a sufficient condition for the existence of such a homotopy embedding in a given codimension when 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 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 whose Lusternik-Schnirelmann category and cone-length are not affected by removing one point of .

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