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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-02-03030-1
Published electronically: June 10, 2002
MathSciNet review: 1926862
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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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Additional Information

Pascal Lambrechts
Affiliation: Laboratoire de Géométrie-Algèbre “LaboGA” de l’Université d’Artois
Address at time of publication: Institut Mathématique, 2 Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium
Email: lambrechts@math.ucl.ac.be

Don Stanley
Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: stanley@math.ualberta.ca

Lucile Vandembroucq
Affiliation: Universidade do Minho, CMAT, Departamento de Matemática, 4710 Braga, Portugal
Email: lucile@math.uminho.pt

DOI: https://doi.org/10.1090/S0002-9947-02-03030-1
Keywords: Two-cone, embedding, cone-length, homotopical boundary
Received by editor(s): February 22, 2000
Received by editor(s) in revised form: June 1, 2001
Published electronically: June 10, 2002
Additional Notes: P.L. is chercheur qualifié au F.N.R.S
D.S. was supported by CNRS at UMR 8524 “AGAT”, Université de Lille 1.
L.V. was supported by a Lavoisier fellowship and an Alexander von Humboldt fellowship.
Article copyright: © Copyright 2002 American Mathematical Society

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