Transactions of the American Mathematical Society

Author(s): Pascal Lambrechts; Don Stanley; Lucile Vandembroucq
Journal: Trans. Amer. Math. Soc. 354 (2002), 3973-4013.
MSC (2000): Primary 57R40, 55P25, 55Q25, 55M30
Posted: June 10, 2002
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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

1.: H. Ando, On the generalized Whitehead products and the generalized Hopf invariant of a composition element. Tôhoku Math. J. (2) 20 (1968), 516-553 MR 39:2164
2.: M. Arkowitz, The generalized Whitehead product, Pacific J. Math., 12 (1962), 7-23 MR 27:5262
3.: H.J. Baues, Whitehead Produkte und Hindernisse in dem Produkt von Abbilkdungskegeln. Archiv der Math. (Basel) 25 (1974), 184-197 MR 49:11509
4.: H.J. Baues, Relationen für primäre Homotopieoperationen und eine verallgemeinerte EHP Sequenz, Ann. Sci. Ecole Normale Sup. 4, 8 (1975), 509-533 MR 53:1586
5.: H.J. Baues, Rationale Homotopietypen, Manuscripta Math. 20 (1977), no. 2, 119-131 MR 56:1297
6.: H.J. Baues, Obstruction theory, Lecture Notes in Mathematics, vol. 628, Springer-Verlag, 1977 MR 57:7600
7.: H.J. Baues, Commutator Calculus and groups of homotopy classes, London Math. Soc. Lecture Notes Series, vol. 50, Cambridge University Press, 1981 MR 83b:55012
8.: W. Browder, Embedding 1-connected manifolds, Bull. Amer. Math. Soc. 72 (1966), 225-231 MR 32:6467
9.: W. Browder, Surgery on simply-connected manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 65. Springer-Verlag, 1972 MR 50:11272
10.: F. Connolly, B. Williams, Embedding up to homotopy type and geometric suspensions of manifolds, Quarterly J. Math. Oxford (2), 29 (1978), 385-401 MR 83a:57022
11.: G. Cooke, Embedding certain complexes up to homotopy type in Euclidean space, Annals of Math. (2), 90 (1969), 144-156 MR 39:3486
12.: G. Cooke, Thickenings of CW complexes of the form $S\sp{m}\cup \sb{\alpha }e\sp{n}$ , Trans. Amer. Math. Soc. 247 (1979), 177-210 MR 81c:55016
13.: O. Cornea, Cone-Length and Lusternik-Schnirelmann category, Topology 33 (1994), 95-111 MR 95d:55008
14.: O. Cornea, Y. Félix, J.-M. Lemaire, Rational LS-category and cone-length of Poincaré duality complexes, Topology 37 (1998), 743-748 MR 99a:55022
15.: T. Ganea A generalization of the homology and homotopy suspension, Comment. Math. Helveticii 39 (1965), 295-322 MR 31:4033
16.: N. Habegger, Embedding up to homotopy type--the first obstruction, Topology Appl. 17 (1984), no. 2, 131-143 MR 85e:57022
17.: P. Hilton, E. Spanier, On the embeddability of certain complexes in the euclidean space, Proc. Amer. Math. Soc. 11 (1960), 523-526 MR 23:A2211
18.: N. Iwase, Ganea's conjecture on Lusternik-Schnirelmann category, Bull. London Math. Soc. 30 (1998), no. 6, 623-634 MR 99j:55003
19.: N. Iwase, Lusternik-Schnirelmann category of a sphere bundle over a sphere, preprint 2001
20.: I.M. James, Reduced product spaces, Annals of Math. (2), 62 (1955), 170-197 MR 17:396b
21.: I.M. James, Note on cup product, Proc. Amer. Math. Soc. 8 (1957), 374-383 MR 19:974a
22.: J. Klein, On the homotopy embeddability of complexes in euclidean space. I. The weak thickening theorem, Math. Z. 213 (1993), no. 1, 145-161 MR 95e:57042
23.: P. Lambrechts, Cochain models of thickenings and applications to rational LS-category, Manuscripta Math. 103 (2000), no. 2, 143-160 MR 2001m:55009
24.: P. Lambrechts and L. Vandembroucq, Modeles de Quillen des bords homotopiques, http://gauss.math.ucl.ac.be/~ lambrech/publications.html, preprint 1999
25.: M. Mahowald, A new infinite family in ${}_{2}\pi_{*}^S$ , Topology 16 (1977), no. 3, 249-256 MR 56:3838
26.: M. Mahowald, Private communication
27.: F.P. Peterson, Some non-embedding problems, Bol. Soc. Mat. Mexicana (2) 2 (1957), 9-15 MR 19:440d
28.: F.P. Peterson and N. Stein, The dual of a secondary operation, Illinois J. Math. 4 (1960), 397-404 MR 27:1948
29.: J.J. Rivadeneyra-Perez, On $\mbox{\rm cat\,}(X\setminus p)$ , International J. Math. and Math. Sci. 15 (1992), 465-468 MR 93g:55005a
30.: D. Ravenel, Private communication
31.: C.P. Rourke, B.J. Sanderson, Introduction to Piecewise-Linear Topology, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69, Springer Verlag, 1972 MR 50:3236
32.: A. Shapiro, Obstructions to the embedding of a complex in a euclidian space. I. The first obstruction, Ann. of Math. (2) 66 (1957), 256-269 MR 19:671a
33.: E. Spanier, Function spaces and duality, Ann. of Math. (2) 70 (1959), 338-378 MR 21:6584
34.: D. Stanley, Spaces with Lusternik-Schnirelmann category and cone-length , Topology 39 (2000), no. 5, 985-1019 MR 2001e:55004
35.: N. Steenrod and D. Epstein, Cohomology operations, Annals of Math. studies, Vol. 50, Princeton University Press, 1962 MR 26:3056
36.: J. Stallings, The embedding of homotopy types into manifolds, mimeo notes, Princeton University, 1966
37.: D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. No. 47, (1977), 269-331 MR 58:31119
38.: R. Switzer, Algebraic topology--homotopy and homology, Die Grundlehren der mathematischen Wissenschaften, Band 212. Springer-Verlag, New York-Heidelberg, 1975. xii+526 pp. MR 52:6695
39.: R. Thom, Espaces fibrés en sphères et carrés de Steenrod, Ann. Sci. Ecole Normale Sup. 69 (1952), pp. 109-182 MR 14:1004c
40.: C.T.C. Wall, Classification problems in differential topology IV: thickenings, Topology 5 (1966), pp. 73-94 MR 33:734
41.: G. Whitehead, Elements of Homotopy Theory, Graduate Texts in Math., vol. 61, Springer-Verlag, 1978 MR 80b:55001

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57R40, 55P25, 55Q25, 55M30

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: 10.1090/S0002-9947-02-03030-1
PII: S 0002-9947(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
Posted: 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.
Copyright of article: Copyright 2002, American Mathematical Society

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