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
HTML articles powered by AMS MathViewer

by Pascal Lambrechts, Don Stanley and Lucile Vandembroucq

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

DOI: https://doi.org/10.1090/S0002-9947-02-03030-1

Published electronically: June 10, 2002

PDF | 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 $M$.

References

Hideo Ando, On the generalized Whitehead products and the generalized Hopf invariant of a composition element, Tohoku Math. J. (2) 20 (1968), 516–553. MR 240819, DOI 10.2748/tmj/1178243076
Martin Arkowitz, The generalized Whitehead product, Pacific J. Math. 12 (1962), 7–23. MR 155328, DOI 10.2140/pjm.1962.12.7
Hans Joachim Baues, Whitehead Produkte und Hindernisse in dem Produkt von Abbildungskegeln, Arch. Math. (Basel) 25 (1974), 184–197 (German). MR 346785, DOI 10.1007/BF01238662
Hans Joachim Baues, Relationen für primäre Homotopieoperationen und eine verallgemeinerte EHP-Sequenz, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 4, 509–533. MR 397728, DOI 10.24033/asens.1300
Hans Joachim Baues, Rationale Homotopietypen, Manuscripta Math. 20 (1977), no. 2, 119–131 (German, with English summary). MR 442922, DOI 10.1007/BF01170720
Hans J. Baues, Obstruction theory on homotopy classification of maps, Lecture Notes in Mathematics, Vol. 628, Springer-Verlag, Berlin-New York, 1977. MR 0467748, DOI 10.1007/BFb0065144
Hans Joachim Baues, Commutator calculus and groups of homotopy classes, London Mathematical Society Lecture Note Series, vol. 50, Cambridge University Press, Cambridge-New York, 1981. MR 634675, DOI 10.1017/CBO9780511662706
William Browder, Embedding $1$-connected manifolds, Bull. Amer. Math. Soc. 72 (1966), 225–231; erratum: Bull. Amer. Math. Soc. 72 (1966), 736. MR 0189040, DOI 10.1090/S0002-9904-1966-11477-5
William Browder, Surgery on simply-connected manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 65, Springer-Verlag, New York-Heidelberg, 1972. MR 0358813, DOI 10.1007/978-3-642-50020-6
Francis X. Connolly and Bruce Williams, Embeddings up to homotopy type and geometric suspensions of manifolds, Quart. J. Math. Oxford Ser. (2) 29 (1978), no. 116, 385–401. MR 517733, DOI 10.1093/qmath/29.4.385
George Cooke, Embedding certain complexes up to homotopy type in euclidean space, Ann. of Math. (2) 90 (1969), 144–156. MR 242152, DOI 10.2307/1970685
George Cooke, Thickenings of CW complexes of the form $S^{m}\cup _{\alpha }e^{n}$, Trans. Amer. Math. Soc. 247 (1979), 177–210. MR 517691, DOI 10.1090/S0002-9947-1979-0517691-0
Octavian Cornea, Cone-length and Lusternik-Schnirelmann category, Topology 33 (1994), no. 1, 95–111. MR 1259517, DOI 10.1016/0040-9383(94)90037-X
Lawrence M. Graves, The Weierstrass condition for multiple integral variation problems, Duke Math. J. 5 (1939), 656–660. MR 99
T. Ganea, A generalization of the homology and homotopy suspension, Comment. Math. Helv. 39 (1965), 295–322. MR 179791, DOI 10.1007/BF02566956
Nathan Habegger, Embedding up to homotopy type—the first obstruction, Topology Appl. 17 (1984), no. 2, 131–143. MR 738942, DOI 10.1016/0166-8641(84)90037-3
P. J. Hilton and E. H. Spanier, On the imbeddability of certain complexes in euclidean spaces, Proc. Amer. Math. Soc. 11 (1960), 523–526. MR 124902, DOI 10.1090/S0002-9939-1960-0124902-3
Norio Iwase, Ganea’s conjecture on Lusternik-Schnirelmann category, Bull. London Math. Soc. 30 (1998), no. 6, 623–634. MR 1642747, DOI 10.1112/S0024609398004548
N. Iwase, Lusternik-Schnirelmann category of a sphere bundle over a sphere, preprint 2001
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
John R. Klein, On the homotopy embeddability of complexes in Euclidean space. I. The weak thickening theorem, Math. Z. 213 (1993), no. 1, 145–161. MR 1217676, DOI 10.1007/BF03025714
Pascal Lambrechts, Cochain model for thickenings and its application to rational LS-category, Manuscripta Math. 103 (2000), no. 2, 143–160. MR 1796311, DOI 10.1007/s002290070017
P. Lambrechts and L. Vandembroucq, Modeles de Quillen des bords homotopiques, http://gauss.math.ucl.ac.be/~lambrech/publications.html, preprint 1999
Mark Mahowald, A new infinite family in ${}_{2}\pi _{*}{}^s$, Topology 16 (1977), no. 3, 249–256. MR 445498, DOI 10.1016/0040-9383(77)90005-2
M. Mahowald, Private communication
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
F. P. Peterson and N. Stein, The dual of a secondary cohomology operation, Illinois J. Math. 4 (1960), 397–404. MR 151967, DOI 10.1215/ijm/1255456056
Juan Julian Rivadeneyra Perez, On $\textrm {cat}(X\sbs p)$, Internat. J. Math. Math. Sci. 15 (1992), no. 3, 465–468. MR 1169811, DOI 10.1155/S0161171292000620
D. Ravenel, Private communication
C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69, Springer-Verlag, New York-Heidelberg, 1972. MR 0350744, DOI 10.1007/978-3-642-81735-9
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
E. H. Spanier, Function spaces and duality, Ann. of Math. (2) 70 (1959), 338–378. MR 107862, DOI 10.2307/1970107
Donald Stanley, Spaces and Lusternik-Schnirelmann category $n$ and cone length $n+1$, Topology 39 (2000), no. 5, 985–1019. MR 1763960, DOI 10.1016/S0040-9383(99)00047-6
N. E. Steenrod, Cohomology operations, Annals of Mathematics Studies, No. 50, Princeton University Press, Princeton, N.J., 1962. Lectures by N. E. Steenrod written and revised by D. B. A. Epstein. MR 0145525
J. Stallings, The embedding of homotopy types into manifolds, mimeo notes, Princeton University, 1966
Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331 (1978). MR 646078, DOI 10.1007/BF02684341
Robert M. Switzer, Algebraic topology—homotopy and homology, Die Grundlehren der mathematischen Wissenschaften, Band 212, Springer-Verlag, New York-Heidelberg, 1975. MR 0385836, DOI 10.1007/978-3-642-61923-6
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
C. T. C. Wall, Classification problems in differential topology. IV. Thickenings, Topology 5 (1966), 73–94. MR 192509, DOI 10.1016/0040-9383(66)90005-X
George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508, DOI 10.1007/978-1-4612-6318-0