Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-15T10:40:33.509Z Has data issue: false hasContentIssue false
  • English
  • Français

Cone length of the exterior join

Published online by Cambridge University Press:  18 May 2009

Howard J. Marcum
Howard J. Marcum
Affiliation:
The Ohio State University at Newark 1179 University Drive, Newark Ohio 43055, USA E-mail: marcum@math.ohio-state.edu
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The cone length Cl(f) of a map f: X → Y is defined to be the least number of attaching maps possible in a conic (or iterated mapping cone) structure for f. Cone length is a homotopy invariant in the sense that if φ: X → X and ρ: Y → Y are homotopy equivalences then Cl (ρ°f°φ) = Cl(f). Furthermore Cl(f) depends only on the homotopy class of f. It was shown by Ganea [8] that the cone length of the map * → X coincides with the strong Lusternik-Schnirelmann category of X as a space (see Proposition 1.6 below). Recent work of Cornea ([3]–[6]) is much concerned with cone length and its role in critical point theory. For example, let f be a smooth real valued function on a manifold triad (M; V0, V1) with V0 ≠ θ. Under certain conditions, if f has only “reasonable” critical points then it must have at least Cl(V0↪M) of them (see [6]).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 40 , Issue 3 , September 1998 , pp. 445 - 461
Copyright
Copyright © Glasgow Mathematical Journal Trust 1998

References

REFERENCES

1.Arkowitz, M., The generalized Whitehead product, Pacific J. Math. 12 (1962), 7–23.CrossRefGoogle Scholar
2.Blakers, A. L. and Massey, W. S., Products in homotopy theory, Ann. of Math. (2) 58 (1953), 295–324.CrossRefGoogle Scholar
3.Corena, O., Cone-length and Lusternik-Schnirelmann category, Topology 33 (1994), 95–111.CrossRefGoogle Scholar
4.Corena, O., Strong LS category equals cone-length, Topology 34 (1995), 377–381.CrossRefGoogle Scholar
5.Corena, O., Lusternik-Schnirelmann-categorical sections, Ann. Scient. Ěc. Norm. Sup. 28 (1995), 689–704.CrossRefGoogle Scholar
6.Corena, O., Cone-decompositions and degenerate critical points, Proc. London Math. Soc. 77 (1998), 437–461.CrossRefGoogle Scholar
7.Dula, G. and Marcum, H. J., Hopf invariants of the Berstein-Hilton-Ganea kind, Topology Appl. 65 (1995), 179–203.CrossRefGoogle Scholar
8.Ganea, T., Lusternik-Schnirelmann category and strong category, Illinois J. Math. 11 (1967), 417–427.CrossRefGoogle Scholar
9.Hardie, K. A. and Porter, G. J., The slash product homotopy operation, Proc. London Math. Soc. (3) 34 (1977), 505–519.CrossRefGoogle Scholar
10.James, I. M., Lusternik-Schnirelmann category, in Handbook of Algebraic Topology (North-Holland, 1995), 1293–1310.CrossRefGoogle Scholar
11.Marcum, H. J., Homotopy decompositions for product spaces in Alas do Décimo Primeiro Coloquio Brasileiro de Matematica, vol. II, (Instituto de Matematica Pura e aplicada (IMPA), Rio de Janeiro, 1978), pp. 665–680.Google Scholar
12.Marcum, H. J., Fibrations over double mapping cylinders, Illinois J. Math. 24 (1980), 344–358.CrossRefGoogle Scholar
13.Marcum, H. J., Two results on cofibers, Pacific J. Math. 95 (1981), 133–142.CrossRefGoogle Scholar
14.Marcum, H. J., Obstructions for a map to be cyclic, in Algebraic Topology: Oaxtepec 1991, Contemporary Mathematics Series, vol. 146, (American Mathematical Society, 1993), 277–295.CrossRefGoogle Scholar
15.Marcum, H. J., Functional properties of the Hopf invariant I, Quaestiones Math. 19 (1996), 537–587; II, preprint.CrossRefGoogle Scholar
16.Mather, M., Pull-backs in homotopy theory, Canad. J. Math. 28 (1976), 225–263.CrossRefGoogle Scholar
17.Nomura, Y. and Nagase, T., On homotopy commutative squares and cubes, Sci. Rep., Col. Gen. Educ. Osaka Univ. 30 (1982), 91–117.Google Scholar
18.Takens, F., The Lusternik-Schnirelman categories of a product space, Compositio Math. 22 (1970), 175–180.Google Scholar