Abstract
For a closed manifold M, denote by C(M) the minimal number of balls which suffice to cover M. It is shown that C(M) coincides with the Ljusternik-Schnirelmann category cat M if the latter is not too low compared with the dimension of M. In this case it follows in particular that C(M) is an invariant of the homotopy type of M. One of the applications of this result is the following: Let M be a closed manifold of sufficiently high category. Then cat(M×S1)=cat M+1. This is a partial affirmative answer to a long-standing conjecture.
Similar content being viewed by others
References
ARMSTRONG, M. A.: Transversality for polyhedra. Ann. of Math. 86, 172–191 (1967)
ARMSTRONG, M. A., ZEEMAN, E. C.: Transversality for piecewise linear manifolds. Topology 6, 433–466 (1967)
BERSTEIN, I.: On the Lusternik-Schnirelmann category of Grassmannians. Math. Proc. Cambridge Philos. Soc. 79, 129–134 (1976)
BROWN, M.: A proof of the generalized Schoenflies theorem. Bull. Amer. Math. Soc. 66, 74–76 (1960)
FOX, R. H.: On the Lusternik-Schnirelmann category. Ann. of Math. 42, 333–370 (1941)
GANEA, T.: Lusternik-Schnirelmann category and strong category. Illinois J. Math. 11, 417–427 (1967)
GANEA, T.: Some problems on numerical homotopy invariants. In: Symposium on Algebraic Topology, 23–30. Springer-Verlag, Lecture Notes in Mathematics 249 (1971)
HIRSCH, M. W., ZEEMAN, E. C.: Engulfing. Bull. Amer. Math. Soc. 72, 113–115 (1966)
LUFT, E.: Covering manifolds with open discs. Illinois J. Math. 13, 321–326 (1969)
MUNKRES, J.: Obstructions to the smoothing of piecewise-differentiable homeomorphisms. Ann. of Math. 72, 521–554 (1960)
OSBORNE, R. P., STERN, J. L.: Covering manifolds with cells. Pac. J. of Math. 30, 201–207 (1969)
ROURKE, C. P., SANDERSON, B. J.: Introduction to piecewiselinear topology. Springer-Verlag (1972)
ROURKE, C. P., SANDERSON, B. J.: Block bundles I. Ann. of Math. 87, 1–28 (1968)
SINGHOF, W.: Generalized higher order cohomology operations induced by the diagonal mapping. Math. Z. 126, 7–26 (1978)
SMALE, S.: On the structure of manifolds. Amer. J. of Math. 84, 387–399 (1962)
TAKENS, F.: The minimal number of critical points of a function on a compact manifold and the Lusternik-Schnirelmann category. Invent. Math. 6, 197–244 (1968)
THOM, R.: Les structures différentiables des boules et des sphères. In: Colloque de Géométrie Différentielle Globale, Bruxelles (1958)
WHITEHEAD, J. H. C.: Simplicial spaces, nuclei, and m-groups. Proc. London Math. Soc. (2)45, 243–327 (1939)
WHITEHEAD, J. H. C.: On C1-complexes. Ann. of Math. 41, 809–824 (1940)
ZEEMAN, E. C.: Seminar on combinatorial topology (Mimeographed notes). I. H. E. S., Paris, and Univ. of Warwick (1963–66)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Singhof, W. Minimal coverings of manifolds with balls. Manuscripta Math 29, 385–415 (1979). https://doi.org/10.1007/BF01303636
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01303636