Abstract
For a topological space X, the Lusternik-Schnirelmann category, cat(X), is the smallest number N such that X can be covered by N open subsets each of which is contractible in X. W. Singhof [6] proved that the minimal number of n-balls which suffice to cover a closed PL n-manifold M coincides with the Lusternik-Schnirelmann category if the latter is not too small compared with the dimension of M. As a consequence, one obtains in this case that cat(M×1)=cat(M)+1, thus establishing a special case of a long-standing conjecture. The purpose of this paper is to give a quick proof of Singhof 's results by exploiting the linear structure between the k-skeleton of a polyhedron and its dual skeleton.
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References
FOX R. H.: On the Lusternik-Schnirelmann category, Ann. Math. 42, 333–370 (1941)
GANEA T.: Lusternik-Schnirelmann category and strong category, Proc. Lond. Math. Soc. 10(3). 623–639 (1960)
HUSCH L. S., IVANSIC I.: Embedding compacta up to shape, Shape Theory and Geometric Topology, Lecture Notes in Math. 870. 119–134, Berlin-Heidelberg-New York: Springer (1981)
JAMES I.M.: On category, in the sense of Lusternik-Schnirelmann, Topology 17, 331–348 (1978)
RUSHING T. B.: Topological Embeddings, New York: Academic Press 1973.
SINGHOF W.: Minimal coverings of manifolds with balls, manuscripta math. 29, 385–415 (1979)
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Montejano, L. A quick proof of Singhof's cat(M ×S1)=cat(M)+1 theorem. Manuscripta Math 42, 49–52 (1983). https://doi.org/10.1007/BF01171745
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DOI: https://doi.org/10.1007/BF01171745