Lusternik-Schnirelmann category and cobordism

Singh, Harpreet

doi:10.1090/S0002-9939-1988-0915741-1

Lusternik-Schnirelmann category and cobordism
HTML articles powered by AMS MathViewer

by Harpreet Singh PDF

Proc. Amer. Math. Soc. 102 (1988), 183-190 Request permission

Abstract:

The Lusternik-Schnirelmann category of a manifold $M$ is the smallest integer $k$ that $M$ can be covered by $k$ open sets each of which is contractible in $M$. It is an upper bound for the length of nonzero products of Stiefel-Whitney classes of $M$. The object of this paper is to use this restriction, on the length of nonzero products, to investigate the cobordism classes of manifolds with category less than or equal to three.

References

P. E. Conner and E. E. Floyd, Torsion in $\textrm {SU}$-bordism, Mem. Amer. Math. Soc. 60 (1966), 74. MR 189044
P. E. Conner and E. E. Floyd, Fibring within a cobordism class, Michigan Math. J. 12 (1965), 33–47. MR 179796
Albrecht Dold, Erzeugende der Thomschen Algebra ${\mathfrak {N}}$, Math. Z. 65 (1956), 25–35 (German). MR 79269, DOI 10.1007/BF01473868
I. M. James, On category, in the sense of Lusternik-Schnirelmann, Topology 17 (1978), no. 4, 331–348. MR 516214, DOI 10.1016/0040-9383(78)90002-2
M. V. Mielke, Cobordism properties of manifolds of small category, Proc. Amer. Math. Soc. 21 (1969), 332–334. MR 236939, DOI 10.1090/S0002-9939-1969-0236939-2
J. Milnor, On the Stiefel-Whitney numbers of complex manifolds and of spin manifolds, Topology 3 (1965), 223–230. MR 180977, DOI 10.1016/0040-9383(65)90055-8
John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554
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
Robert E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. Mathematical notes. MR 0248858
R. E. Stong, Cobordism and Stiefel-Whitney numbers, Topology 4 (1965), 241–256. MR 205262, DOI 10.1016/0040-9383(65)90009-1
René Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86 (French). MR 61823, DOI 10.1007/BF02566923
C. T. C. Wall, Determination of the cobordism ring, Ann. of Math. (2) 72 (1960), 292–311. MR 120654, DOI 10.2307/1970136
Wen-tsün Wu, Les $i$-carrés dans une variété grassmannienne, C. R. Acad. Sci. Paris 230 (1950), 918–920 (French). MR 35993