# The cohomology algebras of orientable Seifert manifolds and applications to Lusternik-Schnirelmann category

Banach Center Publications (1998)

- Volume: 45, Issue: 1, page 25-39
- ISSN: 0137-6934

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topBryden, J., and Zvengrowski, P.. "The cohomology algebras of orientable Seifert manifolds and applications to Lusternik-Schnirelmann category." Banach Center Publications 45.1 (1998): 25-39. <http://eudml.org/doc/208907>.

@article{Bryden1998,

abstract = {This note gives a complete description of the cohomology algebra of any orientable Seifert manifold with ℤ/p coefficients, for an arbitrary prime p. As an application, the existence of a degree one map from an orientable Seifert manifold onto a lens space is completely determined. A second application shows that the Lusternik-Schnirelmann category for a large class of Seifert manifolds is equal to 3, which in turn is used to verify the Ganea conjecture for these Seifert manifolds.},

author = {Bryden, J., Zvengrowski, P.},

journal = {Banach Center Publications},

keywords = {cup products; cup product length; Lusternik-Schnirelmann category; degree one maps; diagonal map; Seifert manifolds; cohomology algebra; Seifert manifold; Lyusternik-Shnirel'man category},

language = {eng},

number = {1},

pages = {25-39},

title = {The cohomology algebras of orientable Seifert manifolds and applications to Lusternik-Schnirelmann category},

url = {http://eudml.org/doc/208907},

volume = {45},

year = {1998},

}

TY - JOUR

AU - Bryden, J.

AU - Zvengrowski, P.

TI - The cohomology algebras of orientable Seifert manifolds and applications to Lusternik-Schnirelmann category

JO - Banach Center Publications

PY - 1998

VL - 45

IS - 1

SP - 25

EP - 39

AB - This note gives a complete description of the cohomology algebra of any orientable Seifert manifold with ℤ/p coefficients, for an arbitrary prime p. As an application, the existence of a degree one map from an orientable Seifert manifold onto a lens space is completely determined. A second application shows that the Lusternik-Schnirelmann category for a large class of Seifert manifolds is equal to 3, which in turn is used to verify the Ganea conjecture for these Seifert manifolds.

LA - eng

KW - cup products; cup product length; Lusternik-Schnirelmann category; degree one maps; diagonal map; Seifert manifolds; cohomology algebra; Seifert manifold; Lyusternik-Shnirel'man category

UR - http://eudml.org/doc/208907

ER -

## References

top- [1] J. Bryden, C. Hayat-Legrand, H. Zieschang and P. Zvengrowski, L'anneau de cohomologie d'une variété de Seifert, C. R. Acad. Sci. Paris 324, Sér. I (1997), 323-326.
- [2] J. Bryden, C. Hayat-Legrand, H. Zieschang and P. Zvengrowski, The cohomology ring of a class of Seifert manifolds, Top. and its Appl., to appear. Zbl0964.57019
- [3] J. Bryden and P. Zvengrowski, The cohomology ring of the orientable Seifert manifolds II, preprint. Zbl1025.57031
- [4] S. Eilenberg and T. Ganea, On the Lusternik-Schnirelmann category of abstract groups, Ann. of Math. 65 (1957), 517-518. Zbl0079.25401
- [5] R. H. Fox, Free differential calculus. I. Derivations in the free group ring, Ann. of Math. 57 (1953), 547-560. Zbl0050.25602
- [6] R. H. Fox, On the Lusternik-Schnirelmann category, Ann. of Math. 42 (1941), 333-370. Zbl67.0741.02
- [7] C. Hayat-Legrand, S. Wang and H. Zieschang, Degree-one maps onto lens spaces, Pac. J. Math. 176 (1996), 19-32. Zbl0877.57007
- [8] J. Hempel, 3-Manifolds, Annals of Math. Studies, vol. 86, Princeton University Press, Princeton, New Jersey 1976, 115-135.
- [9] N. Iwase, Ganea's conjecture on Lusternik-Schnirelmann category, preprint. Zbl0947.55006
- [10] I. M. James, On category, in the sense of Lusternik-Schnirelmann, Topology 17 (1978), 331-348. Zbl0408.55008
- [11] S. MacLane, Homology, Springer-Verlag, Berlin, 1963.
- [12] J. M. Montesinos, Classical Tesselations and Three-Manifolds, Springer-Verlag, Berlin, 1987.
- [13] P. Orlik, Seifert Manifolds, Lecture Notes in Math. 291, Springer-Verlag, Berlin, 1972.
- [14] K. Reidemeister, Homotopieringe und Linsenräume, Abh. Math. Sem. Univ. Hamburg 11 (1935), 102-109.
- [15] Y. B. Rudyak, On category weight and its applications, preprint.
- [16] P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 No. 56 (1983), 401-487. Zbl0561.57001
- [17] H. Seifert, Topologie dreidimensionaler gefaserter Räume, Acta Math. 60 (1932), 147-238. Zbl0006.08304
- [18] H. Seifert and W. Threlfall, A Textbook of Topology, Academic Press, 1980.
- [19] A. R. Shastri, J. G. Williams and P. Zvengrowski, Kinks in general relativity, International Journal of Theoretical Physics 19 (1980), 1-23. Zbl0448.55009
- [20] E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
- [21] N. Steenrod and D. B. A. Epstein, Cohomology Operations, The University of Princeton Press, Princeton, N.J., 1962.

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