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The ℤ₂-cohomology cup-length of real flag manifolds

Július Korbaš, Juraj Lörinc (2003)

Fundamenta Mathematicae

Using fiberings, we determine the cup-length and the Lyusternik-Shnirel’man category for some infinite families of real flag manifolds O ( n ₁ + . . . + n q ) / O ( n ₁ ) × . . . × O ( n q ) , q ≥ 3. We also give, or describe ways to obtain, interesting estimates for the cup-length of any O ( n ₁ + . . . + n q ) / O ( n ₁ ) × . . . × O ( n q ) , q ≥ 3. To present another approach (combining well with the “method of fiberings”), we generalize to the real flag manifolds Stong’s approach used for calculations in the ℤ₂-cohomology algebra of the Grassmann manifolds.

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

J. Bryden, P. Zvengrowski (1998)

Banach Center Publications

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.

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