The Cup-Length of the Oriented Grassmannians vs a New Bound for Zero-Cobordant Manifolds (Report‪)‬

1 Introduction and statement of results The [[Z].sub.2] -cup-length, cup(X), of a compact path connected topological space X is defined to be the maximum of all numbers c such that there exist, in positive degrees, cohomology classes [a.sub.1], ..., [a.sub.c] [member of] [H.sup.*](X; [Z.sub.2]) such that their cup product [a.sub.1] [universal] ... [universal] [a.sub.c] is nonzero. Instead of the usual notation a [universal] b, we shall mostly write a x b or just ab. For applications, the well known Elsholz inequality