Application of Gröbner Bases to the Cup-length of Oriented Grassmann Manifolds

Let R be a commutative ring. The cup-length of R is defined by the greatest number n such that there exist x1, . . . , xn ∈ R \ R with x1 · · · xn , 0. We denote the cup-length of R by cup(R). In particular, for a space X and a commutative ring A, the cup-length of X with the coefficient A, is defined by cup(H̃(X; A)). We denote it by cupA(X). It is well-known that cupA(X) is a lower bound for the LS-category of X. The aim of this paper is to study cup Z/2(G̃n,3), where G̃n,k is the oriented Grassmann manifold SO(n + k)/SO(n) × SO(k). Note that G̃n,k is (nk)-dimensional. While the cohomology of G̃n,2 is well-known, that of G̃n,3 is in vague. However, Korbaš [Kor06] gave rough estimations for cup Z/2(G̃n,3) by considering the height of w2 ∈ H (G̃n,3;Z/2), where w2 is the second Stiefel-Whitney class. The author studies H(G̃n,3;Z/2) by considering Gröbner bases associated with a certain subring of H(G̃n,3;Z/2). It seems that, in principle, the method of Gröbner bases works better in such complicated calculations than that of usual algebraic topology. The author employs a computer and carries a huge amount of calculations for finding the above Gröbner bases and then he dares to conjecture: