Groebner bases and the cohomology of Grassmann manifolds with application to immersion

Let G(k,n) be the Grassmann manifold of k-planes in Rn+k. Borel showed that H* (G(k,n);Z(2)) = Z(2) [w(1), . . . , w(k)] / I-k,I-n where I-k,I-n is the ideal generated by the dual Stiefel-Whitney classes (w) over bar (n+1), . . . , ($) over bar (n+k). We compute Groebner bases for the ideals I-2,I-2i-3 and I-2,I-2i-4 and use these results along with the theory of modified Postnikov towers to prove immersion results, namely that G immerses in R2 i+2 -15. AS a benefit of the Groebner basis theory G(2, 2)i (- 3) we also obtain a simple description of H* (G(2, 2i -3); Z(2)) and H* (G(2, 2i -4); Z(2)) and use these results to give a simple proof of some non-immersion results of Oproui.