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We estimate the characteristic rank of the canonical $k$–plane bundle over the oriented Grassmann manifold ${\tilde{G}}_{n,k}$. We then use it to compute uniform upper bounds for the ${\mathbb{Z}}_2$–cup-length of ${\tilde{G}}_{n,k}$ for $n$ belonging to certain intervals.

We use known results on the characteristic rank of the canonical $4$–plane bundle over the oriented Grassmann manifold ${\tilde{G}}_{n,4}$ to compute the generators of the ${\mathbb{Z}}_2$–cohomology groups $H^j\left({\tilde{G}}_{n,4}\right)$ for $n=8,9,10,11$. Drawing from the similarities of these examples with the general description of the cohomology rings of ${\tilde{G}}_{n,3}$ we conjecture some predictions.

Extending work of many authors we calculate the higher simple structure sets of lens spaces in the sense of surgery theory with the fundamental group of arbitrary order. As a corollary we also obtain a calculation of the simple structure sets of the products of lens spaces and spheres of dimension grater or equal to $3$.