Let
G
k
(
R
n
+
k
)
{G_k}({\textbf {R}^{n + k}})
denote the grassman manifold of k-planes in real
(
n
+
k
)
(n\, +\, k)
-space and
w
1
∈
H
1
(
G
k
(
R
n
+
k
)
;
Z
2
)
{w_1}\, \in \, {H^1}({G_k}({\textbf {R}^{n + k}});\,{\textbf {Z}_2})
the first Stiefel-Whitney class of the universal bundle. Using Schubert calculus techniques and the cohomology of flag manifolds we estimate the height of
w
1
{w_1}
in the cohomology ring. We then apply this to improve earlier lower bounds on the Lusternik-Schnirelmann category of real grassmanians.