  We present the length, a numerical cohomological index theory, of $ G
$-spaces which are cohomology spheres and $ G $ is a $p$-torus or a torus
group, where $p$ is a prime. As a consequence, we obtain Borsuk-Ulam and
Bourgin-Yang type theorems in this context. A sharper version of the
Bourgin-Yang theorem for topological manifolds is also proved. Also, we give
some general results regarding the upper and lower bound for the length.
