Let $M$ be an $m$-manifold whose cohomology is known. Let $TM$ be the tangent bundle of $M$ and $\xi$ be the fibre-wise one-point compactification of $TM$. Then $\xi$ is a $m$-sphere bundle over $M$. Let $\Gamma(\xi)$ be the space of cross-sections of $\xi$ with compact open topology.
Question: How to compute the cohomology ring $$ H^*(\Gamma(\xi);\mathbb{Z}_2)? $$
My attempt: I want to construct fibrations and use Serre spectral sequence. But the fibration $$ S^m\to E\xi\to M $$ only concerns the space $E\xi$, but not $\Gamma(\xi)$. Moreover, could the Serre spectral sequence give the explicit ring structure of $$ H^*(\Gamma(\xi);\mathbb{Z}_2) $$ completely, but not partial information about the cohomology?
