Given an $m$-dimensional manifold $M$, let $TM$ be the tangent bundle of $M$ and $SM$ be the $m$-sphere bundle over $M$ obtained by fibre-wise one point compactification of $TM$. Let $\Gamma(SM)$ be the space of cross-sections of $SM$ equipped with compact-open topology. Suppose the cohomology ring $H^*(M)$ is known. How to compute the cohomology ring $$ H^*(\Gamma(SM);\mathbb{Z}_p) $$ for primes $p\geq 2$?
2026-03-25 09:29:53.1774430993
cohomology ring of cross-section space of fibre-bundles
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