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{title2=$H^*(STS^{2n};\mathbb Z)$}

The Euler class of $TS^{2n}$ evaluates to $2$. Its <Gysin sequence of a sphere bundle> gives
$$
H^q(STS^{2n};\mathbb Z)\cong
\begin{cases}
\mathbb Z,&q=0,4n-1,\\
\mathbb Z/2,&q=2n,\\
0,&\text{otherwise}.
\end{cases}
$$
All products of positive-degree classes vanish.

Cohomology of the unit tangent bundle of an even-dimensional sphere

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 114 / 5 / Solution

Cohomology of the unit tangent bundle of an even-dimensional sphere ( $H^{*} (ST S^{2 n}; Z)$ )

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Cohomology of the unit tangent bundle of an even-dimensional sphere (H∗(STS2n;Z))

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Cohomology of the unit tangent bundle of an even-dimensional sphere ( $H^{*} (ST S^{2 n}; Z)$ )

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