= Stabilized tangent bundle of real projective space
{title2=$T\mathbb{RP}^m\oplus\varepsilon^1\cong(m+1)\gamma$}
For the real <tautological bundle> $\gamma$ over <Real projective space>, the tangent space at a line is $\operatorname{Hom}(\gamma,\gamma^\perp)$. Adding $\operatorname{Hom}(\gamma,\gamma)\cong\varepsilon^1$ gives $\operatorname{Hom}(\gamma,\varepsilon^{m+1})\cong(m+1)\gamma$, using a metric to identify a real line with its dual. Hence its total <Stiefel–Whitney class> is $(1+a)^{m+1}$, with $a=w_1(\gamma)$.
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