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===Characteristic classes===
Since <math>\mathbb{HP}^1=S^4</math>, its tangent bundle is stably trivial. The tangent bundles of the rest have nontrivial [[Stiefel–Whitney class|Stiefel–Whitney]] and [[Pontryagin class]]es. The total classes are given by the following
:<math>w(\mathbb{HP}^n) = (1+u)^{n+1}</math>
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