Weights at fixed points of S1 actions on compact oriented manifolds with isolated fixed points occur in pairs, proved without assuming orientability of isotropy submanifolds.
Jang: Circle actions on oriented manifolds with discrete fixed point sets and classification in dimension 4
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Weights of circle actions on oriented manifolds with isolated fixed points
Weights at fixed points of S1 actions on compact oriented manifolds with isolated fixed points occur in pairs, proved without assuming orientability of isotropy submanifolds.