This analysis reveals the possible dimensions of smooth manifolds with fixed points, suggesting implications for orientability and spin structures.
We study smooth, closed orientable S¹ S 1 -manifolds M M with exactly $$3$$ 3 fixed points. We show that the dimension of M M is of the form 4· 2ᵃ 4 · 2 a or 8· (2ᵃ+2ᵇ) 8 · ( 2 a + 2 b ) with a,b≥ 0 a , b ≥ 0 and a≠ b a ≠ b . Moreover, under the extra assumption that M M is spin or unitary, we determine exactly the possible dimensions of M.
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Michael Wiemeler (2025) studied this question.