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Circle actions on simply connected 5-manifolds. (English) Zbl 1094.57020
The author studies necessary and sufficient conditions for the existence of a fixed point free circle action on a simply connected 5-manifold. For a manifold $$M$$, write its second integral homology as a direct sum of cyclic groups of prime power order: $H_2(M, \mathbb Z) = \mathbb Z^k + \sum_{p,i} \left( \mathbb Z/p^i \right)^{c(p^i)}.\tag{1}$ Here $$k = k(M)$$ and $$c(p^i) = c(p^i,M)$$ depend on $$M$$. One can choose the decomposition in (1) such that the second Stiefel-Whitney class map $w_2:H_2(M,\mathbb Z) \to \mathbb Z/2$ is zero on all but one summand $$\mathbb Z/2^n$$. This value $$n$$ is unique and it is denoted by $$i(M)$$. It is the smallest $$n$$ such that there is an $$\alpha \in H_2(M,\mathbb Z)$$ such that $$w_2(\alpha) \neq 0$$ and $$\alpha$$ has order $$2^n$$.
Theorem 1. Let $$L$$ be a compact 5-manifold with $$H_1(M,\mathbb Z) = 0$$ which admits a fixed point free differentiable circle action. Then
(i) For every prime $$p$$, we have at most $$k+1$$ nonzero $$c(p^i)$$ in (1).
(ii) One can arrange that $$w_2:H_2(M,\mathbb Z) \to \mathbb Z/2$$ is the zero map on all but the $$\mathbb Z^k + (\mathbb Z/2)^{c(2)}$$ summands in (1). That is, $$i(L) \in \{ 0, 1, \infty \}$$.
(iii) If $$i(L) = \infty$$, then $$\# \{ i \mid c(2^i) >0 \} \leq k$$.
These conditions are sufficient for simply connected manifolds.
Theorem 2. Let $$L$$ be a compact, simply connected 5-manifold. Then $$L$$ admits a fixed point free differentiable circle action if and only if $$w_2:H_2(M,\mathbb Z) \to \mathbb Z/2$$ satisfies conditions (1)–(3) in Theorem 1.
It helps that simply connected 5-manifolds are well understood due to work of S. Smale [Ann. Math. (2) 75, 38–46 (1962; Zbl 0101.16103)] and D. Barden [Ann. Math. (2) 82, 365–385 (1965; Zbl 0136.20602)]. For 3-manifolds the problem was studied by H. Seifert [“Topologie dreidimensionaler gefaserter Räume”, Acta Math. 60, 147–238 (1933; Zbl 0006.08304)] and for 4-manifolds by R. Fintushel [Trans. Am. Math. Soc. 230, 147–171 (1977; Zbl 0362.57014)].

##### MSC:
 57M60 Group actions on manifolds and cell complexes in low dimensions 57S10 Compact groups of homeomorphisms
##### Keywords:
circle actions; 5-manifold; Seifert bundle; Sasakian geometry
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