Circle actions on simply connected 5-manifolds.

*(English)*Zbl 1094.57020The 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)].

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)].

Reviewer: Karl Heinz Dovermann (Honolulu)