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**Pseudofree orbifolds.**
*(English)*
Zbl 0602.57013

The authors give an equivariant version of the work presented in their paper [J. Differ. Geom. 20, 523-539 (1984; Zbl 0562.53023)] applying the solution spaces of certain Yang-Mills equations to obtain non-existence results for certain 4-manifolds. In particular, they describe an invariant \(R(a_ 1,...,a_ n)\) of the Seifert homology 3-sphere \(\Sigma =\Sigma (a_ 1,...,a_ n)\) which must be non-positive if \(\Sigma\) bounds a positive definite 4-manifold M with no 2-torsion in \(H_ 1(M).\)

(This invariant is described as a trigonometric sum, but has been given a simpler description in terms of a plumbing diagram for \(\Sigma\) by the reviewer and D. Zagier [Lect. Notes Math. 1167, 241-244 (1985; Zbl 0589.57016)].)

As one application they show that the Poincaré homology sphere \(\Sigma\) (2,3,5) has infinite order in the group \(\Theta^ H_ 3\) of homology bordism classes of homology 3-spheres - answering an old question - and, moreover, that this group contains \({\mathbb{Z}}\oplus {\mathbb{Z}}/2k\) for some \(k\geq 0\). Other consequences are Alexander polynomial 1 knots which are not slice, non-uniqueness of simplicial triangulations of manifolds up to concordance, and a new proof of K. Kuga’s theorem [Topology 23, 133-137 (1984; Zbl 0551.57019)] on the non-representability of homology classes by smooth 2-spheres.

(This invariant is described as a trigonometric sum, but has been given a simpler description in terms of a plumbing diagram for \(\Sigma\) by the reviewer and D. Zagier [Lect. Notes Math. 1167, 241-244 (1985; Zbl 0589.57016)].)

As one application they show that the Poincaré homology sphere \(\Sigma\) (2,3,5) has infinite order in the group \(\Theta^ H_ 3\) of homology bordism classes of homology 3-spheres - answering an old question - and, moreover, that this group contains \({\mathbb{Z}}\oplus {\mathbb{Z}}/2k\) for some \(k\geq 0\). Other consequences are Alexander polynomial 1 knots which are not slice, non-uniqueness of simplicial triangulations of manifolds up to concordance, and a new proof of K. Kuga’s theorem [Topology 23, 133-137 (1984; Zbl 0551.57019)] on the non-representability of homology classes by smooth 2-spheres.

Reviewer: W.D.Neumann

### MSC:

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

58J20 | Index theory and related fixed-point theorems on manifolds |

58J10 | Differential complexes |

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

57R95 | Realizing cycles by submanifolds |