Invariants for families of Brieskorn varieties. (English) Zbl 0624.57015
R. Fintushel and the reviewer have defined a sequence of invariants \(R_ i(H^ 3)\) for Seifert fibered homology 3-spheres \(H^ 3\) [Ann. Math., II. Ser. 122, 335-364 (1985; Zbl 0602.57013)]. If \(R_ 1\) is positive, they show that the homology 3-sphere cannot bound an acyclic 4- manifold. R. Fintushel and the author [Topology Appl. 23, 305-312 (1986)] showed that under certain additional assumptions, if \(R_ i\) is positive for certain i, then the same conclusion holds. These invariants are defined in terms of sums of trigonometric functions.
W. D. Neumann and D. Zagier [Lect. Notes Math. 1167, 241-244 (1985; Zbl 0589.57016)] showed that \(R_ 1(H^ 3)=2b-3\), where b is the central Euler class of the canonical resolution of \(H^ 3\). The paper under review gives a number-theoretic formula for all the \(R_ i\) and demonstrates the behavior of this invariant for certain families of these \(H^ 3\).
W. D. Neumann and D. Zagier [Lect. Notes Math. 1167, 241-244 (1985; Zbl 0589.57016)] showed that \(R_ 1(H^ 3)=2b-3\), where b is the central Euler class of the canonical resolution of \(H^ 3\). The paper under review gives a number-theoretic formula for all the \(R_ i\) and demonstrates the behavior of this invariant for certain families of these \(H^ 3\).
Reviewer: R.Stern
MSC:
| 57N10 | Topology of general \(3\)-manifolds (MSC2010) |
| 57R20 | Characteristic classes and numbers in differential topology |
| 57R90 | Other types of cobordism |
| 55R55 | Fiberings with singularities in algebraic topology |
References:
| [1] | Ronald Fintushel and Terry Lawson, Compactness of moduli spaces for orbifold instantons, Topology Appl. 23 (1986), no. 3, 305 – 312. · Zbl 0664.57006 · doi:10.1016/0166-8641(85)90048-3 |
| [2] | Ronald Fintushel and Ronald J. Stern, Pseudofree orbifolds, Ann. of Math. (2) 122 (1985), no. 2, 335 – 364. · Zbl 0602.57013 · doi:10.2307/1971306 |
| [3] | -, \( O(2)\) actions on the \( 5\)-sphere, Invent. Math. (to appear). · Zbl 0613.57018 |
| [4] | Terry Lawson, Representing homology classes of almost definite 4-manifolds, Michigan Math. J. 34 (1987), no. 1, 85 – 91. · Zbl 0624.57019 · doi:10.1307/mmj/1029003485 |
| [5] | Walter D. Neumann, An invariant of plumbed homology spheres, Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen, Siegen, 1979), Lecture Notes in Math., vol. 788, Springer, Berlin, 1980, pp. 125 – 144. |
| [6] | Walter D. Neumann and Don Zagier, A note on an invariant of Fintushel and Stern, Geometry and topology (College Park, Md., 1983/84) Lecture Notes in Math., vol. 1167, Springer, Berlin, 1985, pp. 241 – 244. · Zbl 0589.57016 · doi:10.1007/BFb0075227 |
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