Rational homology cobordisms of Seifert fibred rational homology three spheres.
The author applies gauge theory to homology cobordisms of homology 3-spheres. Especially he treats the case where the first homology contains 2-torsion, i.e., \({\mathbb Q}\)-homology cobordisms of Seifert fibered \({\mathbb Q}\)-homology 3-spheres. This is done by applying the results of Donaldson and Fintushel-Stern. The main result of this paper concerns Seifert fibered \({\mathbb Q}\)-homology 3-spheres with Seifert invariants denoted by \(\{0;(1,-b),(\alpha_1,\beta_1), \ldots,(\alpha_n,\beta_n)\}\) where \(\alpha_1,\ldots,\alpha_n\) are pairwise relatively prime integers \(\geq 2\). (notations of Neumann and Raymond). Such a manifold \(M\) is oriented as the link of an algebraic singularity, hence \(M\) bounds a simply connected negative definite 4-manifold. Then let \(\alpha\) be the product of the \(\alpha_i\) for \(i=1,\ldots,n\), \(c=\alpha(-b+\sum^n_{i=1}(\beta_i/\alpha_i))\), and \[ R(M) = \frac{2c}{\alpha} -3 +n +\sum^n_{i= 1}\frac{2}{a_i}\sum^{a_i-1}_{k=1} \operatorname{cot} \frac{\pi\alpha c_i^\ast k}{\alpha_i^2} \operatorname{cot} \frac{\pi k}{\alpha_i} \sin^2\frac{\pi k}{\alpha_i}, \] where \(c_ic_i^\ast \equiv 1 \mod \alpha_i\) for \(i=1,\ldots, n\). Using these notations the theorem is the following: {If \(R(M)>0\) and \(c/\alpha<4/\alpha_i\;(i=1,\ldots,n)\), then \(M^3\) cannot bound an oriented, compact, smooth, positive definite 4-manifold \(V^4\). In particular, any connected sum of \(M^3\) cannot bound a \({\mathbb Q}\)-homology 4-ball. } When \(M^3\) is a \({\mathbb Z}\)-homology 3-sphere and \(H_1(V^4;{\mathbb Z})\) contains no 2-torsion, the theorem coincides with Fintushel-Stern's result. Next, the author gives a similar result in terms of orbifolds, the same proof used to prove this theorem is valid. Contrary to Fintushel-Stern the author allows \(H_1(M^3;{\mathbb Z})\) and \(H_1(V^4;{\mathbb Z})\) to contain 2-torsion and hence must use integral cohomology classes of moduli spaces introduced by Donaldson.
scientific article; zbMATH DE number 1998381