Rational homology 3-spheres and simply connected definite bounding.

*(English)*Zbl 07195378For a smooth, compact and oriented 3-manifold \(Y\), define \(\mathcal{I}(Y)\) to be the set of all negative definite bilinear forms realized by the intersection form of a smooth oriented 4-manifold with boundary \(Y\), up to stable-equivalence. Define \(\mathcal{I}_s(Y)\) similarly, but only considering simple connected 4-manifolds. The results of this paper shows that there is a big difference between these two sets. In fact, the main result is the following: Let \(Y\) be a rational homology 3-sphere such that \(\mathcal{I}_s(Y)\not= \emptyset\) and \(\mathcal{I}_s(-Y)\not= \emptyset\), then there exists infinitely many rational homology spheres \(Y_k\), \(1\leq k < \infty\), which satisfy the properties: (1) \(Y_k\) is homology cobordant to \(Y\); (2) \(\mathcal{I}(Y_k) = \mathcal{I}(Y) \not= \emptyset\) and \(\mathcal{I}(-Y_k) = \mathcal{I}(-Y) \not= \emptyset\); (3) \(\mathcal{I}_s(Y_k) = \emptyset\) and \(\mathcal{I}_s(-Y_k) = \emptyset\); (4) if \(k \not= k'\), then \(Y_k\) is not diffeomorphic to \(Y_k\) not to \(-Y_k\) ; (5) each \(Y_k\) s irreducible and toroidal.

This result has interesting applications in Dehn surgery on knots. For instance, in combination with know results, it is shown that for any relative prime integers \(p\) and \(q\), there exist infinitely many irreducible rational homology 3-spheres which are homology cobordant to the lens space \(L(p,q)\), but that cannot be obtained by surgery on a knot in \(S^3\).

This result has interesting applications in Dehn surgery on knots. For instance, in combination with know results, it is shown that for any relative prime integers \(p\) and \(q\), there exist infinitely many irreducible rational homology 3-spheres which are homology cobordant to the lens space \(L(p,q)\), but that cannot be obtained by surgery on a knot in \(S^3\).

Reviewer: Mario Eudave-Muñoz (Ciudad de México)