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Rational homology 3-spheres and simply connected definite bounding. (English) Zbl 07195378
For 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 $$S^3$$.

##### MSC:
 57K30 General topology of 3-manifolds 57K35 Other geometric structures on 3-manifolds
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