We consider the rational vector space generated by all rational homology spheres up to orientation-preserving homeomorphism, and the filtration defined on this space by Lagrangian-preserving rational homology handlebody replacements. We identify the graded space associated with this filtration with a graded space of augmented Jacobi diagrams.
Gay and Kirby introduced trisections which describe any closed oriented smooth 4manifold X as a union of three four-dimensional handlebodies. A trisection is encoded in a diagram, namely three collections of curves in a closed oriented surface Σ, guiding the gluing of the handlebodies. Any morphism ϕ from π 1 (X) to a finitely generated free abelian group induces a morphism on π 1 (Σ). We express the twisted homology and Reidemeister torsion of (X; ϕ) in terms of the first homology of (Σ; ϕ) and the three subspaces generated by the collections of curves. We also express the intersection form of (X; ϕ) in terms of the intersection form of (Σ; ϕ).
In the setting of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries, there are two candidates to be universal invariants, defined respectively by Kricker and Lescop. In a previous paper, the second author defined maps between spaces of Jacobi diagrams. Injectivity for these maps would imply that Kricker and Lescop invariants are indeed universal invariants; this would prove in particular that these two invariants are equivalent. In the present paper, we investigate the injectivity status of these maps for degree 2 invariants, in the case of knots whose Blanchfield modules are direct sums of isomorphic Blanchfield modules of Q-dimension two. We prove that they are always injective except in one case, for which we determine explicitly the kernel.
MSC: 57M27
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