Realizing isomorphisms between first homology groups of closed 3-manifolds by borromean surgeries

We refine Matveev's result asserting that any two closed oriented 3-manifolds can be related by a sequence of borromean surgeries if and only if they have isomorphic first homology groups and linking pairings. Indeed, a borromean surgery induces a canonical isomorphism between the first homology groups of the involved 3-manifolds, which preserves the linking pairing. We prove that any such isomorphism is induced by a sequence of borromean surgeries. As an intermediate result, we prove that a given algebraic square finite presentation of the first homology group of a 3-manifold, which encodes the linking pairing, can always be obtained from a surgery presentation of the manifold

Finite type invariants of rational homology 3-spheres

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

Slice genus, TT-genus and 44-dimensional clasp number

The $T$-genus of a knot is the minimal number of borromean-type triple points on a normal singular disk with no clasp bounded by the knot; it is an upper bound for the slice genus. Kawauchi, Shibuya and Suzuki characterized the slice knots by the vanishing of their $T$-genus. We generalize this to provide a $3$-dimensional characterization of the slice genus. Further, we prove that the $T$-genus majors the $4$-dimensional positive clasp number and we deduce that the difference between the $T$-genus and the slice genus can be arbitrarily large. We introduce the ribbon counterpart of the $T$-genus and prove that it is an upper bound for the ribbon genus. Interpreting the $T$-genera in terms of $\Delta$-distance, we show that the $T$-genus and the ribbon $T$-genus coincide for all knots if and only if all slice knots are ribbon. We work in the more general setting of algebraically split links and we also discuss the case of colored links. Finally, we express Milnor's triple linking number of an algebraically split $3$-component link as the algebraic intersection number of three immersed disks bounded by the three components.Comment: 19 pages, 21 figures, comments welcom

Torsions and intersection forms of 4-manifolds from trisection diagrams

Gay and Kirby introduced trisections which describe any closed oriented smooth 4-manifold $X$ 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 $\Sigma$, guiding the gluing of the handlebodies. Any morphism $\varphi$ from $\pi_1(X)$ to a finitely generated free abelian group induces a morphism on $\pi_1(\Sigma)$. We express the twisted homology and Reidemeister torsion of $(X;\varphi)$ in terms of the first homology of $(\Sigma;\varphi)$ and the three subspaces generated by the collections of curves. We also express the intersection form of $(X;\varphi)$ in terms of the intersection form of $(\Sigma;\varphi)$.Comment: Comments are welcom