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Clover calculus for homology 3-spheres via basic algebraic topology. (English) Zbl 1072.57008

The filtrations of the \(\mathbb Z\)-module freely generated by integral homology spheres and the associated graded \(\mathbb Z\)-module have been studied previously by Garoufalidis, Goussarov and Polyak [S. Garoufalidis, M. Goussarov and M. Polyak, Geom. Topol. 5, 75–108 (2001; Zbl 1066.57015)], and others. Among these filtrations, the well-known one is the Goussarov-Habiro filtration \(({\mathcal F}_n)_{n\in{\mathbb N}}\), where the Matveev-Borromeo surgeries play the role of the crossing changes for knots. In the study of the filtered \({\mathbb Z}\)-module \(({\mathcal F}_n)_{n\in{\mathbb N}}\), Garoufalidis, Goussarov and Polyak have defined a set of generators \(\Psi_n(\Gamma)\) for the degree \(n\) part \({\mathcal G}_n\) of the associated graded \(\mathbb Z\)-module, for Jacobi diagrams \(\Gamma\) with at most \(n\) vertices. They also gave graphical rules (known as clover calculus) that allow to present an element as a combination of their generators. For a homology handlebody \(A\) with boundary \(\partial A\), the Lagrangian \({\mathcal L} _A\) of \(A\) is defined to be the kernel of the map induced by the inclusion from \(H_1(\partial A,{\mathbb Z})\) to \(H_1(A,{\mathbb Z})\). In the present paper, the authors express the elements of \({\mathcal G}_n\) associated to the Lagrangian-preserving surgeries as explicit combinations of the \(\Psi _n(\Gamma)\) in terms of intersection forms and linking numbers. Here a Lagrangian-preserving surgery on a homology sphere \(M\) consists of removing the interior of a homology handlebody \(A\subset M\) and replacing it with another such \(B\) whose boundary \(\partial B\) is identified with \(\partial A\) so that \({\mathcal L}_A={\mathcal L}_B\). This gives a purely algebraic version of the Garoufalidis-Goussarov-Polyak clover calculus. The authors also give an alternative description of the Goussarov-Habiro filtration of the \(\mathbb Z\)-module of integral homology spheres by means of Lagrangian-preserving surgeries.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)

Citations:

Zbl 1066.57015
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References:

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