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Finite type invariants of integral homology 3-spheres. (English) Zbl 0942.57009
In analogy to the definition of finite type or Vassiliev invariants of links the author defines finite type invariants of integral homology 3-spheres. Starting from the combinatorial identity of Vassiliev invariants the author replaces the knots through integral homology 3-spheres \(\mathcal M\), the crossing changes by Dehn surgery along a sublink \({\mathcal L}^\prime\) of any algebraically split unit-framed link \(\mathcal L\) in \(\mathcal M\) and the sign by \((-1)^{\#{\mathcal L}^\prime}\) of the number of components of the sublink. A scalar invariant of \(\mathcal M\) is said to be of at most order \(k\) if this combinatorial sum vanishes for any (homeomorphism class of) integral homology 3-spheres and any algebraically split unit-framed link \(\mathcal L\). It will be shown that the space of finite type invariants of at most order \(k\) is finite dimensional for all \(k\in \text{N}_0\). Furthermore it will be shown that the Casson invariants are finite type invariants of order 3.

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57T05 Hopf algebras (aspects of homology and homotopy of topological groups)
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