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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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