An alternative construction of Kontsevich-Kuperberg-Thurston's universal finite type invariant of homology 3-spheres (Intelligence of Low-dimensional Topology)
達郎 清水 · Kyoto University Research Information Repository (Kyoto University) · 2014
IntroductionKontsevich-Kuperberg-Thurston invariant is one variation of M. Kontsevich's Chern- Simons perturbation theoretic invariant.G. Kuperberg and D. Thurston ([5]) gave the construction of the invariant based on M. Kontsevich's idea in [4] and they showed that this invariant is a universal finite type invariant for integral homology 3-spheres as the LMO is.Kontsevich-Kuperberg-Thurston invariant, denoted by $z^{KKT}$ , is a sequence $\{z_{n}^{KKT}\}_{n\in N}.$ $z_{n}^{KKT}$ is a topological invariant of rational homology 3-spheres taking values in the finite dimensional rational vector space $\mathcal{A}_{n}(\emptyset)$ .$\mathcal{A}_{n}(\emptyset)$ is the quotient space divided by some relations (called IHX, AS relations) from the vector space freely generated by oriented Jacobi diagrams with $2n$ -vertexes.We don't give an explicit definition of this space and Jacobi diagrams (For example, see [5], [6]).In this article we treat only the case of $n=1.$In this case $\mathcal{A}_{1}(\emptyset)$ is isomorphic to the 1-dimensional vector space $\mathbb{Q}$ .So we take and fix such an isomorphism and then we consider $z_{1}^{KKT}$ as $a\mathbb{Q}$ valued invariant.It is known that $z_{1}^{KKT}$ equals to $\frac{4}{3}$ times the Casson-Walker invariant. PreliminaryIn this article, all homology 3-spheres are oriented, smooth and with a metric.The assumptions "smooth", "oriented and "with a metric" are not usual.We will use these structures in the construction of the invariant.The invariant is, however, independent of the choices of these structure (i.e.topological invariant).Let $Y$ be a rational homology 3-sphere.Let $\infty\in Y$ be a base point.Take $N(\infty;Y)\subset Y$ a neighborhood of $\infty$ in $Y$ and let $N(\infty;S^{3})$ be a neighborhood of $\infty$ in $S^{3}=\mathbb{R}^{3}\cup\{\infty\}.$We take an orientation preserving diffeomorphism $\varphi^{\infty}$ : $(N(\infty;Y), \infty)arrow\simeq(N(\infty;S^{3}), \infty)$