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    A link group invariant of closed orientable 3-manifold

    Akio Kawauchi
    ( Click here for pdf version )

    In this talk, we consider the set $\bf X$ of integral vectors of finite length as any well-ordered set, although a canonical well-order is in mind. Then for the set $\bf L$ of (unoriented) types of links in the 3-sphere $S^3$, we construct a map $$\sigma:\bf L\longrightarrow \bf X$$ which is injective modulo split additions of trivial links. By using this map $\sigma$, we can consider $\bf L$ as a well-ordered set. This well-order of $\bf L$ leads to the concept of a minimal link type.

    Let $\bf L^{sm}$ be the subset of $\bf L$ consisting of simple minimal link types. Let $\pi(\bf L^{sm})$ be the set of isomorphism types of simple minimal link groups. Then we can show that the natural map $\pi:\bf L^{sm}\to\pi(\bf L^{sm})$ is bijective. Let $\bf M$ be the set of unoriented types of closed connected orientable 3-manifolds. Our main theorem is stated as follows:

    \phantom{x}

    \noindent{\bf Theorem.}{ } A previously given well-order of $\bf X$ induces an injective map $$\bf M \buildrel {\alpha}\over {\longrightarrow} \bf L^{sm} \buildrel{\pi}\over {\cong} \pi(\bf L^{sm})$$ such that if we write $\alpha[M]=[L_M]$, then we have $[M]=[\chi(L_M,0)]$ for the $0$-surgery manifold $\chi(L_M,0)$ of $S^3$ along the link $L_M$.

    \phantom{x}

    By taking $\bf X$ with the canonical well-order, several types of closed orientable prime 3-manifolds are ordered and identified with the corresponding simple minimal link types.

    The content of this talk is a growing up version of a part of the research announcement {\it Link corresponding to closed 3-manifold.}

    (see http://www.sci.osaka-cu.ac.jp/\~{}kawauchi/index.htm).