A link group invariant of closed orientable 3-manifold
Akio Kawauchi
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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).
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