Pointed Matched Circle

Point Matched Circles

We now pass to compact 3-manifolds with parametrized boundary.

Definition.
A pointed matched circle is a quadruple
$\mathfrak{Z}=(Z,\underline{a},M,z)$
where

• $Z$ is an oriented circle, $z\in Z$ is a basepoint,
• $\underline{a}$ is a finite set of points in $Z-z$,
• $M:\underline{a}\to \underline{a}$ is a free involution,
• the result of surgery along $M$ is connected.

From $\mathfrak{Z}$ we construct a surface $F^o(\mathfrak{Z})$:

1. Thicken $Z$ to $Z\times [0,1]$.
2. Attach strips between matched points.
3. Boundary components: $Z\times \{1\}$ and $Z^{\prime }$.
4. Fill in $Z^{\prime }$ with a disk ${\mathbb{D}}^2$.

Adding a disk gives $F(\mathfrak{Z})$ with a distinguished basepoint.