Bordered 3 Manifold

Definition of Bordered 3 manifold

Bordered 3 manifold $Y$ with connected boundary is $(Y,\phi :F(\mathfrak{Z})\to \partial Y)$ where $\phi$ is a homeomorphism between $F(\mathfrak{Z})$ and $\partial Y$.

Definition: Equivalence of bordered 3 manifolds

Two bordered 3 manifolds $(Y_1,{\phi }_1:F({\mathfrak{Z}}_1)\to \partial Y_1)$ and $(Y_2,{\phi }_2:F({\mathfrak{Z}}_2)\to \partial Y_2)$ are equivalent if there is an orientation-preserving homeomorphism $\psi :Y_1\to Y_2$ such that the following diagram commutes.

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Definition.
A bordered 3-manifold is $(Y,\phi :F(\mathfrak{Z})\to \partial Y)$.

Definition.
Two bordered 3-manifolds $(Y_1,{\phi }_1)$ and $(Y_2,{\phi }_2)$ are equivalent if there is an orientation-preserving homeomorphism $\psi :Y_1\to Y_2$ making the diagram commute.