If $f_{0}:X_{0}\rightarrow B_{0}$ satisfies the above properties, then there exists a topological manifold $X$, $X_{0}\subseteq X$, and a well-behaved $T^{3}$-fibration $f:X\rightarrow B$ such that the diagram

$$\matrix{X_{0}&\hookrightarrow&X\cr\Big\downarrow\hbox to0.0pt{$\vbox{\hbox{$\scriptstyle f_{0}$}}$\hss}&&\Big\downarrow\hbox to0.0pt{$\vbox{\hbox{$\scriptstyle f$}}$\hss}\cr B_{0}&\hookrightarrow&B\cr}$$

commutes.