(1) $n=2$ and $\Delta$ is discrete; or,

(2) $n=3$ and $\Delta$ can be written as a disjoint union of two sets, $\Delta_{g}$ and $\Delta_{d}$, where $\Delta_{d}$ is a discrete set (the “dissident” points) and $\Delta_{g}$ is a locally closed topological 1-submanifold (the “generic” points). For all $b\in\Delta_{g}$, there exists an open neighbourhood $U$ of $b\in B$, a homeomorphism $\alpha:U\rightarrow D\times(0,1)$, $D$ a two-dimensional disk, a homeomorphism $\alpha^{\prime}:f^{-1}(U)\rightarrow X^{\prime}\times(0,1)\times S^{1}$ with $X^{\prime}$ a four-manifold, $g:X^{\prime}\rightarrow D$ a well-behaved $T^{2}$-fibration, such that the diagram

$$\matrix{f^{-1}(U)&\smash{\mathop{\longrightarrow}\limits^{\alpha^{\prime}}}&X^{\prime}\times(0,1)\times S^{1}\cr\Big\downarrow\hbox to0.0pt{$\vbox{\hbox{$\scriptstyle f$}}$\hss}&&\Big\downarrow\hbox to0.0pt{$\vbox{\hbox{$\scriptstyle f^{\prime}$}}$\hss}\cr U&\smash{\mathop{\longrightarrow}\limits^{\alpha}}&D\times(0,1)\cr}$$

is commutative, where $f^{\prime}$ is the composition of projection onto $X^{\prime}\times(0,1)$ and $g\times id$. Also $\alpha(\Delta_{g}\cap U)=\{v\}\times(0,1)$ for some point $v\in D$. Furthermore, for each point $b\in\Delta_{d}$, there is an open neighbourhood $U$ of $\Delta_{d}$ such that there is a commutative diagram

$$\matrix{U-\Delta&\hookrightarrow&U-\Delta_{d}\cr\Big\downarrow\hbox to0.0pt{$\vbox{\hbox{$\scriptstyle\cong$}}$\hss}&&\Big\downarrow\hbox to0.0pt{$\vbox{\hbox{$\scriptstyle\cong$}}$\hss}\cr(S^{2}-\{p_{1},\ldots,p_{m}\})\times(0,1)&\hookrightarrow&S^{2}\times(0,1)\cr}.$$

In addition, there is a five-manifold $X^{\prime}$ and a map $g:X^{\prime}\rightarrow S^{2}$ along with a commutative diagram

$$\matrix{f^{-1}(U-\Delta_{d})&\smash{\mathop{\longrightarrow}\limits^{\cong}}&X^{\prime}\times(0,1)\cr\Big\downarrow\hbox to0.0pt{$\vbox{\hbox{$\scriptstyle f$}}$\hss}&&\Big\downarrow\hbox to0.0pt{$\vbox{\hbox{$\scriptstyle g\times id$}}$\hss}\cr U-\Delta_{d}&\smash{\mathop{\longrightarrow}\limits^{\cong}}&S^{2}\times(0,1)\cr.}$$