This is a consequence of the above construction. To see this, let

(9.160)

$$\mathcal{N}_{w_{1}}^{4}(-T_{1}-1,T_{2}),\ldots,\mathcal{N}_{w_{m}}^{4}(-T_{m}-1,T_{m+1})$$

be the neck regions in (9.159) such that for each $1\leq j\leq m$, the neck region $\mathcal{N}_{w_{j}}^{4}(-T_{j}-1,T_{j+1})$ has exactly $w_{j}$-monopoles which have the same $z$-coordinate. Notice that the degree of the nilmanifold fiber is determined by the ending slope of the Green’s function. Corollary 2.7 implies that the degree of the nilpotent fibers will jump by $w_{j}$ when crossing a singular fiber in $\mathcal{N}_{w_{j}}^{4}(-T_{j}-1,T_{j+1})$. It is also easy to see from the construction that there are $w_{j}$ Taub-NUT bubbles at each singular point $t_{j}\in(0,1),j=1\dots m$. ∎