  M-theory compactification leads one to consider 7-manifolds obtained by
rolling Calabi-Yau threefolds in the web of Calabi-Yau moduli spaces. The
resulting 7-space in general has singularities governed by the extremal
transition undergone. After providing some background in Sec. 1, the simplest
case of conifold transitions is studied in Sec. 2. In Sec. 3 we employ
topological methods, Smale's classification theorem of smooth simply-connected
spin closed 5-manifolds, and a computer code in the Appendix to understand the
5-manifolds that appear as the link of the singularity of a singuler Calabi-Yau
threefolds from a Type II primitive contraction of a smooth one.
  From this we obtain many locally admissible extremal transition pairs of
Calabi-Yau threefolds, listed in Sec. 4. Their global realization will require
further study. As a mathematical byproduct in the pursuit of the subject, we
obtain a formula to compute the topology of the boundary of the tubular
neighborhood of a Gorenstein rational singular del Pezzo surface embedded in a
smooth Calabi-Yau threefold as a divisor.
