On the isolated singularity of a 7-space obtained by rolling Calabi-Yau threefolds through extremal transitions

M-theory suggests the study of 11-dimensional space-times compactified on 7manifolds. From its intimate relation to superstrings, one possible class of such 7manifolds are those that arise from rolling Calabi-Yau threefolds in the web of CalabiYau moduli spaces. The resulting 7-space in general has singularities governed by the extremal transitions undergone. In this article, we employ topological methods and Smale’s classification theorem of smooth simply-connected spin closed 5-manifolds to understand possible extremal transitions of Calabi-Yau threefolds through one with isolated singularities. For those that involve Kähler deformations that pinch some embedded del Pezzo surfaces, the adjunction formula implies that the corresponding isolated singularity in the resulting 7-space is indifferent of the global structure of the Calabi-Yau threefolds and hence can be studied locally in its own right. In this paper, we create many such local examples. Their global realization and the relation of the 6-dimensional link of the associated isolated singularity in the resulting 7-space to the phenomenon of enhanced gauge symmetry 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.