Conifolds are important objects in string theory. Brian Greene explains the physics of conifolds in Chapter 13 of his book "The Elegant Universe" - including the fact that the space can tear near the cone, and its topology can change.
A well-known example of a conifold is obtained as a deformed limit of the quintic - i.e. the quintic hypersurface in the projective space . The space has complex dimension equal to four, and therefore the space defined by the quintic (degree five) equation
for the homogeneous coordinate has complex dimension three; it is the most famous example of a Calabi-Yau manifold. If the complex structure parameter is chosen so that it is equal to one, the manifold described above becomes singular since the derivatives of the quintic polynomial in the equation vanish when all coordinates are equal or their ratios are certain fifth roots of unity. The neighbourhood of this singular point looks like a cone whose base is topologically just .
In the context of string theory, the geometrically singular conifolds can be shown to lead to completely smooth physics of strings. The divergences are "smeared out" by D3-branes wrapped on the shrinking three-sphere, as originally pointed out by Andrew Strominger. Andrew Strominger, together with Dave Morrison and Brian Greene, found that the topology near the conifold singularity can undergo a topological transition. It is believed that nearly all Calabi-Yau manifolds can be connected via these "critical transitions".