I think i am correct in saying that a supernova ($Type$ $II$) is caused by the collapse of the core of a giant star. This contraction of the core is stopped by the Pauli exclusion principle and the core becomes rigid. The outer layers now rebound of the rigid core and are thrown into space as a supernovae. But presumably this leaves behind a neutron star and not a black hole. For a black hole to form the Pauli exclusion principle would not be strong enough to stop the collapse? If so what causes the outer layers to rebound, as there is no ridged object for them to rebound off? What I am basically asking is when a black hole forms what mechanism causes the supernovae?
Yes, and Types Ib and Ic are also due to core collapse.
No, collapse begins when electron degeneracy pressure can no longer sustain the core, in other words when the core exceeds the Chandrasekhar limit. If the collapse is stopped it is due to neutron degeneracy pressure. If the core is greater in mass than the Tolman–Oppenheimer–Volkoff limit, neutron degeneracy pressure cannot prevent further collapse.
A black hole can form suddenly, or in a two step process. In the two step process, first there is a supernova and a neutron star forms, then material falls back into the neutron star and a black hole forms. If the star is extremely large, collapse into a black hole can occur suddenly without a supernova.