p-Permutation equivalences between blocks of group algebras

We extend the notion of a p-permutation equivalence between two p-blocks A and B of finite groups G and H, from the definition in [5] to a virtual p-permutation bimodule whose components have twisted diagonal vertices. It is shown that various invariants of A and B are preserved, including defect groups, fusion systems, and Külshammer-Puig classes. Moreover it is shown that p-permutation equivalences have additional surprising properties. They have only one constituent with maximal vertex and the set of p-permutation equivalences between A and B is finite (possibly empty). The paper uses new methods: a consequent use of module structures on subgroups of $G\times H$ arising from Brauer constructions which in general are not direct product subgroups, the necessary adaptation of the notion of tensor products between bimodules, and a general formula (stated in these new terms) for the Brauer construction of a tensor product of p-permutation bimodules.