In general, an -manifold cannot be parametrized by an open subset of . But such a parametrization is always possible locally, so it is possible to define integrals over arbitrary manifolds by defining them as sums of integrals over collections of local parametrizations. Moreover, it is also possible to define parametrizations of -dimensional subsets for , and this makes it possible to define integrals of -forms. To make this precise, it is convenient to fix a standard domain in , usually a cube or a simplex. A -chain is a formal sum of smooth embeddings . That is, it is a collection of smooth embeddings, each of which is assigned an integer multiplicity. Each smooth embedding determines a -dimensional submanifold of . If the chain is