  Parity is ubiquitous, but not always identified as a simplifying tool for
computations. Using parity, having in mind the example of the bosonic/fermionic
Fock space, and the framework of Z_2-graded (super) algebra, we clarify
relationships between the different definitions of supermanifolds proposed by
various people. In addition, we work with four complexes allowing an invariant
definition of divergence:
  - an ascending complex of forms, and a descending complex of densities on
real variables
  - an ascending complex of forms, and descending complex of densities on Grass
mann variables.
  This study is a step towards an invariant definition of integrals of
superfunctions defined on supermanifolds leading to an extension to infinite
dimensions. An application is given to a construction of supersymmetric Fock
spaces.
