  The paper focuses on various properties and applications of the homotopy
operator, which occurs in the Poincar\'{e} lemma. In the first part, an
abstract operator calculus is constructed, where the exterior derivative is an
abstract derivative and the homotopy operator plays the role of an abstract
integral. This operator calculus can be used to formulate abstract differential
equations. An example of the eigenvalue problem that resembles the fermionic
quantum harmonic oscillator is presented. The second part presents the dual
complex to the Dolbeault bicomplex generated by the homotopy operator on
complex manifolds.
