  We consider quasicomplexes of pseudodifferential operators on a smooth
compact manifold without boundary. To each quasicomplex we associate a complex
of symbols. The quasicomplex is elliptic if this symbol complex is exact away
from the zero section. We prove that elliptic quasicomplexes are Fredholm.
Moreover, we introduce the Euler characteristic for elliptic quasicomplexes and
prove a generalization of the Atiyah-Singer index theorem.
