  We obtain rigidity and gluing results for the Morse complex of a real-valued
Morse function as well as for the Novikov complex of a circle-valued Morse
function. A rigidity result is also proved for the Floer complex of a
hamiltonian defined on a closed symplectic manifold $(M,\omega)$ with
$c_{1}|_{\pi_{2}(M)}=[\omega]|_{\pi_{2}(M)}=0$. The rigidity results for these
complexes show that the complex of a fixed generic function/hamiltonian is a
retract of the Morse (respectively Novikov or Floer) complex of any other
sufficiently $C^{0}$ close generic function/hamiltonian. The gluing result is a
type of Mayer-Vietoris formula for the Morse complex. It is used to express
algebraically the Novikov complex up to isomorphism in terms of the Morse
complex of a fundamental domain. Morse cobordisms are used to compare various
Morse-type complexes without the need of bifurcation theory.
