### The algebraic theory of finiteness obstruction.

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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 ${c}_{1}{{|}_{{\pi}_{2}\left(M\right)}=\left[\omega \right]|}_{{\pi}_{2}\left(M\right)}=0$. 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....

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