Celebratio Mathematica — Ranicki

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Works connected to Octavian Cornea

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O. Cornea and A. Ranicki: “Rigidity and gluing for Morse and Novikov complexes,” J. Eur. Math. Soc. 5 : 4 (2003), pp. 343–394. MR 2017851 Zbl 1052.57052 ArXiv math/0107221 article

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 \[ c_1\vert_{\pi_2(M)} = [\omega]\vert_{\pi_2(M)} = 0 .\] 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.