Abstract
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,ω) with c 1|π2(M)=[ω]|π2(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.
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Cornea, O., Ranicki, A. Rigidity and gluing for Morse and Novikov complexes. J. Eur. Math. Soc. 5, 343–394 (2003). https://doi.org/10.1007/s10097-003-0052-6
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DOI: https://doi.org/10.1007/s10097-003-0052-6