Journal of the European Mathematical Society
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Rigidity and gluing for Morse and Novikov complexes
Octav Cornea (1) and Andrew Ranicki (2)(1) Department of Mathematics and Statistics, University of Montréal, CP 6128 Succ. Centre Ville, H3C 3J7, MONTRÉAL, QC, CANADA
(2) School of Mathematics, University of Edinburgh, Mayfield Road, EH9 3JZ, EDINBURGH, UNITED KINGDOM
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,ohgr) with c1|pgr2(M)=[ohgr]|pgr2(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 C0 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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