Rigidity and gluing for Morse and Novikov complexes

Details Rigidity and gluing for Morse and Novikov complexes Rigidity and gluing for Morse and Novikov complexes

2001-07-30

arxiv.org/abs/math/0107221v3

J. Eur. Math. Soc. 5, 343-394 (2003)

Mathematics

Algebraic Topology

46 pages, LATEX file with XYPIC diagrams, and one .EPS file. Final version, accepted for publication by the Journal of the Eur

Scientific paper

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.

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