Rigidity and gluing for Morse and Novikov complexes. (English) Zbl 1052.57052

Cornea, Octav; Ranicki, Andrew

J. Eur. Math. Soc. (JEMS) 5, No. 4, 343-394 (2003).

The authors 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 \((M, \omega)\) satisfying \(c_1| _{\pi_2(M)}=[\omega]| _{\pi_2(M)} =0\). The rigidity results for these chain complexes state that the complex of a fixed generic function (Hamiltonian) is a retract of the complex of any other sufficiently \(C^0\)-close generic function (Hamiltonian). The gluing result is a Mayer-Vietoris-type formula for the chain complexes.

Reviewer: Michael Farber (Zürich)

MSC:

57R70	Critical points and critical submanifolds in differential topology
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
57R45	Singularities of differentiable mappings in differential topology

Keywords:

Morse complex; Novikov complex; Floer complex; Hamiltonian