MORSE HOMOLOGY (Progress in Mathematics 111)

1995 ◽  
Vol 27 (5) ◽  
pp. 505-507
Author(s):  
Dietmar Salamon
Keyword(s):  
Morse Homology

scholarly journals Homotopy invariance of the Conley index and local Morse homology in Hilbert spaces

2017 ◽  
Vol 263 (11) ◽  
pp. 7162-7186 ◽  
Author(s):  
Marek Izydorek ◽  
Thomas O. Rot ◽  
Maciej Starostka ◽  
Marcin Styborski ◽  
Robert C.A.M. Vandervorst
Keyword(s):  
Hilbert Spaces ◽  
Conley Index ◽  
Homotopy Invariance ◽  
Morse Homology

scholarly journals Piunikhin-Salamon-Schwarz isomorphisms and spectral invariants for conormal bundle

2017 ◽  
Vol 102 (116) ◽  
pp. 17-47
Author(s):  
Jovana Duretic
Keyword(s):  
Morse Function ◽  
Triangle Inequality ◽  
Floer Homology ◽  
Zero Section ◽  
Spectral Invariants ◽  
Conormal Bundle ◽  
Morse Homology

We give a construction of the Piunikhin-Salamon-Schwarz isomorphism between the Morse homology and the Floer homology generated by Hamiltonian orbits starting at the zero section and ending at the conormal bundle. We also prove that this isomorphism is natural in the sense that it commutes with the isomorphisms between the Morse homology for different choices of the Morse function and the Floer homology for different choices of the Hamiltonian. We define a product on the Floer homology and prove triangle inequality for conormal spectral invariants with respect to this product.

scholarly journals Polytope Novikov homology

2021 ◽  
Vol 23 (4) ◽  
Author(s):  
Alessio Pellegrini
Keyword(s):  
Recent Result ◽  
Chain Complex ◽  
Finiteness Condition ◽  
Homotopy Equivalent ◽  
Closed Manifold ◽  
Novel Approach ◽  
Morse Homology ◽  
Chain Homotopy ◽  
Abelian Case ◽  
Novikov Ring

AbstractLet M be a closed manifold and $${\mathcal {A}} \subseteq H^1_{\mathrm {dR}}(M)$$ A ⊆ H dR 1 ( M ) a polytope. For each $$a \in {\mathcal {A}}$$ a ∈ A , we define a Novikov chain complex with a multiple finiteness condition encoded by the polytope $${\mathcal {A}}$$ A . The resulting polytope Novikov homology generalizes the ordinary Novikov homology. We prove that any two cohomology classes in a prescribed polytope give rise to chain homotopy equivalent polytope Novikov complexes over a Novikov ring associated with said polytope. As applications, we present a novel approach to the (twisted) Novikov Morse Homology Theorem and prove a new polytope Novikov Principle. The latter generalizes the ordinary Novikov Principle and a recent result of Pajitnov in the abelian case.

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