scholarly journals Morse–Conley–Floer homology

2014 ◽  
Vol 06 (03) ◽  
pp. 305-338 ◽  
Author(s):  
T. O. Rot ◽  
R. C. A. M. Vandervorst
Keyword(s):  
Morse Function ◽  
Floer Homology ◽  
Gradient Flow ◽  
Homology Theory ◽  
Closed Manifold ◽  
Invariant Sets ◽  
Invariance Properties ◽  
Singular Homology ◽  
Morse Homology ◽  
Smooth Closed Manifold

The gradient flow of a Morse function on a smooth closed manifold generates, under suitable transversality assumptions, the Morse–Smale–Witten complex. The associated Morse homology is an invariant for the manifold, and equals the singular homology, which yields the classical Morse relations. In this paper we define Morse–Conley–Floer homology, which is an analogous homology theory for isolated invariant sets of smooth, not necessarily gradient-like, flows. We prove invariance properties of the Morse–Conley–Floer homology, and show how it gives rise to the Morse–Conley relations.

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.

Framed Stratified Sets in Morse Theory

2002 ◽  
Vol 54 (2) ◽  
pp. 396-416
Author(s):  
André Lebel
Keyword(s):  
Moduli Spaces ◽  
Morse Theory ◽  
Morse Function ◽  
Gradient Flow ◽  
Riemannian Metric ◽  
Closed Manifold ◽  
Flow Lines ◽  
Cw Complex ◽  
Cw Complexes

AbstractIn this paper, we present a smooth framework for some aspects of the “geometry of CW complexes”, in the sense of Buoncristiano, Rourke and Sanderson [3]. We then apply these ideas to Morse theory, in order to generalize results of Franks [5] and Iriye-Kono [8].More precisely, consider a Morse function f on a closed manifold M. We investigate the relations between the attaching maps in a CW complex determined by f, and the moduli spaces of gradient flow lines of f, with respect to some Riemannian metric on M.

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.

scholarly journals Tau invariants for balanced spatial graphs

2020 ◽  
Vol 29 (09) ◽  
pp. 2050066
Author(s):  
Katherine Vance
Keyword(s):  
Lower Bound ◽  
Floer Homology ◽  
Chain Complex ◽  
Homology Theory ◽  
Heegaard Floer Homology ◽  
Combinatorial Proof ◽  
Knot Concordance ◽  
Spatial Graphs ◽  
Heegaard Floer ◽  
Definition Of

In 2003, Ozsváth and Szabó defined the concordance invariant [Formula: see text] for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of [Formula: see text] for knots in [Formula: see text] and a combinatorial proof that [Formula: see text] gives a lower bound for the slice genus of a knot. Recently, Harvey and O’Donnol defined a relatively bigraded combinatorial Heegaard Floer homology theory for transverse spatial graphs in [Formula: see text], extending HFK for knots. We define a [Formula: see text]-filtered chain complex for balanced spatial graphs whose associated graded chain complex has homology determined by Harvey and O’Donnol’s graph Floer homology. We use this to show that there is a well-defined [Formula: see text] invariant for balanced spatial graphs generalizing the [Formula: see text] knot concordance invariant. In particular, this defines a [Formula: see text] invariant for links in [Formula: see text]. Using techniques similar to those of Sarkar, we show that our [Formula: see text] invariant is an obstruction to a link being slice.

Singular homology theory in the category of soft topological spaces

2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Sadi Bayramov ◽  
Cigdem Gündüz (Aras) ◽  
Leonard Mdzinarishvili
Keyword(s):  
Homology Group ◽  
Homology Theory ◽  
Topological Spaces ◽  
Homology Groups ◽  
Relative Homology ◽  
Singular Homology

AbstractIn the category of soft topological spaces, a singular homology group is defined and the homotopic invariance of this group is proved [Internat. J. Engrg. Innovative Tech. (IJEIT) 3 (2013), no. 2, 292–299]. The first aim of this study is to define relative homology groups in the category of pairs of soft topological spaces. For these groups it is proved that the axioms of dimensional and exactness homological sequences hold true. The axiom of excision for singular homology groups is also proved.

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