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.

COMBINATORIAL NOVIKOV–MORSE THEORY

2002 ◽  
Vol 13 (04) ◽  
pp. 333-368 ◽  
Author(s):  
ROBIN FORMAN
Keyword(s):  
Differential Form ◽  
Morse Theory ◽  
Morse Function ◽  
Cw Complex ◽  
The Way

In [7, 8, 9], we developed a combinatorial Morse theory which can be applied to any CW complex. In [25, 26] Novikov presented a generalization of classical Morse theory in which the Morse function is replaced by a closed 1-forms. In this paper we extend our combinatorial Morse theory to include a combinatorial analog of Novikov's theory. Along the way we introduce the notion of a combinatorial differential form which is well-suited to our work, and which may have other applications.

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 ON MODULI SPACES AND CW STRUCTURES ARISING FROM MORSE THEORY ON HILBERT MANIFOLDS

2010 ◽  
Vol 02 (04) ◽  
pp. 469-526 ◽  
Author(s):  
LIZHEN QIN
Keyword(s):  
Dynamical Systems ◽  
Moduli Spaces ◽  
Morse Theory ◽  
Flow Lines ◽  
Morse Functions ◽  
Negative Gradient

This paper proves some results on negative gradient dynamical systems of Morse functions on Hilbert manifolds. It contains the compactness of flow lines, manifold structures of certain compactified moduli spaces, orientation formulas, and CW structures of the underlying manifolds.

scholarly journals Spaces which Invert Weak Homotopy Equivalences

2018 ◽  
Vol 62 (2) ◽  
pp. 553-558
Author(s):  
Jonathan Ariel Barmak
Keyword(s):  
Empty Space ◽  
Weak Equivalence ◽  
Homotopy Equivalence ◽  
Homotopy Classes ◽  
Cw Complex ◽  
Cw Complexes ◽  
Weak Homotopy ◽  
Weak Homotopy Equivalence

AbstractIt is well known that if X is a CW-complex, then for every weak homotopy equivalence f : A → B, the map f* : [X, A] → [X, B] induced in homotopy classes is a bijection. In fact, up to homotopy equivalence, only CW-complexes have that property. Now, for which spaces X is f* : [B, X] → [A, X] a bijection for every weak equivalence f? This question was considered by J. Strom and T. Goodwillie. In this note we prove that a non-empty space inverts weak equivalences if and only if it is contractible.

On the Moduli Space of a Spherical Polygonal Linkage

1999 ◽  
Vol 42 (3) ◽  
pp. 307-320 ◽  
Author(s):  
Michael Kapovich ◽  
John J. Millson
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Level Sets ◽  
Morse Function ◽  
Great Circle ◽  
Winding Number ◽  
Wall Crossing ◽  
Polygonal Linkage ◽  
Spherical Linkages ◽  
Wall Crossing Formula

AbstractWe give a “wall-crossing” formula for computing the topology of the moduli space of a closed n-gon linkage on 𝕊2. We do this by determining the Morse theory of the function ρn on the moduli space of n-gon linkages which is given by the length of the last side—the length of the last side is allowed to vary, the first (n − 1) side-lengths are fixed. We obtain a Morse function on the (n − 2)-torus with level sets moduli spaces of n-gon linkages. The critical points of ρn are the linkages which are contained in a great circle. We give a formula for the signature of the Hessian of ρn at such a linkage in terms of the number of back-tracks and the winding number. We use our formula to determine the moduli spaces of all regular pentagonal spherical linkages.

Floer type cohomology on cosymplectic manifolds

2015 ◽  
Vol 12 (07) ◽  
pp. 1550082 ◽  
Author(s):  
Yong Seung Cho ◽  
Young Do Chai
Keyword(s):  
Moduli Space ◽  
Loop Space ◽  
Type Theorem ◽  
Gradient Flow ◽  
Universal Covering ◽  
Cochain Complex ◽  
Action Functional ◽  
Flow Lines ◽  
Universal Covering Space ◽  
Cosymplectic Manifolds

We investigate a Floer type cohomology on cosymplectic manifolds M. To do this, we study a symplectic type action functional on the universal covering space of the loop space of contractible loops in M and the moduli space of gradient flow lines of the functional. The cochain complex induced by the critical points of the functional produces Floer type cohomology of M which is naturally isomorphic to a quantum type cohomology of M. We have an Arnold type theorem for Hamiltonian cosymplectomorphisms on compact semipositive cosymplectic manifolds. As an example, we consider the product of a Calabi–Yau 3-fold and the unit circle.

scholarly journals Prescribing scalar curvatures: non compactness versus critical points at infinity

2019 ◽  
Vol 4 (1) ◽  
pp. 51-82 ◽  
Author(s):  
Martin Mayer
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Critical Points ◽  
Gradient Flow ◽  
Slight Modification ◽  
Positive Function ◽  
Flow Lines ◽  
Critical Points At Infinity

Abstract We illustrate an example of a generic, positive function K on a Riemannian manifold to be conformally prescribed as the scalar curvature, for which the corresponding Yamabe type L2-gradient flow exhibits non compact flow lines, while a slight modification of it is compact.

Homotopy Groups and CW Complexes

2020 ◽  
pp. 11-20
Author(s):  
Loring W. Tu
Keyword(s):  
Topological Space ◽  
Algebraic Topology ◽  
Weak Topology ◽  
Homotopy Theory ◽  
Fiber Bundles ◽  
Homotopy Groups ◽  
Cw Complex ◽  
Continuous Maps ◽  
Cw Complexes ◽  
Set Of Points

This chapter discusses some results about homotopy groups and CW complexes. Throughout this book, one needs to assume a certain amount of algebraic topology. A CW complex is a topological space built up from a discrete set of points by successively attaching cells one dimension at a time. The name CW complex refers to the two properties satisfied by a CW complex: closure-finiteness and weak topology. With continuous maps as morphisms, the CW complexes form a category. It turns out that this is the most appropriate category in which to do homotopy theory. The chapter also looks at fiber bundles.

scholarly journals The space of sections of a sphere-bundle I

1986 ◽  
Vol 29 (3) ◽  
pp. 383-403 ◽  
Author(s):  
M. C. Crabb ◽  
W. A. Sutherland
Keyword(s):  
Riemannian Metric ◽  
Sphere Bundle ◽  
Systematic Account ◽  
Cw Complex

Throughout this paper X will be a finite connected CW-complex of dimension m, and ξ will be a real (n + l)-plane bundle over X(n >0) equipped with a Riemannian metric. We aim to give a systematic account of the space ГSξ of sections of the sphere-bundle Sξ.

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