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Morse Theory, Discrete Morse Theory and Applications
Theoretical and Natural ScienceBy employing a specific class of smooth functions to study a space, Morse theory establishes deep connections between analysis and topology. It is a classical subject of pure mathematics, originally pioneered by Marston Morse in the 1920s. In this article, we use Morse theory to present a proof of an interesting result on the knots, known as the Fry ...
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COMBINATORIAL NOVIKOV–MORSE THEORY
International Journal of Mathematics, 2002In [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.
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manuscripta mathematica, 1991
It is shown that the assignment of a cell-complex to a Morse function on a compact smooth manifold can be achieved uniquely up to a contractible space of parameters and continuously in an appropriate sense with respect to these parameters among which being the Riemannian metrics on the manifold.
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It is shown that the assignment of a cell-complex to a Morse function on a compact smooth manifold can be achieved uniquely up to a contractible space of parameters and continuously in an appropriate sense with respect to these parameters among which being the Riemannian metrics on the manifold.
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2015
In this chapter we give an introduction to basic Morse theory. We define the notion of the Hessian of a smooth function at a critical point on a smooth manifold and show that if it is nondegenerate then there are local coordinates in which the function is equal to its second derivative. We also prove the Morse isotopy lemma which gives a criterion when
Richard H. Cushman, Larry M. Bates
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In this chapter we give an introduction to basic Morse theory. We define the notion of the Hessian of a smooth function at a critical point on a smooth manifold and show that if it is nondegenerate then there are local coordinates in which the function is equal to its second derivative. We also prove the Morse isotopy lemma which gives a criterion when
Richard H. Cushman, Larry M. Bates
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1988
The reader who is interested only in Morse theory for singular spaces or for nonproper Morse function may skip this chapter. We will consider the Morse theory of a composition $$ X\xrightarrow{R}Z\xrightarrow{f}\mathbb{R} $$ which will eventually be used (in Part II, Sects.
Mark Goresky, Robert MacPherson
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The reader who is interested only in Morse theory for singular spaces or for nonproper Morse function may skip this chapter. We will consider the Morse theory of a composition $$ X\xrightarrow{R}Z\xrightarrow{f}\mathbb{R} $$ which will eventually be used (in Part II, Sects.
Mark Goresky, Robert MacPherson
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1991
This chapter is the heart of the book. Quite a few spectacular theorems will be proved. The main ones are the famous convexity theorem of Atiyah [7] and Guillemin—Sternberg [63] which asserts that the image of a compact connected symplectic manifold under the momentum mapping of a Hamiltonian torus action is a convex polyhedron (this is Theorem ...
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This chapter is the heart of the book. Quite a few spectacular theorems will be proved. The main ones are the famous convexity theorem of Atiyah [7] and Guillemin—Sternberg [63] which asserts that the image of a compact connected symplectic manifold under the momentum mapping of a Hamiltonian torus action is a convex polyhedron (this is Theorem ...
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2004
The main goal of this chapter is to show how to construct a CW-complex that is homotopy equivalent to a given smooth manifold M using some special functions on M called “Morse” functions (Theorem 3.28). The CW-homology of the resulting CW-complex is isomorphic to the singular homology of M by Theorem 2.15, and hence it is independent of the choice of ...
Augustin Banyaga, David Hurtubise
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The main goal of this chapter is to show how to construct a CW-complex that is homotopy equivalent to a given smooth manifold M using some special functions on M called “Morse” functions (Theorem 3.28). The CW-homology of the resulting CW-complex is isomorphic to the singular homology of M by Theorem 2.15, and hence it is independent of the choice of ...
Augustin Banyaga, David Hurtubise
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1993
We introduce Morse homology theory by two main theorems about the existence of a canonical boundary operator associated to a given Morse function f and about the existence of canonical isomorphisms between each pair of such Morse complexes. These theorems appear as the essence from the theory on compactness, gluing and orientation which has been ...
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We introduce Morse homology theory by two main theorems about the existence of a canonical boundary operator associated to a given Morse function f and about the existence of canonical isomorphisms between each pair of such Morse complexes. These theorems appear as the essence from the theory on compactness, gluing and orientation which has been ...
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