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Perturbed Gradient Flow Trees and A∞-algebra Structures in Morse Cohomology

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Perturbed Gradient Flow Trees and A∞-algebra Structures in Morse Cohomology/ by Stephan Mescher.
Author:	Mescher, Stephan.
Description:	XXV, 171 p. 20 illus.online resource. :
Contained By:	Springer Nature eBook
Subject:	Global analysis (Mathematics). -
Online resource:	https://doi.org/10.1007/978-3-319-76584-6
ISBN:	9783319765846

Perturbed Gradient Flow Trees and A∞-algebra Structures in Morse Cohomology
Mescher, Stephan.

Perturbed Gradient Flow Trees and A∞-algebra Structures in Morse Cohomology[electronic resource] /by Stephan Mescher. - 1st ed. 2018. - XXV, 171 p. 20 illus.online resource. - Atlantis Studies in Dynamical Systems ;6. - Atlantis Studies in Dynamical Systems ;5.

1. Basics on Morse homology -- 2. Perturbations of gradient flow trajectories -- 3. Nonlocal generalizations -- 4. Moduli spaces of perturbed Morse ribbon trees -- 5. The convergence behaviour of sequences of perturbed Morse ribbon trees -- 6. Higher order multiplications and the A∞-relations -- 7. A∞-bimodule structures on Morse chain complexes -- A. Orientations and sign computations for perturbed Morse ribbon trees.

This book elaborates on an idea put forward by M. Abouzaid on equipping the Morse cochain complex of a smooth Morse function on a closed oriented manifold with the structure of an A∞-algebra by means of perturbed gradient flow trajectories. This approach is a variation on K. Fukaya’s definition of Morse-A∞-categories for closed oriented manifolds involving families of Morse functions. To make A∞-structures in Morse theory accessible to a broader audience, this book provides a coherent and detailed treatment of Abouzaid’s approach, including a discussion of all relevant analytic notions and results, requiring only a basic grasp of Morse theory. In particular, no advanced algebra skills are required, and the perturbation theory for Morse trajectories is completely self-contained. In addition to its relevance for finite-dimensional Morse homology, this book may be used as a preparation for the study of Fukaya categories in symplectic geometry. It will be of interest to researchers in mathematics (geometry and topology), and to graduate students in mathematics with a basic command of the Morse theory.

ISBN: 9783319765846

Standard No.: 10.1007/978-3-319-76584-6doiSubjects--Topical Terms:

1255807
Global analysis (Mathematics).

LC Class. No.: QA614-614.97

Dewey Class. No.: 514.74

Perturbed Gradient Flow Trees and A∞-algebra Structures in Morse Cohomology
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505 0 $a 1. Basics on Morse homology -- 2. Perturbations of gradient flow trajectories -- 3. Nonlocal generalizations -- 4. Moduli spaces of perturbed Morse ribbon trees -- 5. The convergence behaviour of sequences of perturbed Morse ribbon trees -- 6. Higher order multiplications and the A∞-relations -- 7. A∞-bimodule structures on Morse chain complexes -- A. Orientations and sign computations for perturbed Morse ribbon trees.
520 $a This book elaborates on an idea put forward by M. Abouzaid on equipping the Morse cochain complex of a smooth Morse function on a closed oriented manifold with the structure of an A∞-algebra by means of perturbed gradient flow trajectories. This approach is a variation on K. Fukaya’s definition of Morse-A∞-categories for closed oriented manifolds involving families of Morse functions. To make A∞-structures in Morse theory accessible to a broader audience, this book provides a coherent and detailed treatment of Abouzaid’s approach, including a discussion of all relevant analytic notions and results, requiring only a basic grasp of Morse theory. In particular, no advanced algebra skills are required, and the perturbation theory for Morse trajectories is completely self-contained. In addition to its relevance for finite-dimensional Morse homology, this book may be used as a preparation for the study of Fukaya categories in symplectic geometry. It will be of interest to researchers in mathematics (geometry and topology), and to graduate students in mathematics with a basic command of the Morse theory.
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