Perturbed gradient flow trees and A∞-algebra structures in Morse cohomology
著者
書誌事項
Perturbed gradient flow trees and A∞-algebra structures in Morse cohomology
（Atlantis studies in dynamical systems, v. 6）
Atlantis Press , Springer, c2018
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注記
Includes bibliographical references (p. 165-168) and index
内容説明・目次
内容説明
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.
目次
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.
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