Perturbed Gradient Flow Trees and A∞-algebra Structures in Morse Cohomology (Atlantis Studies in Dynamical Systems)

Mescher, Stephan

Springer Nature Switzerland AG（2018/12発売）

• 製本 Paperback:紙装版/ペーパーバック版／ページ数 171 p.
• 言語 ENG
• 商品コード 9783030095260

Full Description

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 beof interest to researchers in mathematics (geometry and topology), and to graduate students in mathematics with a basic command of the Morse theory.

Contents

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.