Perturbed Gradient Flow Trees and A∞-algebra Structures in Morse Cohomology [electronic resource] /
Mescher, Stephan.author.
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.
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.
Global analysis (Mathematics).
Manifolds (Mathematics).
Dynamics.
Ergodic theory.
Complex manifolds.
Global Analysis and Analysis on Manifolds.
Dynamical Systems and Ergodic Theory.
Manifolds and Cell Complexes (incl. Diff.Topology).