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

By Stephan Mescher

Book Overview

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.

• Format: Paperback
• Language: English
• ISBN: 3030095266
• ISBN13: 9783030095260
• Release Date: December 2018
• Publisher: Springer
• Length: 171 Pages
• Weight: 0.60 lbs.
• Dimensions: 0.4" x 8.4" x 9.3"