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

  • Book
  • © 2018

Overview

Authors:
  1. Stephan Mescher

    1. Mathematisches Institut, Universität Leipzig, Leipzig, Germany

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  • Introduces a self-contained discussion of an aspect of Morse theory that has remained largely overlooked
  • Includes an accessible introduction to a part of Morse theory that is closely related to algebraic topology
  • Offers a useful preparation for the study of Fukaya categories in Lagrangian Floer theory

Part of the book series: Atlantis Studies in Dynamical Systems (ASDS, volume 6)

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-xxv
    Download chapter PDF

  2. Basics on Morse Homology

    • Stephan Mescher
    Pages 1-7

  3. Perturbations of Gradient Flow Trajectories

    • Stephan Mescher
    Pages 9-28

  4. Nonlocal Generalizations

    • Stephan Mescher
    Pages 29-37

  5. Moduli Spaces of Perturbed Morse Ribbon Trees

    • Stephan Mescher
    Pages 39-48

  6. The Convergence Behaviour of Sequences of Perturbed Morse Ribbon Trees

    • Stephan Mescher
    Pages 49-72

  7. Higher Order Multiplications and the \(A_\infty \)-Relations

    • Stephan Mescher
    Pages 73-107

  8. \(A_\infty \)-bimodule Structures on Morse Chain Complexes

    • Stephan Mescher
    Pages 109-120

  9. Back Matter

    Pages 121-171
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Keywords

About this book

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.



Authors and Affiliations

  • Mathematisches Institut, Universität Leipzig, Leipzig, Germany

    Stephan Mescher

About the author

Dr. Stephan Mescher is a Research Fellow at the University of Leipzig. He graduated with a degree in Mathematics from Bielefeld University in 2008 and obtained his Ph.D. at the University of Leipzig in 2017, supervised by Prof. Matthias Schwarz.

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