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

  • Stephan Mescher

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

Table of contents

  1. Front Matter
    Pages i-xxv
  2. Stephan Mescher
    Pages 1-7
  3. Stephan Mescher
    Pages 9-28
  4. Stephan Mescher
    Pages 29-37
  5. Stephan Mescher
    Pages 39-48
  6. Back Matter
    Pages 121-171

About this book

Introduction

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.

Keywords

Morse theory Morse homology Geometric topology A-infinity-algebras Differential topology

Authors and affiliations

  • Stephan Mescher
    • 1
  1. 1.Mathematisches InstitutUniversität LeipzigLeipzigGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-76584-6
  • Copyright Information Springer International Publishing AG, part of Springer Nature 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-76583-9
  • Online ISBN 978-3-319-76584-6
  • Buy this book on publisher's site
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