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

1. Front Matter

   Pages i-xxv

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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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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.

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

• Mathematisches Institut, Universität Leipzig, Leipzig, Germany

  Stephan Mescher

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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