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2018 | Book

Basics on Morse Homology Read chapter Read first chapter

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

Author: Dr. Stephan Mescher

Publisher: Springer International Publishing

Book Series : Atlantis Series in Dynamical Systems

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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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 be of interest to researchers in mathematics (geometry and topology), and to graduate students in mathematics with a basic command of the Morse theory.

Table of Contents

Frontmatter
Chapter 1. Basics on Morse Homology
Abstract
This brief chapter is intended to provide the reader with an overview of the construction of Morse (co)homology for finite-dimensional manifolds. We present the main notions and results in a concise way and give references to detailed presentations and proofs whenever appropriate. Moreover, we establish some notation that will be employed throughout this book. Except for notational conventions, a reader familiar with Morse homology might skip this chapter without disadvantages. There are several detailed and recommendable references on Morse homology, see e.g. the textbooks Schwarz, Morse homology, Birkhäuser, Basel, 1993, [Sch93], Banyaga et al. Lectures on Morse homology, Kluwer Academic Publishers Group, Dordrecht, 2004, [BH04], Jost, Riemannian geometry and geometric analysis, Springer, Berlin, 2008, [Jos08, Chap. 7], Nicolaescu, An invitation to Morse theory, Springer, New York, 2011, [Nic11] or Audin and Damian, Morse theory and Floer homology, Springer, London, [AD14] as well as the set of lecture notes Hutchings, Lecture notes on Morse homology (with an eye towards Floer theory and pseudoholomorphic curves), UC Berkeley, 2002, [Hut02] and the article Weber, Expo Math, 24(2), 127–159, 2006, [Web06].
Stephan Mescher
Chapter 2. Perturbations of Gradient Flow Trajectories
Abstract
As we have mentioned in the introduction, a crucial step towards defining an \(A_\infty \)-algebra structure on the Morse cochain complex of a single Morse function is to consider perturbed Morse trajectories. More precisely, we want to dicuss curves which do not satisfy a negative gradient flow equation, but a perturbed negative gradient flow equation of the form \(\dot{\gamma }(s) + \nabla ^g f\circ \gamma (s) + Z(s,\gamma (s)) = 0 , \)where we pick up the notation from Chap. 1 and where Z is a suitable time-dependent vector field.
Stephan Mescher
Chapter 3. Nonlocal Generalizations
Abstract
In this chapter we derive a more general, and in a certain sense nonlocal, version of the transversality results from the previous chapter. More precisely, we want to derive a transversality theorem, our Theorem 3.10, in which:
  • all three types of trajectories from the previous chapter are considered at once,
  • the perturbations may depend on the interval length parameters of all of the finite-length trajectories involved,
  • the limiting behaviour of the perturbations for length parameters becoming infinitely large can be controlled a priori.
Stephan Mescher
Chapter 4. Moduli Spaces of Perturbed Morse Ribbon Trees
Abstract
In the remainder of of this book, we will discuss perturbed Morse ribbon trees, which can be interpreted as continuous maps from a tree to the manifold M which edgewise fulfill perturbed negative gradient flow equations. In this chapter, we will make this notion precise in terms of the constructions of Chaps. 2 and 3. Moreover, we will apply the nonlocal transversality result Theorem 3.​10 to equip moduli spaces of perturbed Morse ribbon trees with the structures of finite-dimensional manifolds of class \(C^{n+1}\).
Stephan Mescher
Chapter 5. The Convergence Behaviour of Sequences of Perturbed Morse Ribbon Trees
Abstract
Having constructed moduli spaces of perturbed Morse ribbon trees in the previous chapter, we want to investigate sequential compactness properties of these moduli spaces. Our starting point is the consideration of certain sequential compactness results for spaces of perturbed Morse trajectories of the three different types we introduced in Chap. 2. We will show that in all three cases, every sequence in the respective moduli space without a convergent subsequence has a subsequence that converges geometrically against a family of trajectories. The notion of geometric convergence will be made precise in Sects. 5.1 and 5.2.
Stephan Mescher
Chapter 6. Higher Order Multiplications and the -Relations
Abstract
We continue by taking a closer look at zero- and one-dimensional moduli spaces of perturbed Morse ribbon trees. The results from Chap. 5 enable us to show that zero-dimensional moduli spaces are in fact finite sets. This basic observation allows us to define homomorphisms \(C^*(\,f)^{\otimes d} \rightarrow C^*(\, f)\) for every \(d \ge 2\) via counting elements of these zero-dimensional moduli spaces. After constructing these higher order multiplications explicitly, we will study the compactifications of one-dimensional moduli spaces of perturbed Morse ribbon trees. The results from Chap. 5 imply that one-dimensional moduli spaces can be compactified to one-dimensional manifolds with boundary, and we will explicitly describe their boundaries. If we impose an additional consistency condition on the chosen perturbations, the boundary spaces will coincide with product of zero-dimensional moduli spaces of perturbed Morse ribbon trees.
Stephan Mescher
Chapter 7. -bimodule Structures on Morse Chain Complexes
Abstract
Having established \(A_\infty \)-algebra structures on Morse cochain complexes \(C^*(f)\), we consider their duals, namely Morse chain complexes. We will use this brief final chapter to show that these possess the structures of \(A_\infty \)-bimodules over the \(A_\infty \)-algebra \(C^*(f)\).
Stephan Mescher
Backmatter
Metadata
Title
Perturbed Gradient Flow Trees and A∞-algebra Structures in Morse Cohomology
Author
Dr. Stephan Mescher
Copyright Year
2018
Electronic ISBN
978-3-319-76584-6
Print ISBN
978-3-319-76583-9
DOI
https://doi.org/10.1007/978-3-319-76584-6

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