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

Stephan Mescher (Autor)

Buch | Softcover

XXV, 171 Seiten

2018
Springer International Publishing (Verlag)
9783030095260 (ISBN)

Artikel merken

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.

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.

1. Basics on Morse homology.- 2. Perturbations of gradient flow trajectories.- 3. Nonlocal generalizations.- 4. Moduli spaces of perturbed Morse ribbon trees.- 5. The convergence behaviour of sequences of perturbed Morse ribbon trees.- 6. Higher order multiplications and the A -relations.- 7. A -bimodule structures on Morse chain complexes.- A. Orientations and sign computations for perturbed Morse ribbon trees.

Erscheinungsdatum	05.03.2022
Reihe/Serie	Atlantis Studies in Dynamical Systems
Zusatzinfo	XXV, 171 p. 20 illus.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Gewicht	314 g
Themenwelt	Mathematik / Informatik ► Informatik ► Theorie / Studium
	Mathematik / Informatik ► Mathematik ► Analysis
	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	A-infinity-algebras • Differential topology • geometric topology • Morse Homology • Morse Theory
ISBN-13	9783030095260 / 9783030095260
Zustand	Neuware