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

Stephan Mescher (Autor)

Buch | Hardcover

XXV, 171 Seiten

2018
Springer International Publishing (Verlag)
978-3-319-76583-9 (ISBN)

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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.

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	18.05.2018
Reihe/Serie	Atlantis Studies in Dynamical Systems
Zusatzinfo	XXV, 171 p. 20 illus.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Gewicht	467 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-10	3-319-76583-3 / 3319765833
ISBN-13	978-3-319-76583-9 / 9783319765839
Zustand	Neuware