Perturbed Gradient Flow Trees and A∞-algebra Structures in Morse Cohomology: 6 (Atlantis Studies in Dynamical Systems, 6) - Hardcover

Mescher, Stephan

Hardcover

ISBN 10: 3319765833 ISBN 13: 9783319765839

Publisher: Springer, 2018

Synopsis

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.

About this title

About the Author

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.

From the Back Cover

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.

Edition details

• Publisher: Springer
• Publication date: 2018
• Language: English
• ISBN 10: 3319765833
• ISBN 13: 9783319765839
• Binding: Hardcover
• Edition number: 1
• Number of pages: 196