Perturbed gradient flow trees and \(A_\infty\)-algebra structures in Morse cohomology
The aim of the book is to give a detailed and self-contained exposition of \(A_\infty\)-algebra structures on Morse cochain complexes of closed manifolds, following works of Kenji Fukaya amd Mohammed Abouzaid. A detailed construction of higher order multiplications on the Morse cochains of a fixed Morse function using the perturbation methods of Abouzaid is described. Although the subject is rather special, it is a piece of good and important mathematics. There are so many computational details that the author decided to write a book rather than a long article as his first official publication. The origin of the author's interest on the subject is in his Ph.D. dissertation under Matthias Schwarz, well known expert on Morse homology. See also the author's preprints on the arXiv. As the author writes in the introduction: The basic ideas of this book are implicitly contained in [\textit{M. Abouzaid}, J. Differ. Geom. 87, No. 1, 1--80 (2011; Zbl 1228.57015)]. The analytic approach may be helpful for better understanding the underlying phenomena. The book is more or less self-contained (many references are given), however a good knowledge of Morse theory would be helpful. The book consists of 7 chapters: 1. Basics on Morse homology (no proofs here) 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 \(A_\infty\)-relations 7. \(A_\infty\)-bimodule structures on Morse chain complexes and rather long Appendix A: Orientations and sign computations for perturbed Morse ribbon trees.
scientific article; zbMATH DE number 6855995