Canonical models of filtered $A_\infty$-algebras and Morse complexes

Bott-Morse情形下$A_{\infty}$-运算的符号约定

Sign Convention for $A_{\infty}$-Operations in Bott-Morse Case

本文研究了Bott-Morse设定下Lagrangian Floer理论中过滤$A_{\infty}$-公式的符号与定向问题，定义了Fukaya范畴中的运算并验证了相关公式。

扭曲Morse复形

Twisted Morse complexes

本文研究闭有限维光滑流形上的扭曲Morse同调与上同调，证明相关定理与多类同调/上同调的同构关系，并通过多个计算实例验证了扭曲Morse复形的有效性。

Perturbed gradient flow trees and $A_\infty$-algebra structures on Morse cochain complexes

We elaborate on an idea of M. Abouzaid of equipping the Morse cochain complex of a smooth Morse function on a closed oriented manifold with the structure of an $A_\infty$-algebra. This is a variation on K. Fukaya's definition of Morse-$A_\infty$-categories for closed oriented manifolds involving families of Morse functions. The purpose of this article is to provide a coherent and detailed treatment of Abouzaid's approach including a discussion of all relevant analytic notions and results.

Duality in filtered Floer-Novikov complexes

We prove that a certain bilinear pairing (analagous to the Poincare-Lefschetz intersection pairing) between filtered sub- and quotient complexes of a Floer-type chain complex and of its "opposite complex" is always nondegenerate on homology. This implies a duality relation for the Oh-Schwarz-type spectral invariants of these complexes which (in Hamiltonian Floer theory) was established in the special case that the period map has discrete image by Entov and Polterovich. The duality relation served as a key lemma in Entov and Polterovich's construction of a Calabi quasimorphism on certain ration…

过滤Fukaya范畴

Filtered Fukaya categories

本文将Fukaya范畴的自然弱过滤结构升级为真正的过滤结构，主要利用Morse-Bott或“簇”模型及特定摄动数据，并构造了延拓$A_\infty$-函子。

Morse 复合体是一个 ∞-函子

The Morse complex is an $\infty$-functor

本文展示了一个紧致李半群的Morse复合体可以赋予f-bialgebra结构，并证明这一赋值定义了一个∞-函子，从而得到两个将闭合光滑流形映射到其Morse复合体的∞-函子，以及带有紧致李群作用的流形到其Morse复合体的u-bimodule结构。