Hamiltonian Floer homology
Hamilton 系统周期解的 Floer 同调
Meaning in context
In the closed symplectic setting where the symplectic form and first Chern class vanish on sphere classes, the homology over the field with two elements of the complex generated by nondegenerate contractible one-periodic Hamiltonian orbits, whose differential counts isolated finite-energy Floer cylinders modulo translation after regular choices.
在闭辛流形上，若辛形式和第一 Chern 类在球面类上均为零，取非退化的可缩一周期 Hamilton 轨道生成二元域上的链复形；经正则选择后，以模平移的孤立有限能量 Floer 圆柱计数定义边界，其同调即本条所指对象。
Example
For the standard symplectic two-torus and a sufficiently small suitable Morse Hamiltonian, the ungraded Floer homology has total -dimension .
Usage and distinctions
This scoped construction concerns Hamiltonian periodic orbits. Lagrangian Floer homology uses different generators; manifolds with sphere bubbling require a broader construction and additional hypotheses.
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Recommended reading in this subject · Preparation & skip guidance
Definition sources
- Will J. Merry — Floer theory§2, Definition 2.24 and Theorem 2.25; §4, Floer equation; §9, Invariance of Floer homology · Checked 2026-10-14
Translation sources
- Jiang Meiyue — Peking University, （辛）Floer 同调简介 report notice（辛）Floer 同调简介，主要内容 item 3：Hamilton 系统周期解的 Floer 同调; item 4 distinguishes Lagrange submanifold intersection theory · Checked 2026-10-14