Arnold's spectral sequence

In the field of symplectic topology, Arnold's spectral sequence is a fundamental tool that relates the Floer homology of a Hamiltonian diffeomorphism to the quantum cohomology of the underlying symplectic manifold. It arises from a natural filtration of the Floer chain complex by the symplectic action functional. The spectral sequence provides a powerful computational method and reveals deep structural connections between the dynamics of Hamiltonian systems and the enumerative geometry of pseudo-holomorphic curves.

Introduction

The study of fixed points of Hamiltonian diffeomorphisms is a central theme in symplectic geometry, originating with Vladimir Arnold's work in the 1960s. The celebrated Arnold conjecture posits that the number of fixed points of a Hamiltonian diffeomorphism on a closed symplectic manifold ${\displaystyle (M,\omega )}$ is bounded below by the sum of the Betti numbers of ${\displaystyle M}$. To address this conjecture, Andreas Floer developed Floer homology in the late 1980s, an infinite-dimensional analogue of Morse homology. The chain complex in this theory is generated by the fixed points of the diffeomorphism, and its homology, ${\displaystyle HF_{*}(\phi )}$, is an invariant of the Hamiltonian isotopy class of ${\displaystyle \phi }$.

Arnold's spectral sequence emerges by endowing the Floer complex with an additional structure: a filtration based on the action of the periodic orbits corresponding to the fixed points. As is standard in homological algebra, a filtered chain complex gives rise to a spectral sequence, which can be thought of as an iterative machine for computing the homology of the original complex. The profound discovery, which unfolded through the work of several mathematicians in the 1990s, was the identification of the second page of this spectral sequence with the quantum cohomology ring of the manifold, ${\displaystyle QH^{*}(M)}$. Thus, the spectral sequence provides a direct algebraic bridge between dynamics (Floer homology) and quantum invariants (quantum cohomology).

Definition

The construction of the spectral sequence requires defining the filtered chain complex from which it arises.

Setup

Let ${\displaystyle (M,\omega )}$ be a closed (compact and without boundary) symplectic manifold. A Hamiltonian diffeomorphism ${\displaystyle \phi :M\to M}$ is the time-1 map of a flow generated by a time-dependent Hamiltonian function ${\displaystyle H:S^{1}\times M\to \mathbb {R} }$, where ${\displaystyle S^{1}=\mathbb {R} /\mathbb {Z} }$. The flow ${\displaystyle \phi _{t}^{H}}$ is defined by the ordinary differential equation:

${\displaystyle {\frac {d}{dt}}\phi _{t}^{H}(x)=X_{t}(\phi _{t}^{H}(x))}$

where the vector field ${\displaystyle X_{t}}$ is determined by the relation ${\displaystyle i_{X_{t}}\omega =-dH_{t}}$, and ${\displaystyle H_{t}(\cdot )=H(t,\cdot )}$.

The fixed points of ${\displaystyle \phi =\phi _{1}^{H}}$ correspond precisely to the 1-periodic orbits of the flow. Let ${\displaystyle {\mathcal {P}}(\phi )}$ denote the set of such orbits. For the theory to be well-defined, we assume the diffeomorphism ${\displaystyle \phi }$ is non-degenerate, meaning that for every fixed point ${\displaystyle x\in M}$ (i.e., ${\displaystyle \phi (x)=x}$), the derivative ${\displaystyle d\phi _{x}:T_{x}M\to T_{x}M}$ does not have 1 as an eigenvalue. This condition ensures that the set of 1-periodic orbits is discrete.

The Filtered Floer Chain Complex

The Floer chain complex, denoted ${\displaystyle CF_{*}(\phi ,H)}$, is a free abelian group generated by the 1-periodic orbits:

${\displaystyle CF_{k}(\phi ,H)=\bigoplus _{x\in {\mathcal {P}}(\phi ),\mu _{CZ}(x)=k}\mathbb {Z} \cdot x}$

The grading ${\displaystyle k}$ is given by the Conley–Zehnder index ${\displaystyle \mu _{CZ}(x)}$, an integer associated with the linearized flow along the orbit ${\displaystyle x}$.

The differential ${\displaystyle \partial :CF_{k}\to CF_{k-1}}$ is defined by counting rigid pseudo-holomorphic cylinders. Specifically, the coefficient of ${\displaystyle y}$ in ${\displaystyle \partial x}$ is given by the number of solutions ${\displaystyle u:\mathbb {R} \times S^{1}\to M}$ to Floer's equation:

${\displaystyle {\frac {\partial u}{\partial s}}+J_{t}\left({\frac {\partial u}{\partial t}}-X_{t}(u)\right)=0}$

subject to the boundary conditions ${\displaystyle \lim _{s\to +\infty }u(s,t)=x(t)}$ and ${\displaystyle \lim _{s\to -\infty }u(s,t)=y(t)}$. Here, ${\displaystyle J_{t}}$ is a family of ${\displaystyle \omega }$-compatible almost complex structures. For a generic choice of ${\displaystyle J_{t}}$, the moduli space of such cylinders of index 1 is a finite set of points. The homology of this complex, ${\displaystyle HF_{*}(\phi )}$, is independent of the choices of ${\displaystyle H}$ (within a fixed isotopy class) and ${\displaystyle J_{t}}$.

The key ingredient for the spectral sequence is the symplectic action functional${\displaystyle {\mathcal {A}}_{H}:{\mathcal {P}}(\phi )\to \mathbb {R} }$, defined as:

${\displaystyle {\mathcal {A}}_{H}(x)=-\int _{S^{1}}x^{*}\lambda -\int _{0}^{1}H(t,x(t))dt}$

where ${\displaystyle \lambda }$ is a primitive of ${\displaystyle \omega }$ (i.e., ${\displaystyle d\lambda =\omega }$). The differential ${\displaystyle \partial }$ decreases the action; that is, if the coefficient of ${\displaystyle y}$ in ${\displaystyle \partial x}$ is non-zero, then ${\displaystyle {\mathcal {A}}_{H}(y)<{\mathcal {A}}_{H}(x)}$. This allows us to define a filtration on the Floer complex. For any ${\displaystyle a\in \mathbb {R} }$, let

${\displaystyle CF_{*}^{a}(\phi ,H)=\bigoplus _{x\in {\mathcal {P}}(\phi ),{\mathcal {A}}_{H}(x)<a}\mathbb {Z} \cdot x}$

This gives a filtration ${\displaystyle ...\subset CF_{*}^{a}\subset CF_{*}^{b}\subset ...}$ for ${\displaystyle a<b}$.

The Spectral Sequence

A filtered chain complex ${\displaystyle (C_{*},\partial ,F)}$ gives rise to a spectral sequence ${\displaystyle \{E_{r},d_{r}\}}$. For Arnold's spectral sequence:

• The ${\displaystyle E_{1}}$ page is the homology of the associated graded complex. It is given by:

${\displaystyle E_{1}^{p,q}=H_{p+q}(CF_{*}^{p}/CF_{*}^{p-1})}$

where the filtration is indexed by action levels. A fundamental result identifies this page with the singular homology of the manifold ${\displaystyle M}$:

${\displaystyle E_{1}^{p,q}\cong H_{p+q}(M;\mathbb {Z} )}$

(The index ${\displaystyle p}$ is related to the action level, and ${\displaystyle q}$ completes the homological degree). The differential ${\displaystyle d_{1}}$ on this page relates homology classes at different action levels.

• The ${\displaystyle E_{2}}$ page is the homology of the ${\displaystyle E_{1}}$ page with respect to ${\displaystyle d_{1}}$. This page carries profound geometric meaning. It is isomorphic as a ring to the quantum cohomology of ${\displaystyle M}$:

${\displaystyle (E_{2}^{*,*},d_{2})\cong (QH^{*}(M),0)}$

The ring structure on ${\displaystyle E_{2}}$ is induced by the pair-of-pants product on Floer homology and corresponds to the quantum cup product on ${\displaystyle QH^{*}(M)}$. The quantum cup product is a deformation of the classical cup product on ${\displaystyle H^{*}(M)}$ defined by counting pseudo-holomorphic spheres (Gromov–Witten invariants).

• Convergence: The spectral sequence converges to the Floer homology of ${\displaystyle \phi }$. This is written as:

${\displaystyle E_{\infty }^{p,q}\Rightarrow HF_{p+q}(\phi )}$

This means that ${\displaystyle E_{\infty }^{p,q}}$ are the graded components of the associated graded module of a certain filtration on ${\displaystyle HF_{p+q}(\phi )}$. In particular, there is a relationship between the ranks: ${\displaystyle {\text{rank}}(HF_{*})={\text{rank}}(E_{\infty })\leq {\text{rank}}(E_{2})={\text{rank}}(E_{1})=\sum _{k}{\text{dim}}H_{k}(M)}$.

Properties

• Relation to Classical Topology: The ${\displaystyle E_{1}}$ page recovers the ordinary homology of the manifold, showing that at its first level of approximation, the structure is governed by the classical topology of ${\displaystyle M}$.
• Isomorphism with Quantum Cohomology: The identification ${\displaystyle E_{2}\cong QH^{*}(M)}$ is the central property. It implies that the differential ${\displaystyle d_{1}}$ captures exactly the information needed to deform the classical product on ${\displaystyle H^{*}(M)}$ into the quantum product.
• Multiplicative Structure: The spectral sequence is multiplicative. The product structure on each page ${\displaystyle E_{r}}$ is inherited from the pair-of-pants product on Floer homology. The product on ${\displaystyle E_{2}}$ is precisely the quantum cup product.
• Degeneration: For some symplectic manifolds, such as complex projective spaces (${\displaystyle \mathbb {CP} ^{n},\omega _{FS}}$), the spectral sequence degenerates at the ${\displaystyle E_{2}}$ page, meaning ${\displaystyle E_{2}=E_{\infty }}$. In such cases, the Floer homology is directly isomorphic to the quantum cohomology (as modules), making it much easier to compute.
• Higher Differentials: When the spectral sequence does not degenerate, the higher differentials ${\displaystyle d_{r}}$ for ${\displaystyle r\geq 2}$ contain subtle geometric information related to more complex configurations of pseudo-holomorphic curves.

Applications

1. Proof of the Arnold Conjecture: The spectral sequence provides one of the most elegant proofs of the Arnold conjecture. The number of fixed points of a non-degenerate ${\displaystyle \phi }$ is the rank of the Floer chain complex ${\displaystyle CF_{*}(\phi )}$. By basic homological algebra, this is greater than or equal to the rank of its homology, ${\displaystyle HF_{*}(\phi )}$. The spectral sequence gives the following inequalities:

   ${\displaystyle \#{\text{Fix}}(\phi )={\text{rank}}(CF_{*})\geq {\text{rank}}(HF_{*})=\sum _{p,q}{\text{rank}}(E_{\infty }^{p,q})\geq \sum _{p,q}{\text{rank}}(E_{1}^{p,q})=\sum _{k}{\text{dim}}H_{k}(M)}$

This chain of inequalities proves the conjecture.

1. Computation of Floer Homology: For manifolds where the quantum cohomology is known, the spectral sequence is a primary tool for computing or constraining Floer homology groups. The degeneration of the sequence for manifolds like ${\displaystyle \mathbb {CP} ^{n}}$ immediately determines their Hamiltonian Floer homology.
2. Probing Symplectic Invariants: The spectral sequence connects a dynamical invariant (${\displaystyle HF_{*}}$) with a geometric one (${\displaystyle QH^{*}}$). This allows information to be transferred between the two. For example, a non-trivial higher differential ${\displaystyle d_{r}}$ implies the existence of specific pseudo-holomorphic curves that would otherwise be difficult to detect.
3. Homological Mirror Symmetry: Arnold's spectral sequence is a key structure on the "A-side" of Kontsevich's homological mirror symmetry conjecture. Its mirror counterpart on the "B-side" (related to complex geometry) is the Hodge-to-de Rham spectral sequence. This analogy provides deep insight into the structure of mirror symmetry.

History

The intellectual origins of the spectral sequence lie in Vladimir Arnold's pioneering work on Hamiltonian dynamics in the 1960s and his formulation of the Arnold conjecture. The necessary analytical framework was established by Andreas Floer in the late 1980s with the invention of Floer homology. Floer's construction already implicitly contained the action filtration.

The full development of the spectral sequence and the crucial identification of its ${\displaystyle E_{2}}$ page with quantum cohomology was a major achievement of the 1990s, built upon the work of many mathematicians, including Alexander Givental, Helmut Hofer, Dietmar Salamon, and Kenji Fukaya. This discovery was a landmark event, as it unified two powerful but previously separate theories in symplectic geometry: Floer theory, which grew out of dynamics, and quantum cohomology, which grew out of the enumerative geometry of Gromov–Witten theory.

See Also

• Arnold conjecture
• Floer homology
• Hamiltonian mechanics
• Homological mirror symmetry
• Quantum cohomology
• Spectral sequence
• Symplectic manifold

References

Key Publications

• Floer, Andreas (1988). "Morse theory for Lagrangian intersections". Journal of Differential Geometry. **28** (3): 513–547. doi:10.4310/jdg/1214442477.
• Fukaya, Kenji; Oh, Yong-Geun; Ohta, Hiroshi; Ono, Kaoru (2009). Lagrangian Intersection Floer Theory: Anomaly and Obstruction. AMS/IP Studies in Advanced Mathematics. American Mathematical Society. ISBN 978-0-8218-4837-1.
• Givental, Alexander (1995). "Homological geometry and mirror symmetry". Proceedings of the International Congress of Mathematicians, Zürich 1994. Birkhäuser. pp. 472–480. ISBN 978-3-7643-5153-7.
• Hofer, Helmut; Salamon, Dietmar (1995). "Floer homology and Novikov rings". In Hofer, H.; Taubes, C. H.; Weinstein, A.; Zehnder, E. (eds.). The Floer Memorial Volume. Progress in Mathematics. Vol. 133. Birkhäuser. pp. 483–524. ISBN 978-3-7643-5044-8.
• McDuff, Dusa; Salamon, Dietmar (2012). J-holomorphic Curves and Symplectic Topology. American Mathematical Society Colloquium Publications. Vol. 52 (2nd ed.). American Mathematical Society. ISBN 978-0-8218-8746-2.

Further Reading

• **Foundational Concepts**: The formulation and early development of the field are credited to the works of Vladimir Arnold (1960s) on the Arnold conjecture and Andreas Floer (late 1980s) with the invention of Floer homology.
• **Spectral Sequence Development**: The identification and formal construction of the spectral sequence relating Floer homology and quantum cohomology are found in works from the 1990s by mathematicians including Alexander Givental, Helmut Hofer, Dietmar Salamon, and Kenji Fukaya.
• **Modern Expositions**: Detailed backgrounds and modern treatments are presented in texts such as J-holomorphic Curves and Symplectic Topology by McDuff and Salamon, as well as in various research surveys on the connections between symplectic topology and quantum cohomology.
• **Historical Perspectives**: For historical context, see essays such as "The beginnings of symplectic topology in Bochum in the early eighties" by Dietmar Salamon. Recent developments can be tracked in research related to topics like "The Arnold conjecture for singular symplectic manifolds."