Floer Homology¶

Job: infinite-dimensional Morse theory for the symplectic action functional — package periodic orbits and Lagrangian intersections into homology theories that detect Hamiltonian dynamics and prove Arnold-type conjectures.

Core¶

Setup (Hamiltonian Floer)¶

Let $(M,\omega)$ be closed symplectic, $H:S^1\times M\to\mathbb{R}$ nondegenerate (1-periodic orbits nondegenerate), and $J_t$ a compatible almost complex structure. Contractible 1-periodic orbits of $X_{H_t}$ are critical points of the action functional on the cover of the contractible loop space: $$ \mathcal{A}H(\tilde x)=\intH_t(x(t))\,dt, $$ where }}\bar x^*\omega - \int_{S^1$\bar x$ fills $x$ and $\tilde x$ records the homotopy class / Novikov data when $[\omega]$ is not aspherical.

Floer equation (negative gradient flow of $\mathcal{A}_H$ in the $L^2$ metric from $\omega(\cdot,J\cdot)$): $$ \partial_s u + J_t(u)\bigl(\partial_t u - X_{H_t}(u)\bigr)=0,\qquad u:\mathbb{R}\times S^1\to M. $$ Solutions are $J$-holomorphic sections of $\mathbb{R}\times S^1\times M$ with Hamiltonian perturbation — energy $E(u)=\int|\partial_s u|^2$ equals action drop.

Floer chain complex¶

Ingredient	Role
Generators	nondeg. contractible 1-periodic orbits (graded by Conley–Zehnder index $\mu_{\mathrm{CZ}}$)
Differential	count isolated (mod $\mathbb{R}$-translation) index-$1$ Floer cylinders
Coefficients	$\mathbb{Z}/2$ or Novikov ring $\Lambda$ when bubbling / area filtrations require it
Compactness	Gromov–Floer: breaking + sphere bubbling; rule out bubbles via monotonicity / virtual techniques / polyfolds / Kuranishi

Theorem (Floer). Under standard monotonicity or semipositivity (or with virtual fundamental cycles in general), $HF_*(M,H,J)$ is well-defined and independent of generic $(H,J)$ up to canonical isomorphism.

PSS isomorphism¶

Piunikhin–Salamon–Schwarz: for closed monotone (or more generally, with suitable Novikov coefficients) symplectic manifolds, $$ HF_(M,H)\cong H_(M;\Lambda) $$ as modules over the (Novikov) coefficient ring — Floer homology recovers ordinary homology. The PSS map counts spiked disks (Morse flowlines glued to Floer half-cylinders).

Consequence. Existence of enough periodic orbits: rank of $HF$ forces critical points of $\mathcal{A}_H$.

Arnold conjecture (fixed points)¶

Conjecture / theorem package (Arnold). A nondegenerate Hamiltonian diffeomorphism $\varphi_H$ on closed $(M,\omega)$ has at least as many fixed points as a Morse function has critical points: $$ #\operatorname{Fix}(\varphi_H)\ge \sum_i \operatorname{rank} H_i(M;F) $$ (nondegenerate case; cup-length / LS category bounds in degenerate cases).

Floer proof sketch. Fixed points of $\varphi_H$ $\leftrightarrow$ 1-periodic orbits of $H$. Nondegenerate $\Rightarrow$ Floer complex defined; $HF\cong H_*(M)$ (PSS) $\Rightarrow$ at least $\sum b_i$ generators. Degenerate case: cup-length estimates via pair-of-pants product / quantum product structures, or Lusternik–Schnirelmann theory in the action filtration.

Status (orientation). Settled for large classes: monotone, aspherical, complex projective, toric, and — via virtual techniques — broad closed symplectic manifolds (Fukaya–Ono, Liu–Tian, Hofer–Wysocki–Zehnder polyfolds, Pardon, Abouzaid–McLean–Smith, etc., with ongoing foundational refinements). Always cite the exact technical setup when using the strongest statement.

Sibling theories (pointers)¶

Theory	Counts
Lagrangian Floer $HF(L_0,L_1)$	intersections $L_0\cap\varphi(L_1)$; $A_\infty$ Fukaya category
Embedded contact homology / HF$^+$	contact 3-manifold invariants; Taubes–ECH = Seiberg–Witten
Symplectic homology $SH(W)$	Liouville domains; wrapped Fukaya input
Quantum homology	Gromov–Witten deformed product on $H_*(M)$

Lagrangian Floer is the backbone of mirror symmetry (Kontsevich HMS).

See it¶

Area-preserving maps of surfaces. On $(S^2,\omega)$, every Hamiltonian diffeomorphism has at least $2$ fixed points (nondeg.: $\ge 2$; degenerate: still $\ge 2$ by Franks / recent strengthenings). Floer/PSS gives $\ge \sum b_i(S^2)=2$ in the nondegenerate case — matching Morse theory on $S^2$. On $T^{2n}$, Conley–Zehnder and Floer recover the classical Conley–Zehnder theorem: nondeg. Ham. maps have $\ge 2^{2n}$ fixed points.

Action spectrum. Filtered Floer homology $HF^{(a,b)}$ and spectral invariants $c(H,\alpha)$ (Oh, Schwarz, Viterbo) produce numerical symplectic invariants of Hamiltonians used in Hofer geometry and $C^0$ symplectic topology.

Check¶

Why does sphere bubbling obstruct a naive $C^\infty$ compactification of the Floer moduli space?

Energy can escape into nonconstant $J$-holomorphic spheres in classes $A\in\pi_2(M)$ with $\omega(A)>0$. The moduli space of cylinders gains bubble trees; dimensions and orientations must be handled by virtual cycles or by hypotheses (monotonicity: $\omega(A)=\lambda c_1(A)$) that rule out index-threatening bubbles.

How does PSS reduce the Arnold lower bound to Morse theory on $M$?

PSS identifies $HF_*(H)$ with $H_*(M;\Lambda)$. The Floer chain group is freely generated by periodic orbits (fixed points of $\varphi_H$), so $\#\operatorname{Fix}\ge\operatorname{rank} HF=\operatorname{rank} H_*(M)$ when nondegenerate and coefficients are a field (Novikov graded appropriately).

Pitfalls¶

Conley–Zehnder grading depends on a trivialization of $x^*TM$; absolute $\mathbb{Z}$-grading needs $c_1|_{ \pi_2}=0$ or works modulo $2N$ (minimal Chern number).
Novikov rings are mandatory when holomorphic spheres or action windows require formal series $\sum n_A T^{\omega(A)}$.
Transversality for multiply covered curves / superregular $J$ fails in general — virtual techniques are not optional decoration for the most general closed symplectic manifolds.
Hamiltonian vs Reeb dynamics: contact homology / ECH live on contact manifolds; see contact geometry.

Refs¶

Audin–Damian, Morse Theory and Floer Homology
McDuff–Salamon, J-holomorphic Curves and Symplectic Topology
Salamon, “Lectures on Floer homology” (IAS/Park City notes)
Oh, Symplectic Topology and Floer Homology I–II

Next¶

Symplectic foundations · Symplectic index · Contact geometry · CG / ECH & Weinstein — the ECH the contact / Seiberg–Witten Floer · Mirror symmetry · Fukaya category & HMS — the Fukaya category is built from Floer theory; HMS is the categorical mirror · Complex geometry · Mathematical physics · Differential geometry