Floer homology

Definition

This defines Hamiltonian Floer homology with coefficients in $\F_2$ in the simplest setting. Let $(M, \omega)$ be a closed symplectic manifold such that $\int_{S^2} v^* \omega = 0$ for every smooth map $v \colon S^2 \to M$. Let $H \colon (\R / \Z) \times M \to \R$ be smooth, write $H_t = H(t, \cdot)$, and let $X_{H_t}$ be the Hamiltonian vector field of $H_t$.

1. Let $\mathcal{P}(H)$ be the set of contractible loops $x \colon \R / \Z \to M$ with $x'(t) = X_{H_t}(x(t))$, the contractible $1$-periodic orbits. Assume that each of them is nondegenerate: the linearized time-$1$ flow at $x(0)$ does not have $1$ as an eigenvalue.
2. Let $J = (J_t)_{t \in \R / \Z}$ be a family of almost complex structures compatible with $\omega$. For $x, y \in \mathcal{P}(H)$, let $\mathcal{M}(x, y)$ be the set of smooth maps $u \colon \R \times (\R / \Z) \to M$ with $$\partial_s u + J_t(u) \left( \partial_t u - X_{H_t}(u) \right) = 0, \qquad \lim_{s \to -\infty} u(s, t) = x(t), \qquad \lim_{s \to +\infty} u(s, t) = y(t),$$ on which $\R$ acts by translation in $s$.
3. For generic $J$, each set $\mathcal{M}(x, y)$ is a smooth manifold. When it has dimension $1$, let $n(x, y) \in \F_2$ be the number modulo $2$ of elements of $\mathcal{M}(x, y) / \R$. Otherwise let $n(x, y) = 0$.
4. Let $CF(H)$ be the $\F_2$-vector space with basis $\mathcal{P}(H)$, and let $\partial \colon CF(H) \to CF(H)$ be the linear map with $\partial x = \sum_{y} n(x, y)\, y$.

Then $\partial \circ \partial = 0$, so $(CF(H), \partial)$ is a chain complex. The Floer homology of $(H, J)$ is $HF(H, J) = \ker \partial / \im \partial$.