I am trying to understand the differences between several definitions used in many texts in symplectic topology. Let $(M,\omega)$ be a symplectic manifold, and $c_1 \in H_2(M,\mathbb{Z})$ be its first Chern class.
The minimal Chern number $N_M$ of $M$ can be defined as the positive generator of the group $\langle c_1, H_2(M,\mathbb{Z}) \rangle$. On the other hand, I have seen another definition in Salamon's lectures on Floer homology: $$ N_M := \underset{k > 0}{\inf} \lbrace \exists v : S^2 \to M \ | \ \int_{S^2} v^*c_1 = k \rbrace. $$ If $N \neq \infty$, then Salamon writes that $\langle c_1, \pi_2(M) \rangle = N_M \mathbb{Z}$.
I would like to understand the meaning of this definition, and its relation to the first one. In particular, what does $c_1(A)$ means, for an element $A \in \pi_2(M)$ ? Is $\pi_2$ in this setting a notation for its image under the Hurewicz homomorphism ?
Another related definition confuses me as well: a symplectic manifold $(M,\omega)$ is called monotone if there exists $\tau \in \mathbb{R}$ such that $[\omega] = \tau c_1$, where $[\omega] \in H^2(M,\mathbb{R})$ is the cohomology class of $\omega$. In Salamon's lecture, the definition is again different: $$ \exists \tau \in \mathbb{R}, \quad \forall v : S^2 \to M, \ \int_{S^2} v^*\omega = \tau \int_{S^2} v^* c_1. $$ What is the relation between these two definitions ? Is the second equivalent to saying that $\langle \omega, \pi_2(M) \rangle = \tau \langle c_1, \pi_2(M) \rangle$, for a suitable meaning of this last equation ?
Thanks for your help !
1) Yes. Let $f: S^2 \to M$ be a representative of $A \in \pi_2(M)$. Then $$\int_{S^2} f^* c_1 = c_1(A).$$ You may think of Salamon's minimal Chern number as the minimum integer that you get when you evaluate the first Chern class on all spheres.
2) Those are two different definitions of monotone. One is that $\omega$ is proportional to $c_1$ on all of $H^2$, whereas Salamon's definition is only restricted to spheres, which is a weaker requirement.
However, in the case that $M$ is simply connected, the Hurewicz map is an isomorphism, so the two definitions are equivalent.
For Hamiltonian Floer homology, you wish control two things: energy, and bubbling, and it turns out that it suffices to have monotonicity only on $\pi_2$.
Given two holomorphic cylinders $u_1, u_2 \in \mathcal{M}(x,y)$, the Fredholm index difference is given by $$ \mu(u_1) - \mu(u_2) = 2 c_1(u_1 \# u_2)$$ i.e. by the first Chern class evaluated on the resulting torus. The Fredholm index $\mu(u)$ is computed by the difference Conley-Zehdner indices of the two ends of $u$ of a symplectic trivialization. But since the ends of the cylinders are contractible, you end up computing the Chern class on a sphere $A$. Monotonicity implies that the difference in energy is $E(u_1) - E(u_2) = \tau c_1(A)$, so the energy is constant on all moduli spaces between two orbits of a given dimension. This implies compactness (modulo bubbling) for index 1 components of $\mathcal{M}(x,y)$. For $\tau >0$, you kill a second bird, which is the bubbling.