# Evaluating a symplectic form on $\pi_2$ or its image through the Hurewicz map

Let $$(M,\omega)$$ be a symplectic manifold. There are a priori two ways of evaluating $$\omega$$ on an element $$A \in \pi_2(M)$$:

1. we can integrate $$\omega$$ on any representative $$u : S^2 \to M$$ of the class $$A$$: $$\omega(A) := \int_{S^2} u^* \omega$$ Indeed, if $$u$$ and $$u'$$ represent the same class in $$\pi_2(M)$$, then the integrals $$\int_{S^2} u^* \omega$$ and $$\int_{S^2} u^{'*} \omega$$ differ by that of a $$\omega$$ on the boundary of a three dimensional manifold, which vanishes since $$d \omega = 0$$.
2. Let $$\pi_2(M) \overset{\rho_1}{\longrightarrow} H_2(M;\mathbb{Z}) \overset{\rho_2}{\longrightarrow} H_2(M; \mathbb{R})$$ be the composition of the Hurewicz map $$\rho_1$$ and the natural homomorphism $$\rho_2 : H_2(M; \mathbb{Z}) \to H_2(M; \mathbb{R})$$. Then one can define $$\omega(A) := \langle [\omega], \rho_2 \rho_1 (A) \rangle, \quad A \in \pi_2(M),$$ where $$\langle . , . \rangle$$ denotes the natural pairing $$H^2(M; \mathbb{R}) \times H_2(M; \mathbb{R}) \to \mathbb{R}$$.

Suppose that $$\rho_1$$ is an isomorphism $$\pi_2(M) \simeq H_2(M; \mathbb{Z})$$. Then are these two definitions equivalent ? In other words, do we have: $$\langle [\omega], \rho_2 \rho_1 (A) \rangle = \int_{S^2} u^* \omega,$$ for any $$A \in \pi_2(M)$$ and representative $$u : S^2 \to M$$ of $$A$$ ?

Recall that the Hurewicz homomorphism $$\rho_1 : \pi_2(M) \to H_2(M; \mathbb{Z})$$ is given by $$A \mapsto u_*[S^2]$$ where $$A = [u]$$; here $$[S^2] \in H_2(S^2; \mathbb{Z})$$. So

$$\int_{S^2}u^*\omega = \langle [u^*\omega], \rho_2[S^2]\rangle = \langle u^*[\omega], \rho_2[S^2]\rangle = \langle [\omega], u_*\rho_2[S^2]\rangle.$$

Now note that we have the following commutative diagram

$$\require{AMScd} \begin{CD} H_2(S^2; \mathbb{Z}) @>{u_*}>> H(M;\mathbb{Z})\\ @V{\rho_2}VV @VV{\rho_2}V \\ H_2(S^2; \mathbb{R}) @>{u_*}>> H(M;\mathbb{R}) \end{CD}$$

i.e. $$u_*\rho_2 = \rho_2 u_*$$. Using singular homology, this follows because pushforwards are defined on singular simplicies (by composition) and extended linearly.

So we see that

$$\int_{S^2}u^*\omega = \langle [\omega], u_*\rho_2[S^2]\rangle = \langle[\omega], \rho_2 u_*[S^2]\rangle = \langle[\omega], \rho_2\rho_1(A)\rangle.$$

• Thank you very much for the answer @Michael. Commented May 1, 2019 at 15:40