TITLE: Is $\pi_2$ algebraic?
QUESTION [13 upvotes]: Let $X$ be a smooth complex projective surface. Is the Hurewicz image $h(\alpha)\in H_2(X)$ of a homotopy class $\alpha\in\pi_2(X)$ algebraic?

REPLY [16 votes]: A slightly better variant of this question is to ask: is  the Hurewitz image of $\pi_{2}(X)$ in $H_{2}(X)$ a sub Hodge structure? This is in fact an old question of Philippe Eyssidieux. In section 4.3 of this paper we proved that if the fundamental group of $X$ is nice enough (more precisely if $\pi_{1}(X)$ is algebraically good) , then $\text{im}\left[ \pi_{n}(X) \to H_{n}(X)\right]$ is a sub Hodge structure for all $n$.