Let $\Sigma$ be a compact Riemann surface with complex structure $j$. Let $(M,J)$ be an almost complex manifold. A map $u: \Sigma \rightarrow M$ is called $J$-holomorphic if $$ du \circ j = J \circ du.$$ Now $u_*[\Sigma]$ is an element in $H_2(M, \mathbb{Z})$. Is this element in the free part of $H_2(M, \mathbb{Z})$?

I want to know if I can ignore torsion part when we consider Gromov-Witten invariants. All I could see was that when $M$ is symplectic, pure torsion element cannot represent $J$-holomorphic curves.

[edited] As Sam Lisi pointed out, the 'free part' of $H_2(M, \mathbb{Z})$ doesn't make sanse. I changed the question. Suppose $A$ is represented by $J$-holomorphic curves. If $B$ is a torsion element in $H_2(M, \mathbb{Z})$, can $A+B$ be represented by a $J$-holomorphic curve?

Let $\Sigma$ be a compact Riemann surface with complex structure $j$. Let $(M,J)$ be an almost complex manifold. A map $u: \Sigma \rightarrow M$ is called $J$-holomorphic if $$ du \circ j = J \circ du.$$ Now $u_*[\Sigma]$ is an element in $H_2(M, \mathbb{Z})$. Is this element in the free part of $H_2(M, \mathbb{Z})$?

I want to know if I can ignore torsion part when we consider Gromov-Witten invariants. All I could see was that when $M$ is symplectic, pure torsion element cannot represent $J$-holomorphic curves.

[edited] As Sam Lisi pointed out, the 'free part' of $H_2(M, \mathbb{Z})$ doesn't make sanse. I changed my question. Suppose $A$ is represented by $J$-holomorphic curves. If $B$ is a torsion element in $H_2(M, \mathbb{Z})$, can $A+B$ be represented by a $J$-holomorphic curve?

Let $\Sigma$ be a compact Riemann surface with complex structure $j$. Let $(M,J)$ be an almost complex manifold. A map $u: \Sigma \rightarrow M$ is called $J$-holomorphic if $$ du \circ j = J \circ du.$$ Now $u_*[\Sigma]$ is an element in $H_2(M, \mathbb{Z})$. Is this element in the free part of $H_2(M, \mathbb{Z})$?

I want to know if I can ignore torsion part when we consider Gromov-Witten invariants. All I could see was that when $M$ is symplectic, pure torsion element cannot represent $J$-holomorphic curves.

Let $\Sigma$ be a compact Riemann surface with complex structure $j$. Let $(M,J)$ be an almost complex manifold. A map $u: \Sigma \rightarrow M$ is called $J$-holomorphic if $$ du \circ j = J \circ du.$$ Now $u_*[\Sigma]$ is an element in $H_2(M, \mathbb{Z})$. Is this element in the free part of $H_2(M, \mathbb{Z})$?

I want to know if I can ignore torsion part when we consider Gromov-Witten invariants. All I could see was that when $M$ is symplectic, pure torsion element cannot represent $J$-holomorphic curves.

[edited] As Sam Lisi pointed out, the 'free part' of $H_2(M, \mathbb{Z})$ doesn't make sanse. I changed the question. Suppose $A$ is represented by $J$-holomorphic curves. If $B$ is a torsion element in $H_2(M, \mathbb{Z})$, can $A+B$ be represented by a $J$-holomorphic curve?