Let $\Sigma$ be a closed Riemann surface and $X$ a 1-connected topological space. I would like to prove the following fact.
(A) The map $[\Sigma,X] \to H_2(X;\mathbb Z)$ defined by $[f] \mapsto f_*[\Sigma]$ is injective.
Question: How can we prove this statement?
I found this statement in Cieliebak-Gaio-Salamon (Prop 2.1 (ii)). This paper gives a following hint to see the statement.
(B) Every map $\Sigma \to X$ factors, up to homotopy, through a map $\Sigma \to S^2$ of degree 1.
By this hint, $f:\Sigma \to X$ can be given the composition of a map $a:\Sigma \to S^2$ of degree 1 and a map $b:S^2 \to X$, up to homotopy. Since $H_2(X;\mathbb Z)=\pi_2(X)$, $f_*[\Sigma]$ determines a homotopy class of $b$. Therefore if
(C) any two maps $\Sigma \to S^2$ of degree $1$ are homotopic,
then (A) follows. But I can not see both (B) and (C).
- I suspect that the statement is classical. (This is the reason for the tag "reference-request".) But I have not found any references.
- I use the terminology "degree 1" in the following sense: a map $a:\Sigma \to S^2$ is of degree 1 if $a_*[\Sigma]=[S^2]$.
- The 1-conected space $X$ is originally the classifying space $BG$ of a compact and connected Lie group $G$. A proof making use of $X=BG$ is also welcome.