2 fixed typo

1

Extending maps on Riemman surfaces

Suppose you have a map $g:\Sigma \rightarrow G$ from a Riemann surface $\Sigma$ to a compact Lie group $G$. What is the obstruction to finding a $3$-manifold $W$, such that $\partial W = \Sigma$, and an extension of $g$ to a map $\tilde{g}:W\rightarrow G$? In the paper I'm reading they say it lies in $H_2(G,\mathbb{Z})$. Why is this true? I mean, the obstruction class to extending $g$ to $\tilde{g}$ is an element in $H^3(W,\pi_2(G))$, which vanishes since $\pi_2(G)=0$ for compact $G$, right? So does this mean that the obstruction to finding a $3$-manifold $W$ with boundary $\Sigma$ lies in $H_2(G,\mathbb{Z})$? By the way, the paper I'm referring is here http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104180750 , see section 4.1 (page 405). Thanks.