Let $M$ be a compact oriented manifold of dimension $2l-1$, and $\pi:E \to M$ be a principal $G$-bundle. We then have $H^{2l-1}(M,\mathbb{R}/\mathbb{Z}) \simeq \mathbb{R}/\mathbb{Z}$. Consider $\pi^*:H^{2l-1}(M,\mathbb{R}/\mathbb{Z}) \to H^{2l-1}(E,\mathbb{R}/\mathbb{Z})$.
Can we conclude that $ ker( \pi^*)$ is finite ?
Somehow I am not able to see why this should be true. This fact is used at the end of section 3 in the Chern-Simons paper. Thanks !