Let $M$ be a closed smooth manifold. It is well-known that there is a bijection between the cohomology group $H^n(M;G)$ and $[M,K(G,n)]$, where $G$ is a group and $K(G,n)$ is an Eilenberg-Maclane space. In particular there is a bijection between $H^1(M)$ and $[M,S^1]$, and I would like to concretely understand what this is.
In one direction, this is simple: $H^1(S^1)$ has a distinguished generator $u$ given by the identity map $\pi_1(S^1)\to\mathbb{Z}$, so given a map $f:M\to S^1$ we obtain $f^*u\in H^1(M)$ depending only on the homotopy class of $f$.
For the other direction, the way I have found it easiest to construct a map is as follows. We may think of $H^1(M)$ as the group of "integral de Rham classes" of closed 1-forms, which are those closed 1-forms $\alpha$ for which $$\int_c\alpha\in\mathbb{Z}$$ for all smooth 1-cycles $c:S^1\to M$. Given such an integral 1-form we can define the period group $$\Omega_{\alpha}=\left\{\int_c\alpha|c\in H_1(M)\right\}\subset\mathbb{Z}.$$ Now fix any $p\in M$ and notice that $$q\mapsto\int_p^q\alpha$$ is well-defined as a map $M\to\mathbb{R}/\Omega_{\alpha}\cong S^1$. I call this map $\phi_{\alpha}$.
I now have two questions.
- Is the above construction valid? The notion of integral 1-forms is a bit confusing to me.
- If so, are the two assignments above inverses to one another? That is, do we have $[\alpha]=[\phi_{\alpha}^*u]$? If I'm not mistaken, this would mean that for every smooth $c:S^1\to M$ we'd have
$$\int_{S^1}c^*\alpha=\int_{S^1}(\phi_{\alpha}\circ c)^*u=\deg(\phi_{\alpha}\circ c).$$