Let $M$ be a closed, connected, orientable 3-manifold. Here, it is stated that (integer coefficients)
$$H_2(M) \cong H^1(M) \cong \langle M,S^1\rangle.$$
$\langle M,S^1\rangle$ is the group of homotopy classes of maps $M \to S^1$. The first $\cong$ is Poincare duality. What is the second isomorphism? I've not seen it before. Is it discussed in, say, Hatcher, or other standard reference? Can it be visualized? (The linked answer is concerned with embedded, orientable surfaces, which I believe are the means of visualization.)
Best Answer
Letting $\gamma \in H^1(S^1) \approx \mathbb Z$ be a generator, there is a function $\phi : \langle M, S^1 \rangle \to H^1(M)$ defined by $$\phi[f] = f^*(\gamma) $$ for each continuous $f : M \to S^1$, where $[f] \in \langle M,S^1 \rangle$ denotes the homotopy class of $f$, and $f^* : H^1(S^1) \to H^1(M)$ denotes the homomorphism induced by $f$ with respect to the contravariant functor $H^1(\cdot)$.
The theorem you need is that this function $\phi$ is a bijection. As said in the comments, this theorem can be found in Hatcher's book, and also in many other algebraic topology books; I first learned it in Spanier's book.
In the special case that $M$ is an oriented manifold, you can evaluate $\phi[f]$ by first homotoping $f$ to be transverse to a point $p \in S^1$, in which case $f^{-1}(p)$ is a codimension 1 transversely oriented submanifold of $M$. By pairing that transverse orientation with the orientation of $M$ one obtains an orientation of $f^{-1}(p)$, which therefore represents an element of $H_{n-1}(M)$ where $n$ is the dimension of $M$. In this situation $\phi[f]$ is the Poincaré dual of the homology class of $f^{-1}(p)$. To put it another way, $\phi[f] \in H^1(M)$ is evaluated on each 1-dimensional homology class in $M$ by taking the intersection number of that class (i.e. the algebraic intersection number of any 1-cycle in that class) with $f^{-1}(p)$.
A similar construction of $\phi[f]$ also works for the case that $M$ is a simplicial complex: choose $f$ to be in general position with respect to $p$, in which case $f^{-1}(p)$ is a transversely oriented subcomplex with a product neighborhood. In this case $\phi[f] \in H^1(M)$ is defined by evaluating it on an arbitrary cycle in $M$ to be the algebraic intersection number of that cycle with $f^{-1}(p)$.