In singular cohomology, cocycles (i.e., cochains $\varphi$ such that $d\varphi = 0$) are the same as cochains which vanish on boundaries. For example, a $1$-cocycle on $\mathbb{R}^2 \setminus \{0\}$ would vanish on every loop that does not contain the origin, but need not vanish on a loop that contains the origin.
I'm trying to find a similar description for coboundaries (i.e., cochains of the form $d\psi$). It is immediate that coboundaries vanish on cycles, so for instance, a $1$-coboundary on $\mathbb{R}^2 \setminus \{0\}$ would vanish on all loops. However, the converse is not necessarily true. By the universal coefficient theorem, we have an exact sequence
$$
0 \to \mathrm{Ext}(H_{n – 1}(X), G) \to H^n(X; G) \xrightarrow{h} \mathrm{Hom}(H_n(X), G) \to 0,
$$
where $h$ sends a cocycle $\varphi : C_n(X) \to G$ to the induced map $H_n(X) = Z_n(X) / B_n(X) \to G$. The kernel of $h$ consists precisely of the cocycles that vanish on $Z_n(X)$. The point of the universal coefficient theorem is that $\ker h$ is not the same as the set of coboundaries. If $R$ is a field, the $\mathrm{Ext}$ term vanishes, so in this case coboundaries are the same as cocycles that vanish on all cycles.
Here is an example of a cocycle vanishing on cycles which is not a coboundary. Consider $X = \mathbb{R}P^2$. For $n = 2$ and $G = \mathbb{Z}$, the above exact sequence is of the form
$$
0 \to \mathbb{Z}_2 \to \mathbb{Z}_2 \to 0 \to 0.
$$
In terms of cellular cohomology, the generator of $H^2(X; \mathbb{Z}) \cong \mathbb{Z}_2$ is the $2$-cochain sending the $2$-cell of $\mathbb{R}P^2$ to $1$. This is not a coboundary since if $\varphi$ is a $1$-cochain, then $d\varphi$ must send the $2$-cell of $\mathbb{R}P^2$ to an even integer, twice of $\varphi$ evaluated on the $1$-cell of $\mathbb{R}P^2$.
Question: Is there an intuitive description of coboundaries, beyond the fact they lie in the image of $d$?
Best Answer
From the universal property of the kernel, $\alpha : C_n \to G$ vanishes on cycles iff it factors as $\beta \circ \partial$ with $\beta : \operatorname{im} \partial \to G$. It is further a coboundary if $\beta$ extends to a homomorphism $C_{n-1} \to G$. In particular if $R$ is a field then every $G$ is injective so these properties are equivalent.