I was trying to follow the construction of the Euler class in Morita's 'Geometry of differential forms' for an $S^1$ bundle over a manifold M; $p: E \rightarrow M$. There the author builds the class as the obstruction to extend a section in the 1-skeleton $K^1$ but he uses the method of "Simple observation" in many proofs where I can't see why is it so obvious. There are two proofs by this method that I don't see.
Suppose we already have a section $s: K^1 \rightarrow E$. We assign to each 2-simplex $\sigma$ a integer defined as the degree of the map $s_s$
$$S^1 \simeq \partial \sigma \rightarrow \pi^{-1}(\sigma) \simeq \sigma \times S^1 \rightarrow S^1$$
Where the first isomorphism is given (up to isotopy) by the orientation in $\partial \sigma$, the first arrow is the section $s$, the second isomorphismm is a trivialization over the simplex $\sigma$ and the second arrow is the projection on the second term. This defines a map $\sigma \mapsto deg(s_s)$ we denote the map by $c_s$.
Lemma 6.17. The map $c_s$ is a cochain, in fact, it is a cocycle.
He shows a picture of a $3$-simplex with all its $2$-faces oriented and claim that proves it. I guess he claims that because orientation on the boundary of those $2$-simplices cancel. I see that it is, by construction, a cochain because it is defined for each 2-simplex and elements in $C^2(K,\mathbb{Z})$ are formal sums of 2-simplices. To show that it is a cocycle we must show that $\delta c_\sigma \equiv 0$ or equivalently that $\delta c_s\sigma (\tau) := c_s (\partial \tau)$ but I feel a little more work than a picture is needed here to conclude. Some property on the degree of a map could be used or quoted. In particular it seems apprpiate to me the following
"Let $\phi:M \rightarrow S^1$ be a continuous map and let $f,g:S^1 \rightarrow M$ be two continous maps such that their images intersect in, at least, one point $p$. Let $\hat f:=\phi \circ f$ (and the same for $g$). Now let $h:S^1 \rightarrow M$ that sends the upper part of $S^1$ to $f(S^1)$ and the lower half to $g(\S^1)$ (sending the equator to $p$) and let $\hat h$ be its composition with $\phi$. Then $deg(\hat f) + deg (\hat g) = deg (\hat h)$"
I guess that proposition should be true and I could certainly conclude from there.
Also I have troubles with the next lemma which says that $c_s$ and $c_{s'}$ are cohomologous for two different sections $s$ and $s'$. The author builds a 1-cycle $d$ and claims that $c_{s'}(\sigma)= c_s(\sigma) + d(\partial \sigma)$.
If anyone can help me with the first part, maybe the second will turn obvious from there.