I'm self-studying Munkres's Elements of Algebraic Topology, but I'm struggling with cohomology and cocycles of cell complexes.
Here, he demonstrates how to compute the cohomology group of a torus and a klein bottle, both of which can be obtained by taking quotients of a rectangular region, but he defines $\lambda: D_p(X)\to \Bbb{Z}$ s.t. $\lambda(\gamma)=\lambda(g_\text{#}(d))=1$ where $d=\sum_i(\text{oriented 2-simplices})$. I thought that this follows from the fact that $$Hom(D_p(X),\Bbb{Z})\leftarrow^\tilde{i}Hom(C_p(X),\Bbb{Z})\implies D_p(X)\to^i C_p(X)\to^\phi\Bbb{Z}$$ But it doesn't seem to be reasonable, so:
Q1: Why is $\lambda$ defined to be $\gamma\mapsto 1$? (the same question for $\phi$)
He states that there is no general way to find such cocycles, which means whenever we're asked to find the cocycle of some space, we have to guess first? In the example of torus and klein bottle, he already knows that the cocycles are those zigzag lines so what he needs to do is just check if they really belong to $Z^p(X)$ (i.e. the group of cocycles), but what if we're given something like an $n$-fold dunce cap which could be obtained by identifying edges of an $n$-gon or other strange spaces like $D^3\setminus(S^1\times[0,1])$? I can't help to ask because Munkres says in the book that this concept is extremely important for cup products.
Q2: Is there a way to find the cocycles by computation instead of guessing (maybe I misunderstand the word "general procedure" but I don't see any paragraph that explains how we know the cocycles of the torus and klein bottle)? and how to find the cocycles of $n$-fold dunce cap?
I managed to find the cohomology group of it: $$ H^p(\text{n-fold dunce cap};\Bbb{Z})\cong \begin{cases} \Bbb{Z} & p=0\\ \Bbb{Z}/n & p=2\\ 0 & \text{otherwise} \end{cases} $$
Edit: My attempt to solve the n-fold dunce cap case...
Let $L$ be the n-gon with edges labelled (all counterclockwise), then use line segments to join the vertices and the center of this $n$-gon (labelled towards the center). Denote the ounter edges $e_0$ and those line segments in the $n$-gon $e_i$, where $1\le i \le n$. Similarly, denote every 2-simplex by $\sigma_i$ and orient them counterclockwise.
As what @Brevan Ellefsen noted in the comment, using some intuition, I can see that the zero-cochain $v_0+v_1$ is the only cocycle in 0-dim because $\delta(v_0+v_1)=n(-e_0+e_0)-\sum_i e_i+\sum_i e_i=0$. In 1-dimension, any cochain containing $e_0$ is not a cocycle because it's impossible to cancel its value; but $\delta(\sum_i e_i)=\sum_{i=2}^n(\sigma_{i-1}-\sigma_i)+\sigma_n-\sigma_1$ which means $\sum_ie_i$ is a cocycle. Besides this cocycle is cohomologous to $0$ since its equal to $\delta(v_1)$. In 2-dimension, any 2-cochain is a cocycle because $\delta:C^2(X)\to C^3(X)$ is a trivial homomorphism. ($\Bbb{Z}$ coefficient)
Change the coefficient to $\Bbb{Z}/n$ won't change the fact that any 2-cochain is a 2-cocycle and $v_0+v_1$ is a 0-cocycle. But in dimension one we see that $\sum e_0$ becomes a cocycle because $\delta(\sum e_0)=0$.
Am I correct?
I think I should simply summarize my question because this is a long post... (sorry for that):
Q1: Why is $\lambda$ defined to be $\gamma\mapsto 1$? (the same question for $\phi$)
Q2: Is there a way to find the cocycles by computation instead of guessing (maybe I misunderstand the word "general procedure" but I don't see any paragraph that explains how we know the cocycles of the torus and klein bottle)? and how to find the cocycles of $n$-fold dunce cap?
Q3: Is my attempt to find the cocycles in $n$-fold dunce cap correct?
Thank you very much for your time and efforts!!! :)