TITLE: Hatcher pg 187: Idea of Cohomology
QUESTION [8 upvotes]: I am readying Hatcher page 178 where he tries to give the idea of cohomology in the case where our space $X$ is a graph. Now in the 4th paragraph from the top he says this:


The cohomology group $H^1(X,G) = \Delta^1(X;G)/\textrm{Im} \delta$ will be trivial iff the equation $\delta \varphi = \psi$ has a solution $\varphi \in \Delta^0 (X;G)$ for each $\psi \in \Delta^1 (X;G)$. Solving this equation means deciding whether specifying the change in $\varphi$ across each edge of $X$ determines an actual function $\varphi \in \Delta^0(X;G)$.


What does he mean by that last sentence in bold in the paragraph above? 
Thanks.

REPLY [11 votes]: I like to think of this as follows.  If you're hiking in the mountains and you keep track of your changes in elevation, then if you ever return to the same spot your total change in elevation will be $0$.  So, $H^1$ detects whether it's possible to have a sensible notion of "local change in elevation" such that you don't necessarily return to $0$ when you come around to your starting point.  A great illustration of this possibility is the "never-ending staircase";

note that this is only possible because we're not required to say what happens with our elevation anywhere in the center hole.  (That is, $H^1(S^1)\not= 0$ but $H^1(D^2)=0$.)
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