TITLE: Motivating Cohomology
QUESTION [29 upvotes]: Question: Are there intuitive ways to introduce cohomology?  Pretend you're talking to a high school student; how could we use pictures and easy (even trivial!) examples to illustrate cohomology?
Why do I care: For a number of math kids I know, doing algebraic topology is fine until we get to homology, and then it begins to get a bit hazy: why does all this quotienting out work, why do we make spaces up from other spaces, how do we define attaching maps, etc, etc.  I try to help my peers do basic homological calculations through a sequence of easy examples (much like the ones Hatcher begins with: taking a circle and showing how "filling it in" with a disk will make the "hole" disappear --- ) and then begin talking about what kinds of axioms would be nice to have in a theory like this.  I have attempted to begin studying co-homology through "From Calculus to Cohomology" and Hatcher's text, but I cannot see the "picture" or imagine easy examples of cohomology to start with.

REPLY [5 votes]: There is a nice and, to my opinion, more natural way to motivate cohomology - a geometric one, rather than an analytical one. Please read carefully the following question and answer in math.stackexchange:
Intuitive Approach to de Rham Cohomology