I am starting to think that the issue here is that I do not have a firm grasp on what homology is "counting" in general (beyond of course "cycles modulo boundaries") and why I might want to move to a different group other than comptuational ease.
(a) One way to visualize homology is to think about cellular homology. Here, we form a chain complex which is free abelian on the number of cells in each dimension, and our differential is determined by the degrees of the various attaching maps. Somehow this measures how twisted up and far away our CW-complex is from being just a wedge of spheres. In the case of $\mathbb{R}P^{n}$, for example, there's only one cell in each dimension, and the degrees of the attaching maps are either $0$ or $2$; in particular, they are always even! So if we take cellular homology with $\mathbb{Z}/2$ coefficients, the maps all become zero. In the case of $\mathbb{C}P^n$, there's too much space in between dimensions for anything to happen, so it "looks" like a wedge of spheres, according to homology. The upshot is that the algebra simplifies greatly for certain choices of coefficients because you can simplify or get rid of differentials in the cellular chain complex.
(b) Often I tend to think of $H_*(-, \mathbb{Z})$ as being singular homology, and other choices of coefficients just a convenient way to organize the algebra (this is justified by the universal coefficient theorem). Picking coefficients in a certain characteristic can either isolate phenomena in that characteristic, or make certain problems vanish. From the cellular perspective it seems to make things vanish (since you're making it closer to a wedge of spheres)... however, it seems that for other invariants you manage to isolate things. This usually has to do with torsion... It's helpful to work through some examples other than just projective space: I recommend lens spaces.
...if there is some structure in an appropriate group action on something related to some space should I expect to see corresponding structure when viewing homology groups of that space with that group as the coefficients?
(a) It's unfortunate that the two different $\mathbb{Z}/2$'s here are named the same. What's important here is that the group $C_2$ is doing the acting and its order is divisible by $2$, and so things will behave differently with coefficients in $\mathbb{Z}/2$ as opposed to $\mathbb{Z}/p$ for odd primes. This generalizes: if you have a finite group $G$ acting on a space, $X$, then things with coefficients in $\mathbb{Z}/p$ will behave differently depending on whether or not $p$ divides $\vert G\vert$. One hint at this is the following: An action of $G$ descends to an action on the singular chain complex with coefficients in whatever (say, field) $k$ you're using, and so you have a bunch of vector spaces with actions of $G$, i.e. a bunch of representations of $G$. Now, it turns out that representation theory for finite groups behaves very differently in characteristic $p$ depending on whether or not $p$ divides the order of the group.
(b) Part (a) can be made even more precise... It turns out that there is a special homology theory for $G$-spaces called (Borel) equivariant homology. I won't go into it, but it has two properties (1) It has a very close relationship with group homology and (2) sometimes the ordinary homology of $X/G$ serves as a good approximation to the equivariant homology of $X$.
And here are some miscellaneous related comments:
- There is a notion of R-orientability for any ring $R$; see Hatcher. This might help explain some things. In particular, note the sections on Poincare duality, which explains some of the regularity we see when choosing particular coefficients.
- You don't have to be too exotic in your choice of ring because of the following Exercise: Show that a map $f: X \rightarrow Y$ induces an isomorphism on all homology groups if and only if it induces an isomorphism on rational homology and an isomorphism on homology with coefficients of $\mathbb{Z}/p$ for all $p$. (In some sense this point tells you that all different choices of coefficients can do for you is pinpoint torsion at different primality.)
A 2-form is a function that eats a parallelogram (technically it eats 2 vectors, which you should think of as spanning a parallelogram) and spits out a number proportional to its area. A 3-form eats a parallelepiped (the 3-dimensional analog of a parallelogram) and spits out a number proportional to its volume. A 4-form eats a 4-dimensional parallelotope and spits out a number proportional to its hypervolume. A 1-form eats a line segment (which you can think of as a 1-dimensional parallelogram) and spits out a number proportional to its length. A 0-form eats a single point (which you can think of as a 0-dimensional parallelogram) and spits out a number, though there's nothing for it to be proportional to since a point has no extension in space. I think you get the picture. In general an n-form eats n vectors, which you should think of as spanning an n-dimensional parallelotope, and spits out a number proportional to its hypervolume.
Usually books that teach differential forms obscure this. They will define an n-form as a "real-valued multilinear, skew-symmetric function of n vectors". But it means the same thing. Multilinearity and skew-symmetry = output is proportional to length/area/volume/hypervolume. The determinant, which is used to compute the volume of a parallelepiped (and its higher and lower dimensional analogs), has the same two properties.
So why do we require forms to have this property? Well it's just because it's needed for integration. Imagine a curve you want to integrate over. The first step is to approximate it with line segments. Then you apply some function to each line segment in order to get a number. You need that number to shrink as the size of the line segment shrinks otherwise the sum won't converge. Think about it, if the output of the function was independent of the length of the input, then as more segments were added to the approximation the sum would just shoot up to infinity. Now think of a surface you want to integrate over. You can approximate it with parallelograms, imagine the scales of an armadillo. Then for each parallelogram you apply some function that spits out a number. We need the numbers to shrink as the scales do so the sum actually converges. If you want to integrate over some 3-dimensional volume, approximate it with parallelepipeds and again evaluate a function for each parallelepiped. The output of this function needs to shrink with its input for the sum to converge. These functions that we integrate over curves/surfaces/volumes/hypervolumes are forms.
Now let me explain why you write forms as linear combinations of elementary forms. It has to do with the generalized Pythagorean theorem, which I'll just call the GPT. In the same way that the length of a line segment is equal to the sum of the squared lengths of its projections onto the various coordinate axes, the area of an arbitrary parallelogram is equal to the sum of the squared areas of its projections onto the various coordinate planes. And the volume of a parallelepiped is equal to the sum of the squared volumes of its projections onto the various 3-dimensional subspaces. And so on. So the Pythagorean theorem applies to more than just line segments.
So let's look at the example of a 1-form that eats line segments embedded in 3-dimensional space. In general it's gonna look like $adx + bdy + cdz$ (if you forgot, $dx$, $dy$, and $dz$ are just functions that eat a line segment and spit out its projections on the x axis, y axis, and z axis respectively). All that's happening is you're taking the dot product of a vector $(a,b,c)$ with another vector $(dx,dy,dz)$ which equals the projection of $(a,b,c)$ onto $(dx,dy,dz)$ times the length of $(dx,dy,dz)$ (the length of $(dx,dy,dz)$ is $\sqrt{dx^2 + dy^2 + dz^2}$ ie the length of the line segment by the GPT). In other words $adx + bdy + cdz$ is literally just another way of writing: (projection of $(a,b,c)$ onto $(dx,dy,dz)$) times (length of the line segment). Since the length of the line segment is a factor in this product, the function is obviously proportional to the length of the line segment. Any 1-form can be written like this.
Another example: A 2-form that eats parallelograms embedded in 3-dimensional space is gonna have the form $a(dx \wedge dy) + b(dx \wedge dz) + c(dy \wedge dz)$ (if you forgot, $dx \wedge dy$, $dx \wedge dz$, and $dy \wedge dz$ are just functions that eat parallelograms and spit out the areas of their projections on the xy, xz, and yz planes respectively). So this is just another way of writing the dot product of $(a,b,c)$ and $(dx \wedge dy, dx \wedge dz, dy \wedge dz)$ which is just the projection of $(a,b,c)$ onto $(dx \wedge dy, dx \wedge dz, dy \wedge dz)$ times the length of $(dx \wedge dy, dx \wedge dz, dy \wedge dz)$ (which is $\sqrt{(dx \wedge dy)^2 + (dx \wedge dz)^2 + (dy \wedge dz)^2}$ ie the area of the parallelogram by the GPT). In other words the linear combination is just equal to: (projection of $(a,b,c)$ onto $(dx \wedge dy, dx \wedge dz, dy \wedge dz)$) times (area of the parallelogram). Which is clearly a function proportional to the area of the parallelogram.
Another example: A 2-form that eats parallelograms in the plane. It has the general form $a(dx \wedge dy)$. You only need one term because $dx \wedge dy$ already gives you the area of the parallelogram. In the same way $dx$ gives you the length of your line segment if you're only in 1 dimension. It's only when you're in a dimension higher than the dimension of the line segment/parallelogram/parallelepiped/parallelotope that you're gonna have to invoke the GPT ie have a linear combination of multiple elementary forms.
So hopefully you see that differential forms are actually very simple objects. They're merely generalized integrands. Other things in exterior calculus like the exterior derivative, the generalized stokes theorem, etc are similarly very simple when explained properly.
edit: a slightly cleaned up version of this post with some pictures can be found here: https://simplermath.wordpress.com/2020/02/13/understanding-differential-forms/
Best Answer
$\DeclareMathOperator\cl{cl}$In terms of metric spaces, the statement "$x\in\cl(A)$" that you want to think of as "$x$ touches $A$" becomes "the distance of $x$ to $A$ is $0$". Here the distance $x$ has to $A$ is defined as $$ \operatorname{dist}(x,A) = \inf_{a\in A} d(x,a). $$ Now take $x\in\cl(\cl(A))$ and let's figure out what this means in terms of distance. The point $x$ has a distance of $0$ to the set of all points having distance $0$ to $A$. Now in general for three points $x,y,z$ we have $$ d(x,z) \le d(x,y) + d(x,z). $$ So if $x$ has distance $0$ to all points $y$ having distance $0$ to $A$, these distances will at most sum up and $x$ has distance at most $0+0=0$ to $A$. We conclude that $x$ is indeed touching $A$ and hence $$ \cl(\cl(A)) \subseteq \cl(A). $$ The other inclusion is given by property 2.