Homotopic function question - very basic

OK.

1. The theorem that pullbacks via homotopic maps give the same integral: that's for manifolds, not manifolds-with-boundary. (Where "manifolds" really means "manifolds without boundary". See below.)

2. Your second example, with $S^1$, uses the function $f(theta) = theta$, g(theta) = 0$, and presumably the homotopy

$$ H(theta, s) = (1-s) theta $$

Unfortunately, this doesn't actually work. I'm assuming that Ted defines $S^1$ as the manifold consisting of all the points ${ (cos t, sin t) mid 0 le t < 2pi }$, with the topology arising from inclusion in $Bbb R^2$. So what is $f$? $f$ is actually a map $[0, 2pi] to [0, 2pi]$, not a map from $S^1$ to $S^1$. What's hidden here is a map $p$ that identifies a point $$ theta in [0, 2pi) $$ with the point $$ (cos theta, sin theta) in S^1. $$ Because $p$ is a bijection, we can "refer to points in $S^1$ by their polar coordinates," which is how $f$ got defined. But while the function $p$ is continuous, the inverse function $p^{-1}: S^1 to [0, 2pi)$ is discontinuous, which can be a problem. For the real function you want, from $S^1$ to $S^1$, is defined by taking a point of $S^1$, finding its polar coordinate (i.e., applying $p^{-1}$, applying $f$, and then applying $p$ to get back to the unit circle. In short, you're looking at $$ F = p circ f circ p^{-1}, $$ while talking about $f$. Similarly for $g$.

Now $H$, as written above, is a homotopy from $f$ to $g$, so you might hope that $$ K(u, s) = p( H(p^{-1}(u), s ) $$ would be a homotopy on $S^1$. (Here $u$ denotes an arbitrary point of $S^1$; we extract its polar coordinate, apply $H$, and then push back to the unit circle with $p$.)

If $K$ we defined by composing $H$ with some continuous functions, it really would be a homotopy between functions $S^1 to S^1$. But $p^{-1}$ is discontinuous, so the function $K$ that you're implicitly using...you can't use a continuity argument to show it's a homotopy, and in fact, it's not actually a homotopy. So the thing you'd like to conclude, that $$ int_0^2pi F^{*}omega = int_0^{2pi} G^{*} omega tag{1} $$ cannot be "proved" using the cited theorem, because the hypotheses of the theorem are not all satisfied. And that's a good thing, because, as you observed with your own computations, the two sides of the "equality" in Equation 1 are not, in fact, equal.

In short: there's no contradiction to the claim's in Shifrin's book.

Extra note: The whole manifold-without-boundary thing can confuse people, and rightly so. If I say I have a blue truck, people tend to assume I have a truck, and that it's a little special: it's blue. If I have a friend-with-benefits, you assume that I have a friend. But if I say I have a manifold-with-boundary, you're not allowed to assume I have a manifold? What's up with that?

Well...historically, people started talking about manifolds first -- things like a sphere or a torus, etc. And then they realized that they wanted to slightly generalize this, and sometimes allow things like a disk, or a cylinder $S^1 times [0, 1]$ into the same discussions. These things were not literally manifolds, but they had a LOT of the same properties, so someone thought up the clever name "manifold-with-boundary". But lots of things were already known about manifolds that were not true for these new things. (For instance: manifolds have no boundary!) So everyone agreed that m-with-b was a new category, one that included (but was not included in) the old category of "just plain old manifolds," because otherwise they'd have to go rewrite everything from the past replacing "manifold" with "manifold-without-boundary". Ugh!

The choice to include the sphere, say, in the collection of manifolds-with-boundary, even though its boundary is empty, might seem peculiar. But it didn't take long for folks to see that if they DIDN'T do this, then lots of theorems would start out "Let M be a manifold or manifold-with-boundary", i.e., lots and lots of the things proved for manifolds-with-nonempty-boundary turned out to also be true for manifolds with empty boundary (i.e., ordinary manifolds). So they made the new term inclusive rather than exclusive.

Of course, this little bit of history is mostly fiction --- there wasn't a congress of mathematicians who all agreed that henceforth they'd do things THIS way. It took a while to settle out. And in the intervening years, I'm sure there have been folks who decided that the old way was stupid, and they were going to change the world and insist on manifold-without-boundary and manifold-with-boundary, and let the word "manifold" die a well-earned death. I don't believe that's caught on, but I could be wrong.