November 12, 2021

★★☆☆☆

*Galois’ Dream*, originally published in 1968 as *Garoa no yume*, is, and always was, an anachronism. As the back of the book will proudly tell you,

From elementary ideas to cartoons to funny examples (considered “undignified” by many of his colleagues), the author provided his students with a book that was considered “hip” just to own, to be seen reading, and perhaps to be learning from.

Clearly the book was a curiosity in 1960s Japan. But while its *levity* distinguished it in its own time, it is its *seriousness* that sets it apart in ours—seriousness in content, not tone. Because this book is an absolute *slog* to finish. This would come as a surprise to anyone who believed it was appropriate for “first year, undergraduate, mathematics students,” as the back of the book claims. In fact, this book would be a challenge for just about anyone who isn’t already an expert in the subject matter—which is a mix of complex differential equations, algebraic topology, and Riemann surfaces. Has undergraduate education really changed that much since the 1960s? (And going from Japan to America …) Maybe, but I doubt it.

Anyway, the challenging content isn’t a bad thing *per se*, but I think the book really understates the difficulty curve. And it means the first two chapters (called “weeks”), which teach basic facts about sets, functions, and equivalence relations, are totally useless. In fact, I’d say the first *seven* weeks, the first 52 pages of the book, have nothing to offer someone who has any hope of finishing it.

As for the remaining weeks, they compose two conceptually distinct parts:

- The first part, weeks 8–13, deals with the theory of covering maps.
- The second part, weeks 14–19, deals with differential equations.

The book’s purpose is to connect these two parts. Does it succeed at that? Superficially, it does. Roughly speaking, Theorem 19.3 at the end of the book tells us that the solutions to certain differential equations are *simple* (in a sense described in week 17) if and only if the deck transformations of the universal covering space of the domain can be represented as a group of upper triangular matrices. That’s *something*, but I find it underwhelming. Frankly, I think the last few weeks would be easier to appreciate if I understood Riemann surfaces much better than I do. On the bright side, weeks 8–13 are actually quite nice. They also include a definition of normal/regular/Galois coverings that, as far as I can tell, neither Lee nor Hatcher mentions. (But Brown’s *Topology and Groupoids* has it! I forget the page though.)

Besides all that, I have some minor complaints. The pictures are good, but they disappear for the final few weeks, where they are sorely missed. Instead we get some very bad typos! For example, in week 18, the “if” and “only if” proofs of Theorem 18.5 are mislabeled. In that same week, a function

\[ F(X) = X(X-1) + \alpha_0X + \beta_0 \]

is defined without the “\(F(X) =\)” part, and the name must be inferred from the following text. A reference is made to Theorem 18.7, which doesn’t exist. (It should say Theorem 18.6.) There are other mistakes but these are the most egregious.

Kuga (or his translators?) also makes some unfortunate choices of notation. For instance, he uses \(i\) as an index in a series of complex numbers, which forces him to use \(\sqrt{-1}\) for the imaginary unit. He also does a lot of “punning” by using the same symbol for a function and its pullback to a covering space, starting in week 16. This is *very confusing*, especially in the final week, and I don’t think he gains much by doing it.

The handwaving, while understandable, gets annoying after a while. This wouldn’t be so bad if Kuga had included more references to other work, but those are few and far between. For his Lemma 18.3 he simply says “The proof is easy”—which it certainly is not! It would’ve been a better idea, instead of wasting paper talking about sets and equivalence relations, to put some more detail in the last few weeks.

On that note, I have some issues with the organization. It would make much more sense for weeks 10, 12, and 11 to be read in that order. I guess the reason for putting week 11 in the middle is because weeks 10 and 12 are both long and difficult. Perhaps for the same reason, week 9 is just one page with no proofs. And I already said weeks 1–7 are superfluous.

I don’t recommend *Galois’ Dream*, either for education or recreation. It has a certain charm, but that doesn’t make up for its flaws. Despite this, if someone were thinking of turning it into a lecture series, I would recommend stopping at week 13 and consulting Lee’s *Topological Manifolds* for details. That might be doable for advanced undergrads. As for the rest of the book—*fugeddaboutit!*