Twenty five or so years ago I read a library book that was meant to be a gentle introduction to covering spaces.

The basic idea was that given a connected surface, and given a point in the space, the set of paths starting from that point modulo homotopy equivalence was a covering of the original space. These paths were called "tails", I believe. The initial part of the book would have used this concept of 'tails' to build up intuition about covering spaces, universal covers, lifts, deck transformations and such. I'm hoping somebody can tell me what that book was so I can track down a copy. I'd like to revisit this particular piece of exposition.

I also believe the author of the book had a Japanese name, and was with a Japanese university. I'm not sure whether it was translated from Japanese, or originally written in English. Given how long ago I read it, it must have been published earlier than 1995.

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    $\begingroup$ Maybe Michio Kuga's Galois' Dream: Group Theory and Differential Equations? amazon.com/Galois-Dream-Theory-Differential-Equations/dp/… $\endgroup$ Commented Jun 29, 2022 at 0:43
  • $\begingroup$ @QiaochuYuan - wow, thank you, this certainly fits the bill - Japanese author, translation is dated 1993, "everyone has a tail", idiosyncratic presentation, is in the stacks of the library where I browsed it. It’s not like I remember it, but it's hard to imagine that, if this isn't the book, that there is another one. Feel free to turn your comment into an answer - it's likely to become the accepted answer. $\endgroup$
    – brainjam
    Commented Jun 29, 2022 at 4:46

1 Answer 1


Converting my comment into an answer: this might be Michio Kuga's Galois' Dream: Group Theory and Differential Equations. You can see from the table of contents on Springer that it discusses covering spaces quite a bit.


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