What does the word "deck" mean in "deck transformation"? What's the idea behind this name?

  • $\begingroup$ I don't think this word has a deep etymology. The idea is to think of a covering space as something like the floors of a building or a big boat. Each floor you could call a deck, like on a boat. So deck transformation you could also call floor transformation, or level transformation, etc. $\endgroup$ – Ryan Budney May 10 '11 at 20:23
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    $\begingroup$ My guess is that it comes from German "decken" - to cover. I have no evidence, though. $\endgroup$ – t.b. May 10 '11 at 20:24
  • $\begingroup$ @Theo: that's what deck means in English, as well. $\endgroup$ – Ryan Budney May 10 '11 at 20:52
  • $\begingroup$ @Ryan: Thanks, I wasn't aware of that. My Oxford American Dictionary doesn't mention such usage, only the sense "decorate or adorn". $\endgroup$ – t.b. May 10 '11 at 21:21
  • $\begingroup$ These all look like pretty convincing arguments. When I first learned about deck transformations, though, I was told to picture a deck of cards sitting over a base card; then deck transformations just shuffle the deck! $\endgroup$ – Aaron Mazel-Gee May 10 '11 at 23:43

I first quote Dieudonné's History of algebraic topology, p. 296f.

It was only in 1934 that, in their book [...] [Lehrbuch der Topologie, Teubner, Leipzig-Berlin, 1934] Seifert and Threlfall gave an admirable and thorough elaboration of the relations between fundamental groups and covering spaces based on the path lifting theorem: although limited to locally finite simplicial complexes (of any dimension), it is essentially definitive, and can be extended to more general spaces with only minor modifications.

Previous work mentioned by Dieudonné is Weyl's treatment for Riemann surfaces (in Die Idee der Riemannschen Fläche, 1913), while he speculates that Poincaré and Dehn were probably aware of that result. Moreover, Dieudonné mentions an article by Reidemeister, Fundamentalgruppe und Überlagerungsräume, Nachr. Ges. Wiss. Göttingen, 1928, 69-76 (which I was unable to locate). I'm linking to his article in the ICM-proceedings of 1928 instead. In this latter, Reidemeister does not mention the word Decktransformationen (he speaks of transitive permutations by the fundamental group, for example), and as far as I can tell neither does Weyl use the word Deck in his works.

However, Seifert and Threlfall explicitly speak of the Deckbewegungsgruppe. The three pieces of that word are deck(en) - to cover, Bewegung - movement and Gruppe - group.

Finally, the etymology of the word decken. It has ancient Germanic roots and is the same as "deck" in English. I'm quoting the Oxford American Dictionary concerning that:

ORIGIN late Middle English : from Middle Dutch dec ‘covering, roof, cloak,’ dekken ‘to cover.’ Originally denoting canvas used to make a covering (esp. on a ship), the term came to mean the covering itself, later denoting a solid surface serving as roof and floor.

As Zhen Lin points out in the other answer, it seems quite likely that the word Decktransformation was simply taken from there, since Seifert and Threlfall was a long time standard reference in topology. I haven't checked later sources.


My Algebraic Topology lecturer (P.T. Johnstone) claimed that it indeed comes from the German "decken", and that "deck transformation" is a mistranslation. (So he instead says "covering translations".) Assuming this is true, then perhaps the best place to look for evidence is in the German literature, maybe starting with Weyl?


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