Grothendieck developed the theory of the fundamental group of a scheme in SGA 1. In order to do so he used the fact that the fundamental group of a topological space is isomorphic to the group of deck transformations --- fiber permuting isomorphisms of the universal cover --- and as a consequence is isomorphic to the automorphism group a particular functor. This reframing seems necessary because who knows what a "loop in a scheme" is.

My question is: Did Grothendieck reframe the theory of the fundamental group of topological spaces as the group of deck transformations? Or, was this already known? Perhaps it was already known, but no one had phrased it in terms of automorphisms of a fiber functor?


2 Answers 2


The idea of a relation between fundamental groups and permutations of the universal cover long predates Grothendieck and SGA. It appears implicitly already in Riemann's work on complex surfaces in 19th century. In Cauchy and Puiseux: Two Precursors of Riemann Papadopoulos even mentions earlier authors, especially Puiseux:

"Riemann defined these surfaces as ramified coverings of the plane (more precisely, of the Riemann sphere). The work of Puiseux on algebraic functions, interpreted from a topological point of view, contains in essence the combinatorics of such a surface, giving a description of how its sheets are permuted above a ramification point, and establishing the precise relation between this sheet permutation and the nature of the singularities of the algebraic equation. At the same time, Puiseux’s work makes the relation with group theory. At the expense of being anachronical, let us mention that the theory of Puiseux expresses the so-called monodromy homomorphism from the fundamental group of the Riemann sphere with a finite set deleted (the singular set of the algebraic equation) into the permutation group on $d$ symbols."

The key technical step, the path lifting theorem, was made by Weyl in 1913, and full development is due to Seifert and Threlfall's classic Lehrbuch der Topologie. Here is Dieudonné's description from A History of Algebraic and Differential Topology 1900-1960, p.296-7:

"It is quite likely that Poincare and his immediate successors, such as Tietze and Dehn, were aware of that theorem, even if they did not formulate it explicitly (still less try to prove it!). It was stated and proved for surfaces by H. Weyl in 1913 [483]. In 1928 Reidemeister published a short paper on fundamental groups and covering spaces [386] in which he only considered combinatorial manifolds. He deplored the fact that, except for dimension 2, no previous treatment of these questions was available. His own treatment is in fact limited to dimension 3, although he claims it might be generalized to any dimension; it is written in the Poincare style, without genuine proofs, and the path lifting theorem is not mentioned.

It was only in 1934 that, in their book [421], 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.

According to Theo Buehler's answer in Etymology of the name “deck transformation” thread on Math SE, the term "deck" is also due to Seifert and Threlfall. They speak of Deckbewegungsgruppe, literally, group of covering movements (from German decken, to cover). Reidemeister called them transitive permutations by the fundamental group. The "automorphisms of a fiber functor" language is of course specific to Grothendieck.

  • $\begingroup$ thanks for your answer and the references :) --- I always thought deck came from shuffling the points of a fiber like you would a deck of cards... :P $\endgroup$ Commented May 22, 2018 at 3:32
  • $\begingroup$ Dieudonné doesn’t say that a group (or action) appears implicitly in Riemann, does he? p. 17: “whereas mathematicians before him tried to attach numbers invariant under homeomorphism to spaces, Poincaré was the first who introduced the idea of computing with topological objects, not only with numbers. He did this in two different ways, by defining the concepts of homology and of fundamental group.” $\endgroup$ Commented May 23, 2018 at 3:44
  • $\begingroup$ @FrancoisZiegler I added more on "prehistory". $\endgroup$
    – Conifold
    Commented May 24, 2018 at 3:44
  • $\begingroup$ Come on, this is reading new into old at a scale that even Dieudonné doesn’t dare. The paper on Cauchy-Puiseux duly qualifies both occurrences of “fundamental group”: “stated in modern terms”, “at the expense of being anachronical”. Who first described the fundamental group as the group of deck transformations? $\endgroup$ Commented May 24, 2018 at 6:12
  • $\begingroup$ Isn't that what "implicitly" is for? If one so wishes the "relation" can be seen "at work" in hindsight even before the relata are explicitly separated. If not, one can pick between Poincaré, Weyl, Reidemeister and Seifert-Threlfall depending on one's "explicitness" standard. Threads of ideas seem more salient to me than symbolic watersheds people tend to look for, after injecting modern terms into the past no less, no one person was ever responsible for introducing any one idea. $\endgroup$
    – Conifold
    Commented May 24, 2018 at 6:25

The answer to the title question is Poincaré, in the very note Sur l’Analysis situs (1892) where he first introduced the fundamental group. Cf. the description by “Saint-Gervais”:

Now Poincaré gives his definition of the fundamental group. To this end he starts by almost copying his 1883 article which we have just discussed. But now he introduces a group! In other words, he defines the universal covering as before, but the fundamental group makes its appearance as a permutation group of the fiber above a point: the automorphism group of the universal covering.

See also Sarkaria (1999, p. 144), note 20 referring to §12 of Analysis Situs (1895):

Poincaré gives four approaches to his groups $g$ and $G$. Firstly, as all deck transformations of a covering space over $M$ (...) Secondly, his differential equations definition (...) Thirdly, his definition using “loops”, “equivalences” and “lacets” (...) Lastly, in §13, for any $M$ obtained from a polytope by facet conjugations, Poincaré defines $\pi_1(M)$ via some simple and elegant (yet intriguing) cyclic relations.

Note added: Of course, on general spaces Poincaré’s four definitions cease to be equivalent. Grothendieck’s approach of sticking with the first one — thus eschewing loops — likely originates with Chevalley (1946), who used it “following a suggestion of H. Cartan” (Dieudonné 1989, p. 300). It is also adopted in e.g. Ganea (1951), Souriau (1964, §10 + Note II), Kennison (1989), etc.


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