I've spent the last half hour browsing Stillwell's translation of Poincaré's Analysis Situs and Dieudonné's History of Algebraic and Differential Topology, and I haven't found the source of this notation. The "1" in the subscript would lead one to believe this notation was introduced at the same time as $\pi_n(X,x)$ for the higher homotopy groups as well.

In my search, I have learned that the name "Fundamental group" comes straight from Poincaré, and from the "Fundamental region" of a group action. (And, incidentally, that "torsion" in algebra comes from the homology of non-orientable surfaces, which are twisted in on themselves. A bit of a tangent, but I'm still marveling.)

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    $\begingroup$ I would guess that it originates from Hurewicz's papers ("Beiträge zur Topologie der Deformationen I-IV") introducing higher homotopy groups, but I have been unable to find an electronic verson of them to verify this. $\endgroup$ – Eric Wofsey Sep 25 '13 at 6:12
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    $\begingroup$ Small remark: your comment about Torsion was previously mentioned on MathOverflow by Qiaochu Yuan here: mathoverflow.net/a/13090/11540 $\endgroup$ – David White Sep 25 '13 at 10:51
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    $\begingroup$ Possibly notation $\ \pi(X)\ $ predated $\ \pi_n(X),\ $ and $\ \pi_1(X)\ $ in particular. $\endgroup$ – Włodzimierz Holsztyński Sep 25 '13 at 15:22

Notice that in Poincare's setting there is no base point $x$, since he regards the fundamental group as a group of permutations of fundamental regions -- though he also knew the path interpretation, of course, as it arose in integration on Riemann surfaces.

The first occurrence of the notation $\pi_n(X)$ for the $n$-th homotopy group of space $X$ is Hurewicz's paper

Beiträge zur Topologie der Deformationen: Höherdimensionale Homotopiegruppen, Proceedings of Koninklijke Akademie van Wetenschappen te Amsterdam vol. 38 (1935) pp. 112-119.

which is translated in the book Collected Works of Witold Hurewicz. Selected pages of that, including this notation, are on Goggle books. Search "homotopy group."

If you want the base point $x$ as well, then I have failed. It seems not to appear anywhere in Math Reviews before 1945 but I did not find when it did occur.

  • $\begingroup$ This doesn't answer to the question "why should I use pi, not another letter, not another symbol". Does anybody have a clue. $\endgroup$ – fosco Dec 16 '16 at 13:42
  • $\begingroup$ Does it stand for the p of poincaré? $\endgroup$ – fosco Dec 16 '16 at 14:31

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