Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.)

share|improve this question
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. –  Eric Wofsey Sep 25 '13 at 6:12
Small remark: your comment about Torsion was previously mentioned on MathOverflow by Qiaochu Yuan here: mathoverflow.net/a/13090/11540 –  David White Sep 25 '13 at 10:51
Possibly notation $\ \pi(X)\ $ predated $\ \pi_n(X),\ $ and $\ \pi_1(X)\ $ in particular. –  Włodzimierz Holsztyński Sep 25 '13 at 15:22

1 Answer 1

up vote 8 down vote accepted

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.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.