# What was Seifert's contribution to the Seifert-van Kampen theorem?

The Seifert-van Kampen theorem is the classical theorem of algebraic topology that the fundamental group functor $\pi_1$ preserves pushouts; more often than not this is referred to simply as the van Kampen theorem, with no Seifert attached. Curious as to why, I tried looking up the history of the theorem, and (in the few sources at my immediate disposal) could only find mention of van Kampen; the wikipedia page of Herbert Seifert doesn't even mention the theorem.

So my question is: Why does the theorem bear Seifert's name?

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Seifert's name is attached to many things already, so it's less distinctive. If we called it Seifert's theorem we'd always have to say something like "Seifert's fundamental group theorem". But saying Van Kampen is easier, since we don't need to worry people will misunderstand what theorem we're referring to. –  Ryan Budney May 9 at 12:32

According to Jahrbuch der Mathematik this theorem is in:

• Seifert, H. Konstruktion dreidimensionaler geschlossener Räume. (German) JFM 57.0723.01 Berichte Leipzig 83, 26-66. Technische Hochschule Dresden, Diss (1931).

See the (long) review in zbMATH by Erika Pannwitz. The relevant sentence is:

• (Die erforderlichen Hilfsmittel werden in § 2 und 3 der Arbeit zusammengestellt; § 3 enthält insbesondere Aussagen über die Fundamentalgruppe eines Komplexes, der aus zwei Komplexen mit zusammenhängendem Durchschnitt zusammengesetzt ist.)

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The paper Gramain, A. Le th\'eor`eme de van Kampen. Cah. Top. G\'eom. Diff. Cat. 33 (1992) 237--251, gives some more history. Actually van Kampen's paper is quite difficult to follow, and the matter was clarified by Crowell to the modern version. It seemed to me in 1965 that there was an anomaly that this theorem did not calculate the fundamental group of the circle, THE basic example in algebraic topology. This led to the use of groupoids in this area. See the discussion at mathoverflow.net/questions/40945/… –  Ronnie Brown May 9 at 9:58
@Ronnie: The paper of Gramain looks very informative. I seems that Seifert treated only the case of 2 sub complexes of a simplicial complex with connected (from the review I cited above) intersection. –  Peter Michor May 9 at 11:16
Wrt the paper by Seifert, that is also my understanding. The proof by Crowell is modern in that it goes by verifying the universal property; this proof generalises easily to the many base point case, i.e. non connected spaces, so dealing with many not unusual examples, including of course the circle; the proof also extends, with new ideas, to higher dimensions (see our EMS Tract vol 15, 2011, "Nonabelian algebraic topology"). –  Ronnie Brown May 9 at 14:17