It may be completely wrong, but my impression is that Poincare usually
has not calculated fundamental groups via coverings, but via generators
and relations. (But probably he has used the connection with coverings
at the discussion of his "Poincare-sphere").
I suppose, it was Reidemeister in the 20th, who really discussed this
matters in depth and used it to calculate invariants of knots.
(It is interesting that at the same time Alexander has established the
same invariants but without considering fundamental groups and their
relations to coverings. He just calculated homology of coverings of
knot spaces.) So, if a schould guess, it was the german school in the
20th. who really introduced the correspondence between coverings and
fundamental groups in the literature of algebraic topology.
If your student is really interested in this matters he should:
1. (if it is available at your library) look at the reviews at "Jahrbuch
Fortschritte der Mathematik", which had a section on Analysis situs
almost from the beginning (around 1870).
2. (if it is available at your library) take a look at the
"Habitilationsschrift" of Heinrich Tietze (1908) published in
"Monatshefte fuer Mathematik und Physik" which was the next big step
in Algebraic Topology after Poincare.
3. Browse through "Archive for the History of Exact sciences" to see
if this topic has been already studied in depth. There is one paper
about the early development of homology-theory (by M.Bollinger), a
paper about invariance of dimension (by D. Johnson).
At least he should consult van den Eynde: "Historical evolution of
the concept of homotopic path" (AHES 45, 1992).
4. for the time before Poincare look at the Dissertation of J.C. Pont.
It would also be interesting to find out if and to what extend Max Dehn
has used the correspondence between coverings and the fundamental group.
Another interesting thing would be to compare the early american text
books (Veblen and Lefschetz) with the german (Seifert-Threllfall)
concerning this topic. I would not be surprised if this reveals some
big differences.
Volker Eisermann
* * * * * * * * * * * * *
Volker Eisermann
Mathematisches Institut
Universitaet Bonn
Beringstrasse 1
53121 Bonn
Tel. 0228/735288
Forwarded message:
>
> I'm curious about historical background about the notion of covering
> spaces, particularly with respect to their relationship with the
> fundamental group. Basically all I've found out is that Poincare
> introduced the fundamental group in 1895, and that covering maps probably
> came out of Riemann's work, but I have no idea when it was first realized
> that the latter could help compute the former.