Subject: History of n-connected covers
Date: Mon, 13 May 2002 14:17:55 -0400 (Eastern Daylight Time)
From: "Nicholas J. Kuhn"
To: Don Davis
CC: njk4x@virginia.edu
Thanks to all who have weighed in with comments on the X notation.
I've had a little fun doing a bit of old fashioned library research on
this. Though I still don't know where the X notation was used
first, I seem to have found the person guilty of the first notational
discrepancy regarding n-connected coverings. My findings...
1. The story starts with
Eilenberg, Samuel Singular homology theory. Ann. of Math. (2) 45,
(1944). 407--447.
This paper defined singular homology! Chapter VI is entitled
"Relations with homotopy groups" and features the following definition.
Let X be a path connected space with basepoint x. Define S_n(X) to be
the subcomplex of the singular complex S(X) generated by the singular
simplices sending all faces of dimension < n to x. It is proved that
if pi_k(X) = 0 for k < n, then H_*(S_n(X)) = H_(S(X)).
2. A list of problems from a 1947 conference was published as
Eilenberg, Samuel On the problems of topology. Ann. of Math. (2) 50,
(1949). 247--260.
Problem 32, attributed to Hurewitz, essentially asks if n-connected
covers exist (as a fiber bundle).
3. Two independent short notes were simultaneously published in 1952:
Cartan, Henri; Serre, Jean-Pierre Espaces fibrés et
groupes d'homotopie. I. Constructions générales. (French) C. R. Acad.
Sci. Paris 234, (1952). 288--290.
Whitehead, George W. Fiber spaces and the Eilenberg homology groups.
Proc. Nat. Acad. Sci. U. S. A. 38, (1952). 426--430.
In both of these, the main points are basically that
(i) n connected covers exist.
(ii) If Y --> X is an n-connected cover, then H_*(Y) = H_*(S_{n+1}(X)),
where I have used Eilenberg's S_n(X) notation.
As for notation...
The Cartan-Serre paper uses the slightly eccentric notation (X,n+1) for
"any space that kills pi_i(X) for i less than or equal to n."
[Translated from the French] Thus (X,n+1) is n connected and H_*(X,n+1)
= H_*(S_{n+1}(X)), so they are in synch with Eilenberg.
By contrast, Whitehead shifts Eilenberg's notation by 1: S_n(X)
is the subcomplex generated by singular simplices sending the n
skeleton to the basepoint. (He never introduces notation for the n
connected cover.)
Thus I conclude that George Whitehead is to blame for muddying these
particular notational waters.
Nick Kuhn
PS Here at U Virginia, I was able to read all of the above, except for
Cartan-Serre, online with JSTOR, which is a pretty cool research tool.
(Finding Cartan-Serre had me mucking around in the bowels of our
scientific periodicals library, where I discovered we have Contes
Rendues back to 1835.)