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.)