3 responses to co-H-space questions............DMD From: "John R. Klein" Subject: Re: co-H-space problem Date: Thu, 23 Mar 2000 13:37:48 -0500 (1) Question one has a negative answer: the inclusion f: S^1 --> S^2 is 1-connected. but f b f : S^1 b S^1 --> S^2 b S^2 has a countable wedge of circles as it's domain. So the domain is 0-connected but not one connected. Now S^2 b S^2 is 2-connected. So f b f is no better than 1-connected. (2) The answer to question (2) for spaces (not maps) is contained in Cor 1 of: Golasi\'nski, Marek; Klein, John R. On maps into a co-$H$-space. Hiroshima Math. J. 28 (1998), no. 2, 321--327. On Thu, 23 Mar 2000, you wrote: > From: "Jianzhong Pan" > Subject: co-H-space problem > Date: Thu, 23 Mar 2000 11:54:38 CST > > Here are two problems in homotopy theory. > 1.Let $XbX$ be the space of paths in $X\timesX$ beginning at > $X \bigvee X$ and ending at base point. > If a map $f$ is n-connected, does it follows that > $fbf$ is (2n+1)-connected. > 2. Another related question is : > Can the maps and spaces in the homology decomposition of > a co-H-map be choosen to be co-H-maps and co-H-spaces? > Thanks in advance for any useful information! > Best wishes > Pan Jianzhong > _____________________________________________________________ Subject: Re: co-H-space problem Date: Thu, 23 Mar 2000 14:27:57 -0500 (EST) From: "Tom Goodwillie,304 Kassar,863-2590,617-926-3565" > 1.Let $XbX$ be the space of paths in $X\timesX$ beginning at > $X \bigvee X$ and ending at base point. > If a map $f$ is n-connected, does it follows that > $fbf$ is (2n+1)-connected. No. For an n-connected map between k-connected spaces where k is less than n, this will usually be just about (n+k)-connected. Note that $XbX$ is the join of $\Omega X$ with itself. Let X be k-connected and Y be n-connected. We have an (n+1)-connected map from Z=X\times Y to X. The fiber of the resulting map from ZbZ to XbX will contain (\Omega X)*(\Omega Y) as a retract. Tom Goodwillie ___________________________________________________________ Date: 23 Mar 2000 15:35:56 EST From: Martin.A.Arkowitz@Dartmouth.EDU (Martin A. Arkowitz) Subject: Re: co-H-space problem An answer to the second question appears in my paper "Induced mappings of hom ology decompositions", Banach Center Publications vol. 45, Homotopy and Geometry (J. Oprea and A. Tralle, eds.), 1998 pp. 225-233. See also M. Golasinski and J. Klein, "On maps into a co-H-space", Hiroshima Math. J. 28 (1998), pp. 321-32 7. Sincerely, Martin