Subject: Re: two postings Date: Thu, 29 May 2003 10:21:52 +0900 From: Norio IWASE To: Don Davis Dear Nick, Unfortunately, I found some counter-examples in "Co-H-spaces and the Ganea conjecture", Topology 40 (2001), 223-234, which are counter-examples also to your Lemma. In fact, your Lemma implies the Ganea conjecture on co-Hopf spaces (Ganea's Problem 10, 1971) - not on L-S cat (Ganea's Problem 2, 1971). Anyway, I suppose that the answer to the original question by zhang is yes, if every space is finite. Regards, Norio Iwase -- On 2003.5.29, at 06:15 AM, Don Davis wrote: > Two postings: Another on wedges of spheres and update on > Lehigh conference...........DMD > ________________________________________ > > Subject: for the list > Date: Wed, 28 May 2003 11:02:20 -0400 (Eastern Daylight Time) > From: "Nicholas J. Kuhn" > > I think the proof of the wedge decomposition question can be finished > off rather formally, using the following lemma, which allows the S^1's > to be separated from the higher spheres. > > Lemma Suppose Y = W wedge Y', where W is a wedge of circles and Y' is > simply connected. If X is a retract of Y then X = V wedge X', where V > is a wedge of circles and X' is simply connected. > > sketch proof: > (i) pi_1(X) is a free group, so there exists j:V --> X giving an iso on > pi_1. > (ii) Let X' be the cofiber of j. By Van Kampen, it is simply > connected. > > (iii) The composite X -> Y -> Y' -> Y -> X is zero on pi_1, so can be > written as X --> X' -f-> X, for some map f. Since the above composite > is id on pi_i with i>1, f is epic on pi_i, for i>1. > (iv) By construction X' -f-> X --> X' is the identity on homology, so > is a homotopy equivalence. Thus f is monic on pi_*. > (v) It follows that j wedge f: V wedge X' --> X is an iso on pi_*, and > thus a homotopy equivalence. > > Nick Kuhn > > Subject: wedges of spheres > Date: Tue, 27 May 2003 17:26:26 -0400 > From: Tom Goodwillie > To: Don Davis > >>> If X and Y are topological spaces, not homotopy equivalent to >>> singletons, >>> such that X wedge Y >>> has the homotopy type of a wedge of spheres, does it follow that X >>> and Y homotopy equivalent to wedges of spheres ?? > >> I guess it's not hard to see in the simply connected case: >> Prop: If X is a retract, in the homotopy category, of a wedge of >> spheres of dimension [more than] 1, then X is homotopy equivalent to > a wedge of > spheres. > _____________________________________________________ -- _/_/ Norio IWASE _/_/