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 _/_/