Subject: Re: four postings : S(X x Y) Date: Mon, 20 Oct 2003 14:26:42 +0100 From: Ronald Brown With regard to question on S(X x Y), you can find a proof with pictures as 7.5.9 of my book Topology, Ellis Horwood, 1988. (Out of print) Actually this was in the 1968 edition (McGraw Hill). However the proof is there simplified (?) by assuming spaces are compact Hausdorff, as the relevant theory of compactly generated spaces was not worked into the exposition, but you can easily do this. Note that the proof relates initially to the join X * Y. Ronnie Brown r.brown@bangor.ac.uk >Date: Sun, 19 Oct 2003 21:11:59 -0400 (EDT) >From: Yuli Rudyak > > >I have a question for the list: > >Does somebody know a good source (a book) with a proof of the following >fact: >the suspension of the product is the wedge of suspensions, $S(X\times >Y)=SX\vee >SY\vee S(X\wedge Y)$? > >(Do not send me the proof, I need a reference). > >Yuli >______ ______________________________________________________________ Subject: reply to Rudyak query Date: Mon, 20 Oct 2003 10:03:56 -0400 From: Allen Hatcher I can't resist: When X and Y are CW complexes this is Proposition 4I.1 on page 467 of my Algebraic Topology book. The proof is an elementary geometric argument that could have appeared much earlier in the book, but for convenience I just stuck it here at the beginning of a short section on stable splittings of spaces. The proof applies more generally than to CW complexes -- one just needs some modest homotopy extension properties. I got the proof from some other book or paper, but I've forgotten where now. Maybe someone else can give an earlier reference. Allen Hatcher ________________________________________________________ Subject: RE: four postings Date: Mon, 20 Oct 2003 15:11:52 +0100 From: "Neil Strickland" Husemoller's "Fibre bundles". In the third edition, it is Proposition 3.1 in Appendix A. Neil Strickland ________________________________________________________ Subject: Re: four postings Date: Mon, 20 Oct 2003 12:33:38 -0400 (Eastern Daylight Time) From: "Nicholas J. Kuhn" Yuli's question caused me to peruse the textbooks I own, from Hilton-Wylie to JPMay. I don't own Brayton Gray's book, and Brayton may well prove this, but of the books I own, I struck out with all except Paul Selick's Introduction to Homotopy Theory, where this appears as Prop. 7.7.6 in a section called Ganea's Theorem. [He has the hypothesis that X and Y be connected CW complexes, but I don't think this is needed: the key point is two pages back as Prop. 7.7. This is the statement that if X and Y have nondegenerate basepoints, then the map from the join X*Y to S(X smash Y) - which evidently factors through S(X x Y) - is a homotopy equivalence. This provides the needed section to the cofibration sequence.] The natural home for this result is after a discussion of Puppe sequences, and indeed, Puppe's original paper on such long exact sequences - Math Zeit 69 (1958), 299-344 - features this result as an application. His proof, in section 5, is perfectly modern and elegant, and is easy to follow even if, like me, you don't know German. Nick Kuhn