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
>______
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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
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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
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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