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http://www.math.uchicago.edu/~amwright/HomotopyGroupsOfSoheres.pdf

I'm trying to understand this theorem on page 6. Apparantly you can use that to prove the Freudenthal suspension theorem for spheres. However, I think it's wrong particularly $\pi_{i}(A,C) \rightarrow \pi_{i}(X,B)$, shouldn't it be A instead of B in the co-domain. Does anyone know any references to this proof?

Also, is there any texts on this lemma, as I'm trying to find the lemma and it seems to not exists. So I was wondering can someone tell me if the proof is correct and provide sources. As I understand everything upto this point in his project.

This seems to be a simplified theorem to the one in Hatcher.

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In the Book "Elements of homotopy theory" by whitehead there is a Theorem of blakers and massey, also called homotopy excision theorem. It implies Freudenthals suspension theorem, so you might want to look there.

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Thank you. I looked at the preface online and it mention proving it for spheres and theorem of blackers massey. I will try and see if I can get a copy in the library or if a lecturer has a copy of it. Thanks –  simplicity May 11 '12 at 1:00

The theorem of Blakers and Massey is also a consequence of a van Kampen theorem for diagrams of spaces: see the references in

R. Brown, ``Triadic Van Kampen theorems and Hurewicz theorems'', Algebraic Topology, Proc. Int. Conf. March 1988, Edited M.Mahowald and S.Priddy, Cont. Math. 96 (1989) 39-57.

This method has the advantage of determining the critical triad homotopy group, even in some non-simply connected situations.

The proof of the general theorem van Kampen is quite sophisticated, though!

May 30: I have now made an updated copy of the above paper available at

http://pages.bangor.ac.uk/~mas010/pdffiles/VKTEVAN2.pdf

Part of the point is that one brings in triad homotopy groups, as they are part of exact sequences which show them as the obstruction to excision. So the problem is that of calculating these groups, and of seeing when they are zero. The paper shows how a structure of crossed square is involved in such calculations.

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