* re: Whitehead's problem
2004-07-27 13:34 Whitehead's problem Colin McLarty
@ 2004-07-29 10:15 ` Robert Knighten
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From: Robert Knighten @ 2004-07-29 10:15 UTC (permalink / raw)
To: categories
Colin McLarty writes:
> Does anyone here know the origin of the "Whitehead conjecture"? I mean the
> one saying every group G with Ext(G,Z)=0 is free. Did Whitehead himself
> consider this, or did it arise by generalizing his ideas? Did he expect it
> to be true, or just wonder about it?
>
> best, Colin
>
This isn't definitive, but I think it is approximately correct. The
connection between Ext(G,Z)=0 and G being free seems to have first appeared as
a reformulation by Henri Cartan of a result of Karl Stein which arose from his
work on the "second Cousin problem" in several complex variables. The
reference to the paper and review is:
@article {MR13:224f,
AUTHOR = {Stein, Karl},
TITLE = {Analytische {F}unktionen mehrerer komplexer {V}er\"anderlichen
zu vorgegebenen {P}eriodizit\"atsmoduln und das zweite
{C}ousinsche {P}roblem},
JOURNAL = {Math. Ann.},
VOLUME = {123},
YEAR = {1951},
PAGES = {201--222},
MRCLASS = {30.0X},
MRNUMBER = {13,224f},
MRREVIEWER = {H. Cartan},
}
Whitehead appears to have first posed the question (it doesn't seem to have
been a conjecture) at a conference in Warsaw in 1952. The Whitehead problem
seems to have been first published by A. Ehrenfeucht in 1955:
@article {MR16:994b,
AUTHOR = {Ehrenfeucht, A.},
TITLE = {On a problem of {J}. {H}. {C}. {W}hitehead concerning
{A}belian groups},
JOURNAL = {Bull. Acad. Polon. Sci. Cl. III.},
VOLUME = {3},
YEAR = {1955},
PAGES = {127--128},
MRCLASS = {20.0X},
MRNUMBER = {16,994b},
MRREVIEWER = {P. A. Smith},
}
-- Bob
--
Robert L. Knighten
Robert@Knighten.org
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