List of Whitehead-like Problems - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T22:45:17Z http://mathoverflow.net/feeds/question/101212 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101212/list-of-whitehead-like-problems List of Whitehead-like Problems Asaf Karagila 2012-07-03T06:42:34Z 2012-07-03T06:56:47Z <p>Whitehead problem is a rather well known problem: </p> <blockquote> <p>Suppose that $G$ is an abelian group and $\mathrm{Ext}^1(G,\Bbb Z)=0$, is $G$ free?</p> </blockquote> <p>It wasn't long before it was proved that if $G$ is countable (and thus countably-generated) then the answer is positive. However the question was open for uncountable groups, and it took more than a decade until Shelah came up with the interesting answer: <em>In ZFC the problem is undecidable</em>.</p> <hr> <p>I was wondering what other open problems have somewhat of a Whitehead-like nature, namely they were proved for objects with a countable nature (e.g. separable topological spaces, countably generated), but the question remains open for the general case.</p> <p>One example which comes to mind is <a href="http://en.wikipedia.org/wiki/Naimark%27s_problem" rel="nofollow">Naimark's problem</a> which was solved for separable $C^\ast$-algebras, and was partially solved in the general case in the sense that it is consistent to have a negative answer.</p> <p>(As with CW big-lists, please post one problem per answer.)</p>