A while ago I stumbled across a paper of Thurston: Some Simple Examples of Symplectic Manifolds, where Thurston constructs closed symplectic manifolds with no Kaehler structure. My question is: What are the most rewarding and beautiful (potential) manifolds that one might seek to construct, that have not been constructed thus far? (Does there exist [insert]?). Obviously potential constructions are not restricted to manifolds , however I am mostly interested in constructions of manifolds or spaces in the most current and/or new areas of research in geometry and topology. I would also be interested in conjectures (interconnected or otherwise) relating to manifolds that can be resolved via constructions, what constructions?
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$\begingroup$ Why the functional analysis and k-theory tags? $\endgroup$– Michael AlbaneseCommented Jan 31, 2018 at 22:00
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$\begingroup$ This is way too broad and vague. I've voted to close. $\endgroup$– Andy PutmanCommented Feb 1, 2018 at 2:18
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