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I've recently been studying some Manifold Theory and got very interested in their topological as well as geometric properties. From my understanding of the current literature, most the big and important open problems involving $3$-manifolds have been resolved in recent times following Perelman's remarkable proof of the Poincare Conjecture. In fact, many of the open problems listed in this MO post have now been resolved.

However, $4$-manifold theory still seems to be a promising area of research (including topics from mathematical physics). So I was hoping people here could briefly explain some of the main open questions and programs that are motivating current research on "$4$-manifolds and/or their interplay with Topology."

Additionally, I wanted to know if the Smooth $4$-dimensional Poincare conjecture is still considered open. I could not find anything convincing on the internet!

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    $\begingroup$ not for nothing, but the study of 3-manifolds "still seems to be a promising area of research" too :) $\endgroup$
    – Robbie Lyman
    Commented Jun 30, 2023 at 22:20
  • $\begingroup$ @RyleeLyman : My apologies for the harsh remark. I was not trying to disprove your statement in any way. Just mentioning the fact that many important $3$-manifolds problems have now been solved. $\endgroup$
    – math-physicist
    Commented Jun 30, 2023 at 22:22
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    $\begingroup$ The smooth 4D Poincare conjecture is definitely still open. See for example Ciprian Manolescu's expository article, Four-dimensional topology. As with any other famous open problem, you can find people claiming to have solved it, but none of these claims have been generally accepted. $\endgroup$
    – Timothy Chow
    Commented Jun 30, 2023 at 22:25
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    $\begingroup$ Start by reading Kirby's problem list. $\endgroup$
    – Moishe Kohan
    Commented Jun 30, 2023 at 23:14
  • $\begingroup$ @MoisheKohanonstrike : Are all problem statuses regularly updated on Kirby's list? I came across this nice collection of problems but don't know which are still open. $\endgroup$
    – math-physicist
    Commented Jul 1, 2023 at 5:39

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