6
$\begingroup$

Background: a well-known conjecture in $4$-dimensional topology states that if $M$ is a $4$-dimensional oriented closed spin smooth manifold, then the second Betti number of $M$ is bounded from below by $\frac{11}{8}$ times the absolute value of the signature of $M$. The most fundamental progress was made by Furuta, who used finite-dimensional approximations of the Seiberg-Witten equation and inputs from algebraic topology to derive inequalities of a similar type. There are follow-up works that sharpen or generalize Furuta's result. On the other hand, it is also known that Seiberg-Witten theory and Heegaard Floer theory more or less give the same output on the homological side.

Question: is it possible to prove Furuta's result using (a homotopical refinement of) invariants from Heegaard Floer theory?

Improve this question
$\endgroup$
2
  • $\begingroup$ Are you asking if we can prove something with a tool that doesn't exist and we wouldn't know what a homotopical refinement would even be? $\endgroup$
    – Chris Gerig
    Commented Dec 31, 2022 at 20:50
  • 1
    $\begingroup$ It looks like we've already known how this tool would roughly look based on recent advances, and I also put "homotopical" in parenthesis in case it might cause confusion. $\endgroup$
    – Shaoyun Bai
    Commented Dec 31, 2022 at 20:59

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .