open problems in Seiberg-Witten Theory on 4-Manifolds - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T17:37:53Z http://mathoverflow.net/feeds/question/120819 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120819/open-problems-in-seiberg-witten-theory-on-4-manifolds open problems in Seiberg-Witten Theory on 4-Manifolds Koushik 2013-02-05T01:28:44Z 2013-02-07T06:17:25Z <p>What are some of the open problems in Seiberg-Witten Theory on 4-Manifolds.I tried googling but couldn't any. I tried googling it, but couldn't find any resources.The places where I can a survey or review of them would be welcome. </p> http://mathoverflow.net/questions/120819/open-problems-in-seiberg-witten-theory-on-4-manifolds/120826#120826 Answer by Henry T. Horton for open problems in Seiberg-Witten Theory on 4-Manifolds Henry T. Horton 2013-02-05T03:09:39Z 2013-02-05T03:09:39Z <p>One basic problem is determining the relationship between Seiberg-Witten invariants and Donaldson invariants of $4$-manifolds. Witten himself proposed the precise relationship between the two in the original paper <em>Monopoles and 4-Manifolds</em>, but as far as I know the relationship has not been proven in general. Witten's conjecture has been proven in many cases, however. See the answer to <a href="http://mathoverflow.net/questions/112233/relation-of-sw-and-donaldson-invariant" rel="nofollow">this question</a> for a good overview of the current status of this problem.</p> http://mathoverflow.net/questions/120819/open-problems-in-seiberg-witten-theory-on-4-manifolds/120827#120827 Answer by Tim Perutz for open problems in Seiberg-Witten Theory on 4-Manifolds Tim Perutz 2013-02-05T03:24:43Z 2013-02-05T03:24:43Z <p>One basic structural problem about the SW invariants is the question of <i>simple type</i>: suppose that $X$ is a simply connected 4-manifold with $b^+>1$, and $\mathfrak{s}$ a $\mathrm{Spin}^c$-structure such that $SW_X(\mathfrak{s})\neq 0$. Must $\mathfrak{s}$ arise from an almost complex structure? This is true when $X$ is symplectic (Taubes in "$SW\Rightarrow Gr$") but open in general.</p> <p>The 11/8-conjecture (that for a closed Spin 4-manifold $X$ of signature $\sigma$, one has $b_2(X)\geq 11|\sigma|/8$) is open. SW theory has yielded strong results in this direction (Furuta's 10/8 theorem); proving the conjecture via SW theory is very hard but might be possible. </p> <p>Essentially all of the fundamental questions about the classification of smooth 4-manifolds, or about the existence and uniqueness of symplectic structures on them, are open. We do not know how much Seiberg-Witten theory sees. For instance:</p> <p>Suppose $X$ is a closed 4-manifold with an almost complex structure $J$. Let $w\in H^2(X;\mathbb{R})$ be a class with $w^2>0$. Is there a symplectic form $\omega$ with compatible almost complex structure homotopic to $J$ and symplectic class $w$? The "Taubes constraints" are the following necessary conditions, which constrain the SW invariants in terms of $w$ and $c=c_1(TX,J)$ (see e.g. <a href="http://www.ams.org/journals/bull/1996-33-01/S0273-0979-96-00625-8/" rel="nofollow">Donaldson's survey</a> on the SW equations): (i) $SW(\mathfrak{s}_{can})=\pm 1$ (the sign can be made precise) where <code>$\mathfrak{s}_{can}$</code> is the $\mathrm{Spin}^c$-structure arising from $J$; (ii) $-c\cdot w\geq 0$; and (iii) if $SW(\mathfrak{s})\neq 0$ then $|c_1(\mathfrak{s})\cdot [\omega]| \leq -c \cdot [\omega]$, with equality iff $\mathfrak{s}$ is isomorphic to $\mathfrak{s}_{can}$ or its conjugate. The question is: if $X$ is simply connected, are these sufficient conditions? (Example: Fintushel-Stern knot surgery on an elliptically fibered K3 surface along a knot with monic Alexander polynomial.)</p> http://mathoverflow.net/questions/120819/open-problems-in-seiberg-witten-theory-on-4-manifolds/120832#120832 Answer by Agol for open problems in Seiberg-Witten Theory on 4-Manifolds Agol 2013-02-05T05:00:53Z 2013-02-05T05:00:53Z <p>It might be useful to generalize <a href="http://www.ams.org/mathscinet-getitem?mr=1032074" rel="nofollow">a theorem of Donaldson and Sullivan</a>, that the Donaldson invariants are defined for quasi-conformal 4-manifolds, to the category of Seiberg-Witten invariants. More generally, one would like to know which smooth invariants of 4-manifolds are defined for quasi-conformal 4-manifolds. </p>