I am interested in seeing examples of two closed 4-manifolds $X\_1,X\_2$ such that $SW(X\_i)=0$ and they are homeomorphic but not diffeomorphic.


So far in the literature I've only found examples which somehow used symplectic geometry and operations on the Seiberg-Witten invariant (knot surgery, rational blow down, star surgery, etc.) to distinguish one from the other. So I am interested in knowing some other (if known) techniques in this direction. The reason I've mentioned $SW(X\_i)=0$ is to ensure that none of them admits symplectic structures (by work of Taubes). Trisection theory is potentially a way but I am not sure if anything is known in that direction or not. Thanks in advance.



Example of two exotic closed 4-manifolds s.t. SW(X)=0
$\mathbb CP^2\#\overline{K3}$ (trivial SW because it is a connected sum with $b^2\_+>0$ for both pieces). This is an example without having to use the Bauer-Furuta invariants (contrast with Kyle's comment), but with the SW invariants of the 4-manifold's *opposite* (reverse $X$'s orientation into $\overline X$). I learned this example from Tim Perutz at some point: The symplectic manifold $X=K3\#\overline{\mathbb CP^2}$ has $b^2\_+=3$ and $b^2\_-=20$ and $b^1=0$ and nontrivial SW invariants (note that $b^2\_++b^1+1$ is even, this quantity is the parity of the various SW moduli spaces and needs to be even for the ordinary SW invariants to possibly be nontrivial). By doing Fintushel-Stern knot surgery on K3 we get exotic copies $X\_K$ (knot $K$) distinguished by the SW invariants. Then $SW(\overline X\_K)=0$ because $b^2\_+(\overline X\_K)+b^1(\overline X\_K)+1=21$ is odd, with each $\overline X\_K$ homeomorphic to the non-symplectic $\overline{K3}\#\mathbb CP^2$. So the "technique" of orientation-reversal should be added to our arsenal -- related discussion is found in Draghici's paper *"Seiberg-Witten invariants when reversing orientation"* and its references, for example.


More interestingly (perhaps), Taubes [studied](https://arxiv.org/abs/1602.01687) a possibly infinite set (if not all diffeomorphic) of smooth 4-manifolds $\lbrace X\_K\rbrace\_K$ all homeomorphic to the non-symplectic $3\mathbb CP^2\#23\overline{\mathbb CP^2}$, obtained from knot surgery on K3 using different hyperbolic knots $K\subset S^3$. They all have vanishing SW invariants (and vanishing Bauer-Furuta invariants). But it is not known whether they are all diffeomorphic... if they are then there are some interesting properties of ASD Weyl curvature metrics on it!


