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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.

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    $\begingroup$ These may not be the spirit of examples you're looking for, but you can produce examples using connected sums. See, for example, Ishida-Sasahira. The point is that Seiberg-Witten invariants vanish under connected sums of 4-manifolds with $b^+_2 \geq 1$, but this isn't true of the stronger Bauer-Furuta invariants. $\endgroup$ – Kyle Hayden May 20 at 21:02
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$\mathbb CP^2\#\overline{K3}$ (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 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!

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  • $\begingroup$ Interesting!! Thank you for sharing :) $\endgroup$ – Anubhav Mukherjee May 22 at 3:01
  • $\begingroup$ One thing is not clear to me, why $\bar{X_K}$ and $\bar{K3} # \mathbb CP^2$ are not diffeomorphic? i.e if X and Y are diffeomorphic does that mean $\bar{X}$ and $\bar{Y}$ are diffeomorphic? $\endgroup$ – Anubhav Mukherjee May 22 at 3:16
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    $\begingroup$ Yes, using the same diffeomorphism (pick your favorite definition of orientation and check). $\endgroup$ – Chris Gerig May 22 at 5:52

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