TITLE: Example of two exotic closed 4-manifolds s.t. SW(X)=0
QUESTION [13 upvotes]: 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.

REPLY [2 votes]: Here is an example as Kyle suggested in his comment. 
Claim is that $K3 \#K3 \# \bar{\mathbb {CP^2}}$ and $\mathbb{ \#_6 CP^2}\#_{39}\mathbb{ \bar{CP^2}}$ are homeomorphic but not diffeomorphic. Notice that they have same intersection form and since they are simply-connected, by Freedman, they are hoeomorphic. Observe that they have trivial Seiberg Witten invariant follows from Taubes result of vanishing Seiberg Witten invariant under connected sum.
In order to distinguish their diffeomorphism type, we are going to use stable cohomotopy Seiberg Witten invariat (ScSW).
Here are two important theorem in this context, one is corresponds to vanishing and the other one for non-vanishing
Theorem 1(Bauer) If a closed smooth connected 4 manifold satisfies one of the following 1) or 2) then they have non vanishing ScSW
1) If $X = X_1 \#X_2$ and $X_1$ has non vanishing ScSW and $b_2^+(X_2)=0$ 
2) If $X= \#_n X_i$ such that $b_2^+(X_i)=3 (mod 4)$, $b_1(X_i)=0$. Each $X_i$ has a compatible spinc structure $s_i$ such that $SW_{s_i}(X_i)=1(mod 2)$ and $1<n<5$ and for $n=4$ we need $b_2^+(X)=4(mod 8)$
By this theorem $K3\#K3\#\mathbb {\bar{CP^2}}$ has non vanishing ScSW (since $K3$ is symplectic and by Taubes, the canonical spinc strure s corresponding to the symplectic structure has $SW_s(K3)=1$). 
Now the vanishing part follows from the adjunction inequality
Theorem 2(Froyshov) Let $X$ be a smooth closed oriented smooth 4 manifold with $b_2^+>1$. Suppose $Y$ is a codim 1 embedded manifold which admists a positive scalar curvature. And the inclusion map induced non trivial map $H^2(X,\mathbb Q)\to H^2(Y,\mathbb Q)$, then ScSW vanishes.
Observe that $\mathbb {CP^2}$ sits in $\mathbb{ \#_6 CP^2}\#_{39}\mathbb{ \bar{CP^2}}$. There is a self intersection 1 sphere in $\mathbb CP^2$. If we blown up once, that will give us a self intersection 0 sphere in $\mathbb{ \#_6 CP^2}\#_{39}\mathbb{ \bar{CP^2}}$. Inface the boundary of such a neighbourhood is $S^1\times S^2$. And the copy of $S^2$ gives a non-torsion element in $H^2$. So by Theorem 2, ScSW=0.
And thus they are EXOTIC.