# Example of two exotic closed 4-manifolds s.t. SW(X)=0

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.

• 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. – Kyle Hayden May 20 at 21:02

## 1 Answer

$$\mathbb CP^2\#\overline{K3}$$ (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!

• Interesting!! Thank you for sharing :) – Anubhav Mukherjee May 22 at 3:01
• 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? – Anubhav Mukherjee May 22 at 3:16
• Yes, using the same diffeomorphism (pick your favorite definition of orientation and check). – Chris Gerig May 22 at 5:52