Pinwheels and nullhomologous surgery on $4$–manifolds with $b^+=1$

Abstract

We present a method for finding embedded nullhomologous tori in standard $4$–manifolds which can be utilized to change their smooth structure. As an application, we show how to obtain infinite families of simply connected smooth $4$–manifolds with $b^{+}=1$ and $b^{-}=2,\dots ,7$, via surgery on nullhomologous tori embedded in the standard manifolds ${ℂ‘\mathbb{P}}^2\#k{\overline{ℂ‘\mathbb{P}}}^2$, $k=2,\dots ,7$.