Trisections of surface complements and the Price twist

2018 
Given a copy $S$ of $\mathbb{R}P^2$ embedded in a $4$-manifold $X^4$ with Euler number $2$ or $-2$, the Price twist is a surgery operation on $\nu(S)$ yielding (up to) $3$ different $4$-manifolds: $X^4,\tau_S(X^4),\Sigma_S(X^4)$. This is of particular interest when $X^4=S^4$, as then $\Sigma_S(X^4)$ is a homotopy 4-sphere which is not obviously diffeomorphic to $S^4$. In this paper, we show how to produce a trisection description of each Price twist on $S\subset X^4$, and how to produce a trisection description of general surface complements in $4$-manifolds.
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