We study trisections of smooth compact nonorientable 4-manifolds and introduce trisections of nonorientable 4-manifolds with boundary. In particular, we prove a nonorientable analogue of a classical theorem of Laudenbach–Poénaru and analogues for some well-known theorems from 3-manifold topology. As a consequence, trisection diagrams and Kirby diagrams of closed nonorientable 4-manifolds exist. We discuss how the theory of trisections may be adapted to the setting of nonorientable 4-manifolds with many examples.
"Trisections of Nonorientable 4-Manifolds." Michigan Math. J. Advance Publication 1 - 45, 2023. https://doi.org/10.1307/mmj/20216127