3-manifolds lying in trisected 4-manifolds

The spine of a trisected 4-manifold is a singular 3-dimensional set from which the trisection itself can be reconstructed. 3-manifolds embedded in the trisected 4–manifold can often be isotoped to lie almost or entirely in the spine of the trisection. We define this notion and show that in fact every 3-manifold can be embedded to lie almost in the spine of a minimal genus trisection of some connect sum of $S^2\tilde{\times }{S}^2$s. This mirrors the known fact that every 3-manifold can be smoothly embedded in a connect sum of $S^2\times S^2$s. Our methods additionally give an upper bound for how many copies of $S^2\tilde{\times }{S}^2$ based on a distance calculated in an appropriately defined graph. For the special case of lens spaces we analyze more closely and obtain more explicit bounds.