Drew Zemke | Searcharxiv

arXiv2017

Calculating the homology and intersection form of a 4-manifold from a trisection diagram

Given a diagram for a trisection of a 4-manifold $X$, we describe the homology and the intersection form of $X$ in terms of the three subgroups of $H_1(Σ;\mathbb{Z})$ generated by the three sets of curves and the intersection pairing on the diagram surface $Σ$. This includes explicit formulas for the second and third homology groups of $X$ as well an algorithm to compute the intersection form. Moreover, we show that all $(g;k,0,0)$-trisections admit "algebraically trivial" diagrams.

math.GT↗

arXiv2016

The simple loop conjecture for 3-manifolds modeled on Sol

The simple loop conjecture for 3-manifolds states that every 2-sided immersion of a closed surface into a 3-manifold is either injective on fundamental groups or admits a compression. This can be viewed as a generalization of the Loop Theorem to immersed surfaces. We prove the conjecture in the case that the target 3-manifold admits a geometric structure modeled on Sol.

math.GT↗