An Improved Tur\'an Exponent for 2-Complexes

The topological Tur\'an number $\mathrm{ex}_{\hom}(n,X)$ of a 2-dimensional simplicial complex $X$ asks for the maximum number of edges in an $n$-vertex 3-uniform hypergraph containing no triangulation of $X$ as a subgraph. We prove that the Tur\'an exponent of any such space $X$ is at most $8/3$, i.e., that $\mathrm{ex}_{\hom}(n,X)\leq Cn^{8/3}$ for some constant $C=C(X)$. This improves on the previous exponent of $3-1/5$, due to Keevash, Long, Narayanan, and Scott. Additionally, we present new streamlined proofs of the asymptotically tight upper bounds for the topological Tur\'an numbers of the torus and real projective plane, which can be used to derive asymptotically tight upper bounds for all surfaces. The key insight is an improved understanding of the placement of 4-cycles $vwv'w'$ that are likely to bound a triangulation of the disk within a randomly-selected subset of vertices.