Spanning spheres in Dirac hypergraphs

We show that a $k$-uniform hypergraph on $n$ vertices has a spanning subgraph homeomorphic to the $(k-1)$-dimensional sphere provided that $H$ has no isolated vertices and each set of $k-1$ vertices supported by an edge is contained in at least $n\mathrm{/}2+o(n)$ edges. This gives a topological extension of Dirac’s theorem and asymptotically confirms a conjecture of Georgakopoulos, Haslegrave, Montgomery, and Narayanan. Unlike typical results in the area, our proof does not rely on the Absorption Method, the Regularity Lemma or the Blow-up Lemma. Instead, we use a recently introduced framework that is based on covering the vertex set of the host graph with a family of complete blow-ups.